four harmonic points in the conic.
S 68. If two conics intersect in four points (they cannot have more
points in common, S 52), there exists one and only one four-point
which is inscribed in both, and therefore one polar-triangle common to
both.
_Theorem._--Two conics which intersect in four points have always one
and only one common polar-triangle; and reciprocally,
Two conics which have four common tangents have always one and only
one common polar-triangle.
DIAMETERS AND AXES OF CONICS
S 69. _Diameters._--The theorems about the harmonic properties of
poles and polars contain, as special cases, a number of important
metrical properties of conics. These are obtained if either the pole
or the polar is moved to infinity,--it being remembered that the
harmonic conjugate to a point at infinity, with regard to two points
A, B, is the middle point of the segment AB. The most important
properties are stated in the following theorems:--
_The middle points of parallel chords of a conic lie in a line--viz.
on the polar to the point at infinity on the parallel chords._
This line is called a _diameter_.
_The polar of every point at infinity is a diameter._
_The tangents at the end points of a diameter are parallel, and are
parallel to the chords bisected by the diameter._
_All diameters pass through a common point, the pole of the line at
infinity._
_All diameters of a parabola are parallel_, the pole to the line at
infinity being the point where the curve touches the line at
infinity.
In case of the ellipse and hyperbola, the pole to the line at infinity
is a finite point called the _centre_ of the curve.
_A centre of a conic bisects every chord through it._
_The centre of an ellipse is within the curve_, for the line at
infinity does not cut the ellipse.
_The centre of an hyperbola is without the curve_, because the line at
infinity cuts the curve. Hence also--
_From the centre of an hyperbola two tangents can be drawn to the
curve which have their point of contact at infinity._ These are called
_Asymptotes_ (S 59).
_To construct a diameter_ of a conic, draw two parallel chords and
join their middle points.
_To find the centre_ of a conic, draw two diameters; their
intersection will be the centre.
S 70. _Conjugate Diameters._--A polar-triangle with one vertex at the
centre will have the opposite side at infinity. The other two sides
pass through the centre, and are called _conjugate diameters_, each
being the polar of the point at infinity on the other.
_Of two conjugate diameters each bisects the chords parallel to the
other, and if one cuts the curve, the tangents at its ends are
parallel to the other diameter._
Further--
_Every parallelogram inscribed in a conic has its sides parallel to
two conjugate diameters_; and
_Every parallelogram circumscribed about a conic has as diagonals two
conjugate diameters._
This will be seen by considering the parallelogram in the first case
as an inscribed four-point, in the other as a circumscribed four-side,
and determining in each case the corresponding polar-triangle. The
first may also be enunciated thus--
_The lines which join any point on an ellipse or an hyperbola to the
ends of a diameter are parallel to two conjugate diameters._
S 71. _If every diameter is perpendicular to its conjugate the conic
is a circle._
For the lines which join the ends of a diameter to any point on the
curve include a right angle.
_A conic which has more than one pair of conjugate diameters at right
angles to each other is a circle._
[Illustration: FIG. 24.]
Let AA' and BB' (fig. 24) be one pair of conjugate diameters at right
angles to each other, CC and DD' a second pair. If we draw through the
end point A of one diameter a chord AP parallel to DD', and join P to
A', then PA and PA' are, according to S 70, parallel to two conjugate
diameters. But PA is parallel to DD', hence PA' is parallel to CC, and
therefore PA and PA' are perpendicular. If we further draw the
tangents to the conic at A and A', these will be perpendicular to AA',
they being parallel to the conjugate diameter BB'. We know thus five
points on the conic, viz. the points A and A' with their tangents, and
the point P. Through these a circle may be drawn having AA' as
diameter; and as through five points one conic only can be drawn, this
circle must coincide with the given conic.
S 72. _Axes._--Conjugate diameters perpendicular to each other are
called _axes_, and the points where they cut the curve _vertices_ of
the conic.
In a circle every diameter is an axis, every point on it is a vertex;
and any two lines at right angles to each other may be taken as a pair
of axes of any circle which has its centre at their intersection.
[Illustration: FIG. 25.]
If we describe on a diameter AB of an ellipse or hyperbola a circle
concentric to the conic, it will cut the latter in A and B (fig. 25).
Each of the semicircles in which it is divided by AB will be partly
within, partly without the curve, and must cut the latter therefore
again in a point. The circle and the conic have thus four points A, B,
C, D, and therefore one polar-triangle, in common (S 68). Of this the
centre is one vertex, for the line at infinity is the polar to this
point, both with regard to the circle and the other conic. The other
two sides are conjugate diameters of both, hence perpendicular to each
other. This gives--
An ellipse as well as an hyperbola has one pair of axes.
This reasoning shows at the same time _how to construct the axis of an
ellipse or of an hyperbola_.
_A parabola has one axis_, if we define an axis as a diameter
perpendicular to the chords which it bisects. It is easily
constructed. The line which bisects any two parallel chords is a
diameter. Chords perpendicular to it will be bisected by a parallel
diameter, and this is the axis.
S 73. The first part of the right-hand theorem in S 64 may be stated
thus: any two conjugate lines through a point P without a conic are
harmonic conjugates with regard to the two tangents that may be drawn
from P to the conic.
If we take instead of P the centre C of an hyperbola, then the
conjugate lines become conjugate diameters, and the tangents
asymptotes. Hence--
_Any two conjugate diameters of an hyperbola are harmonic conjugates
with regard to the asymptotes._
As the axes are conjugate diameters at right angles to one another, it
follows (S 23)--
_The axes of an hyperbola bisect the angles between the asymptotes._
[Illustration: FIG. 26.]
Let O be the centre of the hyperbola (fig. 26), t any secant which
cuts the hyperbola in C, D and the asymptotes in E, F, then the line
OM which bisects the chord CD is a diameter conjugate to the diameter
OK which is parallel to the secant t, so that OK and OM are harmonic
with regard to the asymptotes. The point M therefore bisects EF. But
by construction M bisects CD. It follows that DF = EC, and ED = CF; or
_On any secant of an hyperbola the segments between the curve and the
asymptotes are equal._
If the chord is changed into a tangent, this gives--
_The segment between the asymptotes on any tangent to an hyperbola is
bisected by the point of contact._
The first part allows a simple solution of the problem to find any
number of points on an hyperbola, of which the asymptotes and one
point are given. This is equivalent to three points and the tangents
at two of them. This construction requires measurement.
S 74. For the parabola, too, follow some metrical properties. A
diameter PM (fig. 27) bisects every chord conjugate to it, and the
pole P of such a chord BC lies on the diameter. But a diameter cuts
the parabola once at infinity. Hence--
_The segment PM which joins the middle point M of a chord of a
parabola to the pole P of the chord is bisected by the parabola at A._
S 75. Two asymptotes and any two tangents to an hyperbola may be
considered as a quadrilateral circumscribed about the hyperbola. But
in such a quadrilateral the intersections of the diagonals and the
points of contact of opposite sides lie in a line (S 54). If therefore
DEFG (fig. 28) is such a quadrilateral, then the diagonals DF and GE
will meet on the line which joins the points of contact of the
asymptotes, that is, on the line at infinity; hence they are parallel.
From this the following theorem is a simple deduction:
_All triangles formed by a tangent and the asymptotes of an hyperbola
are equal in area._
[Illustration: FIG. 27.]
[Illustration: FIG. 28.]
If we draw at a point P (fig. 28) on an hyperbola a tangent, the part
HK between the asymptotes is bisected at P. The parallelogram PQOQ'
formed by the asymptotes and lines parallel to them through P will be
half the triangle OHK, and will therefore be constant. If we now take
the asymptotes OX and OY as oblique axes of co-ordinates, the lines OQ
and QP will be the co-ordinates of P, and will satisfy the equation xy
= const. = a^2.
_For the asymptotes as axes of co-ordinates the equation of the
hyperbola is xy = const._
INVOLUTION
[Illustration: FIG. 29.]
S 76. If we have two projective rows, ABC on u and A'B'C' on u', and
place their bases on the same line, then each point in this line
counts twice, once as a point in the row u and once as a point in the
row u'. In fig. 29 we denote the points as points in the one row by
letters above the line A, B, C ..., and as points in the second row by
A', B', C' ... below the line. Let now A and B' be the same point,
then to A will correspond a point A' in the second, and to B' a point
B in the first row. In general these points A' and B will be
different. It may, however, happen that they coincide. Then the
correspondence is a peculiar one, as the following theorem shows:
_If two projective rows lie on the same base, and if it happens that
to one point in the base the same point corresponds, whether we
consider the point as belonging to the first or to the second row,
then the same will happen for every point in the base--that is to say,
to every point in the line corresponds the same point in the first as
in the second row._
[Illustration: FIG. 30.]
In order to determine the correspondence, we may assume three pairs of
corresponding points in two projective rows. Let then A', B', C', in
fig. 30, correspond to A, B, C, so that A and B', and also B and A',
denote the same point. Let us further denote the point C' when
considered as a point in the first row by D; then it is to be proved
that the point D', which corresponds to D, is the same point as C. We
know that the cross-ratio of four points is equal to that of the
corresponding row. Hence
(AB, CD) = (A'B', C'D')
but replacing the dashed letters by those undashed ones which denote
the same points, the second cross-ratio equals (BA, DD'), which,
according to S 15, equals (AB, D'D); so that the equation becomes
(AB, CD) = (AB, D'D).
This requires that C and D' coincide.
S 77. Two projective rows on the same base, which have the above
property, that to every point, whether it be considered as a point in
the one or in the other row, corresponds the same point, are said to
be in _involution_, or to form an _involution_ of points on the line.
We mention, but without proving it, that any two projective rows may
be placed so as to form an involution.
An involution may be said to consist of a row of pairs of points, to
every point A corresponding a point A', and to A' again the point A.
These points are said to be conjugate, or, better, one point is termed
the "mate" of the other.
From the definition, according to which an involution may be
considered as made up of two projective rows, follow at once the
following important properties:
1. The cross-ratio of four points equals that of the four conjugate
points.
2. If we call a point which coincides with its mate a "focus" or
"double point" of the involution, we may say: An involution has either
two foci, or one, or none, and is called respectively a hyperbolic,
parabolic or elliptic involution (S 34).
3. In an hyperbolic involution any two conjugate points are harmonic
conjugates with regard to the two foci.
For if A, A' be two conjugate points, F1, F2 the two foci, then to the
points F1, F2, A, A' in the one row correspond the points F1, F2, A',
A in the other, each focus corresponding to itself. Hence (F1F2, AA')
= (F1F2, A'A)--that is, we may interchange the two points AA' without
altering the value of the cross-ratio, which is the characteristic
property of harmonic conjugates (S 18).
4. The point conjugate to the point at infinity is called the "centre"
of the involution. Every involution has a centre, unless the point at
infinity be a focus, in which case we may say that the centre is at
infinity.
In an hyperbolic involution the centre is the middle point between the
foci.
5. The product of the distances of two conjugate points A, A' from the
centre O is constant: OA . OA' = c.
For let A, A' and B, B' be two pairs of conjugate points, the centre,
I the point at infinity, then
(AB, OI) = (A'B', IO),
or
OA . OA' = OB . OB'.
In order to determine the distances of the foci from the centre, we
write F for A and A' and get
OF^2 = c; OF = [+-][root]c.
Hence if c is positive OF is real, and has two values, equal and
opposite. The involution is hyperbolic.
If c = 0, OF = 0, and the two foci both coincide with the centre. If c
is negative, [root]c becomes imaginary, and there are no foci. Hence
we may write--
In an hyperbolic involution, OA.OA' = k^2,
In a parabolic involution, OA.OA' = 0,
In an elliptic involution, OA.OA' = -k^2.
From these expressions it follows that conjugate points A, A' in an
hyperbolic involution lie on the same side of the centre, and in an
elliptic involution on opposite sides of the centre, and that in a
parabolic involution one coincides with the centre.
In the first case, for instance, OA.OA' is positive; hence OA and OA'
have the same sign.
It also follows that two segments, AA' and BB', between pairs of
conjugate points have the following positions: in an hyperbolic
involution they lie either one altogether within or altogether without
each other; in a parabolic involution they have one point in common;
and in an elliptic involution they overlap, each being partly within
and partly without the other.
_Proof._--We have OA.OA' = OB.OB' = k^2 in case of an hyperbolic
involution. Let A and B be the points in each pair which are nearer to
the centre O. If now A, A' and B, B' lie on the same side of O, and if
B is nearer to O than A, so that OB < OA, then OB' > OA'; hence B' lies
farther away from O than A', or the segment AA' lies within BB'. And so
on for the other cases.
6. An involution is determined--
([alpha]) By two pairs of conjugate points. Hence also
([beta]) By one pair of conjugate points and the centre;
([gamma]) By the two foci;
([delta]) By one focus and one pair of conjugate points;
([epsilon]) By one focus and the centre.
7. The condition that A, B, C and A', B', C' may form an involution
may be written in one of the forms--
(AB, CC') = (A'B', C'C),
or (AB, CA') = (A'B', C'A),
or (AB, C'A') = (A'B', CA),
for each expresses that in the two projective rows in which A, B, C
and A', B', C' are conjugate points two conjugate elements may be
interchanged.
8. Any three pairs. A, A', B, B', C, C', of conjugate points are
connected by the relations:
AB'.BC'.CA' AB'.BC.C'A' AB.B'C'.CA' AB.B'C.C'A'
----------- = ----------- = ----------- = ----------- = -1.
A'B.B'C.C'A A'B.B'C'.CA A'B'.BC.C'A A'B'.BC'.CA
These relations readily follow by working out the relations in (7)
(above).
S 78. _Involution of a quadrangle.--The sides of any four-point are
cut by any line in six points in involution, opposite sides being cut
in conjugate points._
Let A1B1C1D1 (fig. 31) be the four-point. If its sides be cut by the
line p in the points A, A', B, B', C, C', if further, C1D1 cuts the
line A1B1 in C2, and if we project the row A1B1C2C to p once from D1
and once from C1, we get (A'B', C'C) = (BA, C'C).
Interchanging in the last cross-ratio the letters in each pair we get
(A'B', C'C) = (AB, CC'). Hence by S 77 (7) the points are in
involution.
The theorem may also be stated thus:
_The three points in which any line cuts the sides of a triangle and
the projections, from any point in the plane, of the vertices of the
triangle on to the same line are six points in involution._
[Illustration: FIG. 31.]
Or again--
The projections from any point on to any line of the six vertices of a
four-side are six points in involution, the projections of opposite
vertices being conjugate points.
This property gives a simple means to construct, by aid of the
straight edge only, in an involution of which two pairs of conjugate
points are given, to any point its conjugate.
S 79. _Pencils in Involution._--The theory of involution may at once
be extended from the row to the flat and the axial pencil--viz. we say
that there is an involution in a flat or in an axial pencil if any
line cuts the pencil in an involution of points. An involution in a
pencil consists of pairs of conjugate rays or planes; it has two, one
or no _focal rays_ (double lines) or _planes_, but nothing
corresponding to a centre.
An involution in a flat pencil contains always one, and in general
only one, pair of conjugate rays which are perpendicular to one
another. For in two projective flat pencils exist always two
corresponding right angles (S 40).
Each involution in an axial pencil contains in the same manner one
pair of conjugate planes at right angles to one another.
As a rule, there exists but one pair of conjugate lines or planes at
right angles to each other. But it is possible that there are more,
and then there is an infinite number of such pairs. An involution in a
flat pencil, in which every ray is perpendicular to its conjugate ray,
is said to be _circular_. That such involution is possible is easily
seen thus: if in two concentric flat pencils each ray on one is made
to correspond to that ray on the other which is perpendicular to it,
then the two pencils are projective, for if we turn the one pencil
through a right angle each ray in one coincides with its corresponding
ray in the other. But these two projective pencils are in involution.
A circular involution has no focal rays, because no ray in a pencil
coincides with the ray perpendicular to it.
S 80. _Every elliptical involution in a row may be considered as a
section of a circular involution._
In an elliptical involution any two segments AA' and BB' lie partly
within and partly without each other (fig. 32). Hence two circles
described on AA' and BB' as diameters will intersect in two points E
and E'. The line EE' cuts the base of the involution at a point O,
which has the property that OA.OA' = OB . OB', for each is equal to
OE . OE'. The point O is therefore the centre of the involution. If we
wish to construct to any point C the conjugate point C', we may draw
the circle through CEE'. This will cut the base in the required point
C' for OC.OC' = OA.OA'. But EC and EC' are at right angles. Hence the
involution which is obtained by joining E or E' to the points in the
given involution is circular. This may also be expressed thus:
[Illustration: FIG. 32.]
_Every elliptical involution has the property that there are two
definite points in the plane from which any two conjugate points are
seen under a right angle._
At the same time the following problem has been solved:
To determine the centre and also the point corresponding to any given
point in an elliptical involution of which two pairs of conjugate
points are given.
S 81. _Involution Range on a Conic._--By the aid of S 53, the points
on a conic may be made to correspond to those on a line, so that the
row of points on the conic is projective to a row of points on a line.
We may also have two projective rows on the same conic, and these will
be in involution as soon as one point on the conic has the same point
corresponding to it all the same to whatever row it belongs. An
involution of points on a conic will have the property (as follows
from its definition, and from S 53) that the lines which join
conjugate points of the involution to any point on the conic are
conjugate lines of an involution in a pencil, and that a fixed tangent
is cut by the tangents at conjugate points on the conic in points
which are again conjugate points of an involution on the fixed
tangent. For such involution on a conic the following theorem holds:
_The lines which join corresponding points in an involution on a conic
all pass through a fixed point; and reciprocally, the points of
intersection of conjugate lines in an involution among tangents to a
conic lie on a line._
[Illustration: FIG. 33]
We prove the first part only. The involution is determined by two
pairs of conjugate points, say by A, A' and B, B' (fig. 33). Let AA'
and BB' meet in P. If we join the points in involution to any point on
the conic, and the conjugate points to another point on the conic, we
obtain two projective pencils. We take A and A' as centres of these
pencils, so that the pencils A(A'BB') and A'(AB'B) are projective, and
in perspective position, because AA' corresponds to A'A. Hence
corresponding rays meet in a line, of which two points are found by
joining AB' to A'B and AB to A'B'. It follows that the _axis_ of
perspective is the polar of the point P, where AA' and BB' meet. If we
now wish to construct to any other point C on the conic the
corresponding point C', we join C to A' and the point where this line
cuts p to A. The latter line cuts the conic again in C'. But we know
from the theory of pole and polar that the line CC' passes through P.
The point of concurrence is called the "pole of the involution," and
the line of collinearity of the meets is called the "axis of the
involution."
INVOLUTION DETERMINED BY A CONIC ON A LINE.--FOCI
S 82. The polars, with regard to a conic, of points in a row p form a
pencil P projective to the row (S 66). This pencil cuts the base of
the row p in a projective row.
If A is a point in the given row, A' the point where the polar of A
cuts p, then A and A' will be corresponding points. If we take A' a
point in the first row, then the polar of A' will pass through A, so
that A corresponds to A'--in other words, the rows are in involution.
The conjugate points in this involution are conjugate points with
regard to the conic. Conjugate points coincide only if the polar of a
point A passes through A--that is, if A lies on the conic. Hence--
_A conic determines on every line in its plane an involution, in which
those points are conjugate which are also conjugate with regard to the
conic._
_If the line cuts the conic the involution is hyperbolic, the points
of intersection being the foci._
_If the line touches the conic the involution is parabolic, the two
foci coinciding at the point of contact._
_If the line does not cut the conic the involution is elliptic, having
no foci._
If, on the other hand, we take a point P in the plane of a conic, we
get to each line a through P one conjugate line which joins P to the
pole of a. These pairs of conjugate lines through P form an involution
in the pencil at P. The focal rays of this involution are the tangents
drawn from P to the conic. This gives the theorem reciprocal to the
last, viz:--
_A conic determines in every pencil in its plane an involution,
corresponding lines being conjugate lines with regard to the conic._
_If the point is without the conic the involution is hyperbolic, the
tangents from the points being the focal rays._
_If the point lies on the conic the involution is parabolic, the
tangent at the point counting for coincident focal rays._
_If the point is within the conic the involution is elliptic, having
no focal rays._
It will further be seen that the involution determined by a conic on
any line p is a section of the involution, which is determined by the
conic at the pole P of p.
S 83. _Foci._--The centre of a pencil in which the conic determines a
circular involution is called a "focus" of the conic.
In other words, a focus is such a point that every line through it is
perpendicular to its conjugate line. The polar to a focus is called a
_directrix_ of the conic.
From the definition it follows that _every focus lies on an axis_, for
the line joining a focus to the centre of the conic is a diameter to
which the conjugate lines are perpendicular; and _every line joining
two foci is an axis_, for the perpendiculars to this line through the
foci are conjugate to it. These conjugate lines pass through the pole
of the line, the pole lies therefore at infinity, and the line is a
diameter, hence by the last property an axis.
It follows that all _foci lie on one axis_, for no line joining a
point in one axis to a point in the other can be an axis.
As the conic determines in the pencil which has its centre at a focus
a circular involution, no tangents can be drawn from the focus to the
conic. Hence _each focus lies within a conic_; and _a directrix does
not cut the conic_.
Further properties are found by the following considerations:
S 84. Through a point P one line p can be drawn, which is with regard
to a given conic conjugate to a given line q, viz. that line which
joins the point P to the pole of the line q. If the line q is made to
describe a pencil about a point Q, then the line p will describe a
pencil about P. These two pencils will be projective, for the line p
passes through the pole of q, and whilst q describes the pencil Q, its
pole describes a projective row, and this row is perspective to the
pencil P.
We now take the point P on an axis of the conic, draw any line p
through it, and from the pole of p draw a perpendicular q to p. Let q
cut the axis in Q. Then, in the pencils of conjugate lines, which have
their centres at P and Q, the lines p and q are conjugate lines at
right angles to one another. Besides, to the axis as a ray in either
pencil will correspond in the other the perpendicular to the axis (S
72). The conic generated by the intersection of corresponding lines in
the two pencils is therefore the circle on PQ as diameter, _so that
every line in P is perpendicular to its corresponding line in Q_.
To every point P on an axis of a conic corresponds thus a point Q,
such that conjugate lines through P and Q are perpendicular.
We shall show that these _point-pairs_ P, Q _form an involution_. To
do this let us move P along the axis, and with it the line p, keeping
the latter parallel to itself. Then P describes a row, p a perspective
pencil (of parallels), and the pole of p a projective row. At the same
time the line q describes a pencil of parallels perpendicular to p,
and perspective to the row formed by the pole of p. The point Q,
therefore, where q cuts the axis, describes a row projective to the
row of points P. The two points P and Q describe thus two projective
rows on the axis; and not only does P as a point in the first row
correspond to Q, but also Q as a point in the first corresponds to P.
The two rows therefore form an involution. _The centre of this
involution, it is easily seen, is the centre of the conic._
_A focus of this involution has the property that any two conjugate
lines through it are perpendicular; hence, it is a focus to the
conic._
Such involution exists on each axis. But only one of these can have
foci, because all foci lie on the same axis. The involution on one of
the axes is elliptic, and appears (S 80) therefore as the section of
two circular involutions in two pencils whose centres lie in the other
axis. These centres are foci, hence the one axis contains two foci,
the other axis none; _or every central conic has two foci which lie on
one axis equidistant from the centre_.
The axis which contains the foci is called the _principal axis_; in
case of an hyperbola it is the axis which cuts the curve, because the
foci lie within the conic.
In case of the parabola there is but one axis. The involution on this
axis has its centre at infinity. One focus is therefore at infinity,
the one focus only is finite. _A parabola has only one focus._
[Illustration: FIG. 34.]
S 85. If through any point P (fig. 34) on a conic the tangent PT and
the normal PN (i.e. the perpendicular to the tangent through the point
of contact) be drawn, these will be conjugate lines with regard to the
conic, and at right angles to each other. They will therefore cut the
principal axis in two points, which are conjugate in the involution
considered in S 84; hence they are harmonic conjugates with regard to
the foci. If therefore the two foci F1 and F2 be joined to P, these
lines will be harmonic with regard to the tangent and normal. As the
latter are perpendicular, they will bisect the angles between the
other pair. Hence--
_The lines joining any point on a conic to the two foci are equally
inclined to the tangent and normal at that point._
In case of the parabola this becomes--
_The line joining any point on a parabola to the focus and the
diameter through the point, are equally inclined to the tangent and
normal at that point._
From the definition of a focus it follows that--
_The segment of a tangent between the directrix and the point of
contact is seen from the focus belonging to the directrix under a
right angle_, because the lines joining the focus to the ends of this
segment are conjugate with regard to the conic, and therefore
perpendicular.
With equal ease the following theorem is proved:
_The two lines which join the points of contact of two tangents each
to one focus, but not both to the same, are seen from the intersection
of the tangents under equal angles._
S 86. Other focal properties of a conic are obtained by the following
considerations:
[Illustration: FIG. 35.]
Let F (fig. 35) be a focus to a conic, f the corresponding directrix,
A and B the points of contact of two tangents meeting at T, and P the
point where the line AB cuts the directrix. Then TF will be the polar
of P (because polars of F and T meet at P). Hence TF and PF are
conjugate lines through a focus, and therefore perpendicular. They are
further harmonic conjugates with regard to FA and FB (SS 64 and 13),
so that they bisect the angles formed by these lines. This by the way
proves--
_The segments between the point of intersection of two tangents to a
conic and their points of contact are seen from a focus under equal
angles._
If we next draw through A and B lines parallel to TF, then the points
A1, B1 where these cut the directrix will be harmonic conjugates with
regard to P and the point where FT cuts the directrix. The lines FT
and FP bisect therefore also the angles between FA1 and FB1. From this
it follows easily that the triangles FAA1 and FBB1 are equiangular,
and therefore similar, so that FA : AA1 = FB : BB1.
The triangles AA1A2 and BB1B2 formed by drawing perpendiculars from A
and B to the directrix are also similar, so that AA1 : AA2 = = BB1 :
BB2. This, combined with the above proportion, gives FA : AA2 = FB :
BB2. Hence the theorem:
_The ratio of the distances of any point on a conic from a focus and
the corresponding directrix is constant._
To determine this ratio we consider its value for a vertex on the
principal axis. In an ellipse the focus lies between the two vertices
on this axis, hence the focus is nearer to a vertex than to the
corresponding directrix. Similarly, in an hyperbola a vertex is nearer
to the directrix than to the focus. In a parabola the vertex lies
halfway between directrix and focus.
It follows in an ellipse the ratio between the distance of a point
from the focus to that from the directrix is less than unity, in the
parabola it equals unity, and in the hyperbola it is greater than
unity.
It is here the same which focus we take, because the two foci lie
symmetrical to the axis of the conic. If now P is any point on the
conic having the distances r1 and r2 from the foci and the distances
d1 and d2 from the corresponding directrices, then r1/d1 = r2/d2 =
e, where e is constant. Hence also r1 [+-] r2 / d1 [+-] d2 = e.
In the ellipse, which lies between the directrices, d1 + d2 is
constant, therefore also r1 +r2. In the hyperbola on the other hand d1
- d2 is constant, equal to the distance between the directrices,
therefore in this case r1 - r2 is constant.
If we call the distances of a point on a conic from the focus its
focal distances we have the theorem:
_In an ellipse the sum of the focal distances is constant; and in an
hyperbola the difference of the focal distances is constant._
_This constant sum or difference equals in both cases the length of
the principal axis._
PENCIL OF CONICS
S 87. Through four points A, B, C, D in a plane, of which no three lie
in a line, an infinite number of conics may be drawn, viz. through
these four points and any fifth one single conic. This system of
conics is called a pencil of conics. Similarly, all conics touching
four fixed lines form a system such that any fifth tangent determines
one and only one conic. We have here the theorems:
The pairs of points in which The pairs of tangents which
any line is cut by a system of can be drawn from a point to
conics through four fixed points a system of conics touching four
are in involution. fixed lines are in involution.
[Illustration: FIG. 36.]
We prove the first theorem only. Let ABCD (fig. 36) be the four-point,
then any line t will cut two opposite sides AC, BD in the points E,
E', the pair AD, BC in points F, F', and any conic of the system in M,
N, and we have A(CD, MN) = B(CD, MN).
If we cut these pencils by t we get
(EF, MN) = (F'E', MN)
or (EF, MN) = (E'F', NM).
But this is, according to S 77 (7), the condition that M, N are
corresponding points in the involution determined by the point pairs
E, E', F, F' in which the line t cuts pairs of opposite sides of the
four-point ABCD. This involution is independent of the particular
conic chosen.
S 88. There follow several important theorems:
_Through four points two, one, or no conics may be drawn which touch
any given line, according as the involution determined by the given
four-point on the line has real, coincident or imaginary foci._
_Two, one, or no conics may be drawn which touch four given lines and
pass through a given point, according as the involution determined by
the given four-side at the point has real, coincident or imaginary
focal rays._
For the conic through four points which touches a given line has its
point of contact at a focus of the involution determined by the
four-point on the line.
As a special case we get, by taking the line at infinity:
_Through four points of which none is at infinity either two or no
parabolas may be drawn._
The problem of drawing a conic through four points and touching a
given line is solved by determining the points of contact on the line,
that is, by determining the foci of the involution in which the line
cuts the sides of the four-point. The corresponding remark holds for
the problem of drawing the conics which touch four lines and pass
through a given point.
RULED QUADRIC SURFACES
S 89. We have considered hitherto projective rows which lie in the
same plane, in which case lines joining corresponding points envelop a
conic. We shall now consider projective rows whose bases do not meet.
In this case, corresponding points will be joined by lines which do
not lie in a plane, but on some surface, which like every surface
generated by lines is called a _ruled_ surface. This surface clearly
contains the bases of the two rows.
If the points in either row be joined to the base of the other, we
obtain two axial pencils which are also projective, those planes being
corresponding which pass through corresponding points in the given
rows. If A', A be two corresponding points, [alpha], [alpha]' the
planes in the axial pencils passing through them, then AA' will be the
line of intersection of the corresponding planes [alpha], [alpha]' and
also the line joining corresponding points in the rows.
If we cut the whole figure by a plane this will cut the axial pencils
in two projective flat pencils, and the curve of the second order
generated by these will be the curve in which the plane cuts the
surface. Hence
_The locus of lines joining corresponding points in two projective
rows which do not lie in the same plane is a surface which contains
the bases of the rows, and which can also be generated by the lines of
intersection of corresponding planes in two projective axial pencils.
This surface is cut by every plane in a curve of the second order,
hence either in a conic or in a line-pair. No line which does not lie
altogether on the surface can have more than two points in common with
the surface, which is therefore said to be of the second order or is
called a ruled quadric surface._
That no line which does not lie on the surface can cut the surface in
more than two points is seen at once if a plane be drawn through the
line, for this will cut the surface in a conic. It follows also that a
line which contains more than two points of the surface lies
altogether on the surface.
S 90. Through any point in space one line can always be drawn cutting
two given lines which do not themselves meet.
If therefore three lines in space be given of which no two meet, then
through every point in either one line may be drawn cutting the other
two.
_If a line moves so that it always cuts three given lines of which no
two meet, then it generates a ruled quadric surface._
Let a, b, c be the given lines, and p, q, r ... lines cutting them in
the points A, A', A" ...; B, B', B" ...; C, C', C" ... respectively;
then the planes through a containing p, q, r, and the planes through b
containing the same lines, may be taken as corresponding planes in two
axial pencils which are projective, because both pencils cut the line
c in the same row, C, C', C" ...; the surface can therefore be
generated by projective axial pencils.
Of the lines p, q, r ... no two can meet, for otherwise the lines a,
b, c which cut them would also lie in their plane. There is a single
infinite number of them, for one passes through each point of a. These
lines are said to form a set of lines on the surface.
If now three of the lines p, q, r be taken, then every line d cutting
them will have three points in common with the surface, and will
therefore lie altogether on it. This gives rise to a second set of
lines on the surface. From what has been said the theorem follows:
_A ruled quadric surface contains two sets of straight lines. Every
line of one set cuts every line of the other, but no two lines of the
same set meet._
_Any two lines of the same set may be taken as bases of two projective
rows, or of two projective pencils which generate the surface. They
are cut by the lines of the other set in two projective rows._
The plane at infinity like every other plane cuts the surface either
in a conic proper or in a line-pair. In the first case the surface is
called an _Hyperboloid of one sheet_, in the second an _Hyperbolic
Paraboloid_.
The latter may be generated by a line cutting three lines of which one
lies at infinity, that is, cutting two lines and remaining parallel to
a given plane.
QUADRIC SURFACES
S 91. The conics, the cones of the second order, and the ruled quadric
surfaces complete the figures which can be generated by projective
rows or flat and axial pencils, that is, by those aggregates of
elements which are of one dimension (SS 5, 6). We shall now consider
the simpler figures which are generated by aggregates of two
dimensions. The space at our disposal will not, however, allow us to
do more than indicate a few of the results.
S 92. We establish a correspondence between the lines and planes in
pencils in space, or reciprocally between the points and lines in two
or more planes, but consider principally pencils.
In two pencils we may either make planes correspond to planes and
lines to lines, or else planes to lines and lines to planes. If hereby
the condition be satisfied that to a flat, or axial, pencil
corresponds in the first case a projective flat, or axial, pencil, and
in the second a projective axial, or flat, pencil, the pencils are
said to be _projective_ in the first case and _reciprocal_ in the
second.
For instance, two pencils which join two points S1 and S2 to the
different points and lines in a given plane [pi] are projective (and
in perspective position), if those lines and planes be taken as
corresponding which meet the plane [pi] in the same point or in the
same line. In this case every plane through both centres S1 and S2 of
the two pencils will correspond to itself. If these pencils are
brought into any other position they will be projective (but not
perspective).
_The correspondence between two projective pencils is uniquely
determined, if to four rays (or planes) in the one the corresponding
rays (or planes) in the other are given, provided that no three rays
of either set lie in a plane._
Let a, b, c, d be four rays in the one, a', b', c', d' the
corresponding rays in the other pencil. We shall show that we can find
for every ray e in the first a single corresponding ray e' in the
second. To the axial pencil a (b, c, d ...) formed by the planes which
join a to b, c, d ..., respectively corresponds the axial pencil a'
(b', c', d' ... ), and this correspondence is determined. Hence, the
plane a'e' which corresponds to the plane ae is determined. Similarly
the plane b'e' may be found and both together determine the ray e'.
Similarly the correspondence between two reciprocal pencils is
determined if for four rays in the one the corresponding planes in the
other are given.
S 93. We may now combine--
1. Two reciprocal pencils.
Each ray cuts its corresponding plane in a point, the locus of these
points is a quadric surface.
2. Two projective pencils.
Each plane cuts its corresponding plane in a line, but a ray as a
rule does not cut its corresponding ray. The locus of points where a
ray cuts its corresponding ray is a twisted cubic. The lines where a
plane cuts its corresponding plane are secants.
3. Three projective pencils.
The locus of intersection of corresponding planes is a cubic
surface.
Of these we consider only the first two cases.
S 94. If two pencils are reciprocal, then to a plane in either
corresponds a line in the other, to a flat pencil an axial pencil, and
so on. Every line cuts its corresponding plane in a point. If S1 and
S2 be the centres of the two pencils, and P be a point where a line a1
in the first cuts its corresponding plane [alpha]2, _then the line b2
in the pencil S2 which passes through P will meet its corresponding
plane [beta]1 in P_. For b2 is a line in the plane [alpha]2. The
corresponding plane [beta]1 must therefore pass through the line a1,
hence through P.
The points in which the lines in S1 cut the planes corresponding to
them in S2 are therefore the same as the points in which the lines in
S2 cut the planes corresponding to them in S1.
_The locus of these points is a surface which is cut by a plane in a
conic or in a line-pair and by a line in not more than two points
unless it lies altogether on the surface. The surface itself is
therefore called a quadric surface, or a surface of the second order._
To prove this we consider any line p in space.
The flat pencil in S1 which lies in the plane drawn through p and the
corresponding axial pencil in S2 determine on p two projective rows,
and those points in these which coincide with their corresponding
points lie on the surface. But there exist only two, or one, or no
such points, unless every point coincides with its corresponding
point. In the latter case the line lies altogether on the surface.
This proves also that a plane cuts the surface in a curve of the
second order, as no line can have more than two points in common with
it. To show that this is a curve of the same kind as those considered
before, we have to show that it can be generated by projective flat
pencils. We prove first that this is true for any plane through the
centre of one of the pencils, and afterwards that every point on the
surface may be taken as the centre of such pencil. Let then [alpha]1
be a plane through S1. To the flat pencil in S1 which it contains
corresponds in S2 a projective axial pencil with axis a2 and this cuts
[alpha]1 in a second flat pencil. These two flat pencils in [alpha]1
are projective, and, in general, neither concentric nor perspective.
They generate therefore a conic. But if the line a2 passes through S1
the pencils will have S1 as common centre, and may therefore have two,
or one, or no lines united with their corresponding lines. The section
of the surface by the plane [alpha]1 will be accordingly a line-pair
or a single line, or else the plane [alpha]1 will have only the point
S1 in common with the surface.
Every line l1 through S1 cuts the surface in two points, viz. first in
S1 and then at the point where it cuts its corresponding plane. If now
the corresponding plane passes through S1, as in the case just
considered, then the two points where l1 cuts the surface coincide at
S1, and the line is called a tangent to the surface with S1 as point
of contact. Hence if l1 be a tangent, it lies in that plane [tau]1
which corresponds to the line S2S1 as a line in the pencil S2. The
section of this plane has just been considered. It follows that--
_All tangents to quadric surface at the centre of one of the
reciprocal pencils lie in a plane which is called the tangent plane to
the surface at that point as point of contact._
_To the line joining the centres of the two pencils as a line in one
corresponds in the other the tangent plane at its centre._
_The tangent plane to a quadric surface either cuts the surface in two
lines, or it has only a single line, or else only a single point in
common with the surface._
_In the first case the point of contact is said to be hyperbolic, in
the second parabolic, in the third elliptic._
S 95. It remains to be proved that every point S on the surface may be
taken as centre of one of the pencils which generate the surface. Let
S be any point on the surface [Phi]' generated by the reciprocal
pencils S1 and S2. We have to establish a reciprocal correspondence
between the pencils S and S1, so that the surface generated by them is
identical with [Phi]. To do this we draw two planes [alpha]1 and
[beta]1 through S1, cutting the surface [Phi] in two conics which we
also denote by [alpha]1 and [beta]1. These conics meet at S1, and at
some other point T where the line of intersection of [alpha]1 and
[beta]1 cuts the surface.
In the pencil S we draw some plane [sigma] which passes through T, but
not through S1 or S2. It will cut the two conics first at T, and
therefore each at some other point which we call A and B respectively.
These we join to S by lines a and b, and now establish the required
correspondence between the pencils S1 and S as follows:--To S1T shall
correspond the plane [sigma], to the plane [alpha]1 the line a, and to
[beta]1 the line b, hence to the flat pencil in [alpha]1 the axial
pencil a. These pencils are made projective by aid of the conic in
[alpha]1.
In the same manner the flat pencil in [beta]1 is made projective to
the axial pencil b by aid of the conic in [beta]1, corresponding
elements being those which meet on the conic. This determines the
correspondence, for we know for more than four rays in S1 the
corresponding planes in S. The two pencils S and S1 thus made
reciprocal generate a quadric surface [Phi]', which passes through the
point S and through the two conics [alpha]1 and [beta]1.
The two surfaces [Phi] and [Phi]' have therefore the points S and S1
and the conics [alpha]1 and [beta]1 in common. To show that they are
identical, we draw a plane through S and S2, cutting each of the
conics [alpha]1 and [beta]1 in two points, which will always be
possible. This plane cuts [Phi] and [Phi]' in two conics which have
the point S and the points where it cuts [alpha]1 and [beta]1 in
common, that is five points in all. The conics therefore coincide.
This proves that all those points P on [Phi]' lie on [Phi] which have
the property that the plane SS2P cuts the conics [alpha]1, [beta]1 in
two points each. If the plane SS2P has not this property, then we draw
a plane SS1P. This cuts each surface in a conic, and these conics have
in common the points S, S1, one point on each of the conics [alpha]1,
[beta]1, and one point on one of the conics through S and S2 which lie
on both surfaces, hence five points. They are therefore coincident,
and our theorem is proved.
S 96. The following propositions follow:--
_A quadric surface has at every point a tangent plane._
_Every plane section of a quadric surface is a conic or a line-pair._
_Every line which has three points in common with a quadric surface
lies on the surface._
_Every conic which has five points in common with a quadric surface
lies on the surface._
_Through two conics which lie in different planes, but have two points
in common, and through one external point always one quadric surface
may be drawn._
S 97. _Every plane which cuts a quadric surface in a line-pair is a
tangent plane._ For every line in this plane through the centre of the
line-pair (the point of intersection of the two lines) cuts the
surface in two coincident points and is therefore a tangent to the
surface, _the centre of the line-pair being the point of contact_.
_If a quadric surface contains a line, then every plane through this
line cuts the surface in a line-pair (or in two coincident lines)._
For this plane cannot cut the surface in a conic. Hence:--
_If a quadric surface contains one line p then it contains an infinite
number of lines, and through every point Q on the surface, one line q
can be drawn which cuts p._ For the plane through the point Q and the
line p cuts the surface in a line-pair which must pass through Q and
of which p is one line.
_No two such lines q on the surface can meet_. For as both meet p
their plane would contain p and therefore cut the surface in a
triangle.
_Every line which cuts three lines q will be on the surface_; for it
has three points in common with it.
_Hence the quadric surfaces which contain lines are the same as the
ruled quadric surfaces considered in_ SS 89-93, but with one important
exception. In the last investigation we have left out of consideration
the possibility of a plane having only one line (two coincident lines)
in common with a quadric surface.
S 98. To investigate this case we suppose first that there is one
point A on the surface through which two different lines a, b can be
drawn, which lie altogether on the surface.
If P is any other point on the surface which lies neither on a nor b,
then the plane through P and a will cut the surface in a second line
a' which passes through P and which cuts a. Similarly there is a line
b' through P which cuts b. These two lines a' and b' _may_ coincide,
but then they must coincide with PA.
If this happens for one point P, it happens for every other point Q.
For if two different lines could be drawn through Q, then by the same
reasoning the line PQ would be altogether on the surface, hence two
lines would be drawn through P against the assumption. From this
follows:--
_If there is one point on a quadric surface through which one, but
only one, line can be drawn on the surface, then through every point
one line can be drawn, and all these lines meet in a point. The
surface is a cone of the second order_.
_If through one point on a quadric surface, two, and only two, lines
can be drawn on the surface, then through every point two lines may be
drawn, and the surface is ruled quadric surface._
_If through one point on a quadric surface no line on the surface can
be drawn, then the surface contains no lines._
Using the definitions at the end of S 95, we may also say:--
_On a quadric surface the points are all hyperbolic, or all parabolic,
or all elliptic._
As an example of a quadric surface with elliptical points, we mention
the sphere which may be generated by two reciprocal pencils, where to
each line in one corresponds the plane perpendicular to it in the
other.
S 99. _Poles and Polar Planes._--The theory of poles and polars with
regard to a conic is easily extended to quadric surfaces.
Let P be a point in space not on the surface, which we suppose not to
be a cone. On every line through P which cuts the surface in two
points we determine the harmonic conjugate Q of P with regard to the
points of intersection. Through one of these lines we draw two planes
[alpha] and [beta]. The locus of the points Q in [alpha] is a line a,
the polar of P with regard to the conic in which [alpha] cuts the
surface. Similarly the locus of points Q in [beta] is a line b. This
cuts a, because the line of intersection of [alpha] and [beta]
contains but one point Q. The locus of all points Q therefore is a
plane. _This plane is called the polar plane of the point P, with
regard to the quadric surface. If P lies on the surface we take the
tangent plane of P as its polar._
The following propositions hold:--
1. _Every point has a polar plane_, which is constructed by drawing
the polars of the point with regard to the conics in which two planes
through the point cut the surface.
2. _If Q is a point in the polar of P, then P is a point in the polar
of Q_, because this is true with regard to the conic in which a plane
through PQ cuts the surface.
3. _Every plane is the polar plane of one point, which is called the
Pole of the plane._
The pole to a plane is found by constructing the polar planes of three
points in the plane. Their intersection will be the pole.
4. _The points in which the polar plane of P cuts the surface are
points of contact of tangents drawn from P to the surface_, as is
easily seen. Hence:--
5. _The tangents drawn from a point P to a quadric surface form a cone
of the second order_, for the polar plane of P cuts it in a conic.
6. _If the pole describes a line a, its polar plane will turn about
another line a'_, as follows from 2. _These lines a and a' are said to
be conjugate with regard to the surface._
S 100. The pole of the line at infinity is called the _centre_ of the
surface. If it lies at the infinity, the plane at infinity is a
tangent plane, and the surface is called a _paraboloid_.
_The polar plane to any point at infinity passes through the centre,
and is called a diametrical plane._
_A line through the centre is called a diameter. It is bisected at the
centre. The line conjugate to it lies at infinity._
_If a point moves along a diameter its polar plane turns about the
conjugate line at infinity_; that is, _it moves parallel to itself,
its centre moving on the first line._
_The middle points of parallel chords lie in a plane_, viz. in the
polar plane of the point at infinity through which the chords are
drawn.
_The centres of parallel sections lie in a diameter which is a line
conjugate to the line at infinity in which the planes meet._
TWISTED CUBICS
S 101. If two pencils with centres S1 and S2 are made projective, then
to a ray in one corresponds a ray in the other, to a plane a plane, to
a flat or axial pencil a projective flat or axial pencil, and so on.
There is a double infinite number of lines in a pencil. We shall see
that a single infinite number of lines in one pencil meets its
corresponding ray, and that the points of intersection form a curve in
space.
Of the double infinite number of planes in the pencils each will meet
its corresponding plane. This gives a system of a double infinite
number of lines in space. We know (S 5) that there is a quadruple
infinite number of lines in space. From among these we may select
those which satisfy one or more given conditions. The systems of lines
thus obtained were first systematically investigated and classified by
Plucker, in his _Geometrie des Raumes_. He uses the following names:--
A _treble infinite_ number of lines, that is, all lines which satisfy
one condition, are said to form a _complex of lines_; e.g. all lines
cutting a given line, or all lines touching a surface.
A _double infinite_ number of lines, that is, all lines which satisfy
two conditions, or which are common to two complexes, are said to form
a _congruence of lines_; e.g. all lines in a plane, or all lines
cutting two curves, or all lines cutting a given curve twice.
A _single infinite_ number of lines, that is, all lines which satisfy
three conditions, or which belong to three complexes, form a _ruled
surface_; e.g. one set of lines on a ruled quadric surface, or
developable surfaces which are formed by the tangents to a curve.
It follows that all lines in which corresponding planes in two
projective pencils meet form a congruence. We shall see this
congruence consists of all lines which cut a twisted cubic twice, or
of all _secants_ to a twisted cubic.
S 102. Let l1 be the line S1S2 as a line in the pencil S1. To it
corresponds a line l2 in S2. _At each of the centres two corresponding
lines meet._ The two axial pencils with l1 and l2 as axes are
projective, and, as, their axes meet at S2, the intersections of
corresponding planes form a cone of the second order (S 58), with S2
as centre. If [pi]1 and [pi]2 be corresponding planes, then their
intersection will be a line p2 which passes through S2. Corresponding
to it in S1 will be a line p1 which lies in the plane [pi]1, and which
therefore meets p2 at some point P. Conversely, if p2 be any line in
S2 which meets its corresponding line p1 at a point P, then to the
plane l2p2 will correspond the plane l1p1, that is, the plane S1S2P.
These planes intersect in p2, so that p2 is a line on the quadric cone
generated by the axial pencils l1 and l2. Hence:--
_All lines in one pencil which meet their corresponding lines in the
other form a cone of the second order which has its centre at the
centre of the first pencil, and passes through the centre of the
second._
From this follows that the points in which corresponding rays meet lie
on two cones of the second order which have the ray joining their
centres in common, and form therefore, together with the line S1S2 or
l1, the intersection of these cones. Any plane cuts each of the cones
in a conic. These two conics have necessarily that point in common in
which it cuts the line l1, and therefore besides either one or three
other points. It follows that the curve is of the third order as a
plane may cut it in three, but not in more than three, points.
Hence:--
_The locus of points in which corresponding lines on two projective
pencils meet is a curve of the third order or a "twisted cubic" k,
which passes through the centres of the pencils, and which appears as
the intersection of two cones of the second order, which have one line
in common._
_A line belonging to the congruence determined by the pencils is a
secant of the cubic; it has two, or one, or no points in common with
this cubic, and is called accordingly a secant proper, a tangent, or a
secant improper of the cubic._ A secant improper may be considered, to
use the language of coordinate geometry, as a secant with imaginary
points of intersection.
S 103. If a1 and a2 be any two corresponding lines in the two pencils,
then corresponding planes in the axial pencils having a1 and a2 as
axes generate a ruled quadric surface. If P be any point on the cubic
k, and if p1, p2 be the corresponding rays in S1 and S2 which meet at
P, then to the plane a1p1 in S1 corresponds a2p2 in S2. These
therefore meet in a line through P.
This may be stated thus:--
_Those secants of the cubic which cut a ray a1, drawn through the
centre S1 of one pencil, form a ruled quadric surface which passes
through both centres, and which contains the twisted cubic k. Of such
surfaces an infinite number exists. Every ray through S1 or S2 which
is not a secant determines one of them._
If, however, the rays a1 and a2 are secants meeting at A, then the
ruled quadric surface becomes a cone of the second order, having A as
centre. Or _all lines of the congruence which pass through a point on
the twisted cubic k form a cone of the second order_. In other words,
the projection of a twisted cubic from any point in the curve on to
any plane is a conic.
If a1 is not a secant, but made to pass through any point Q in space,
the ruled quadric surface determined by a1 will pass through Q. _There
will therefore be one line of the congruence passing through Q, and
only one._ For if two such lines pass through Q, then the lines S1Q
and S2Q will be corresponding lines; hence Q will be a point on the
cubic k, and an infinite number of secants will pass through it.
Hence:--
_Through every point in space not on the twisted cubic one and only
one secant to the cubic can be drawn._
S 104. The fact that all the secants through a point on the cubic form
a quadric cone shows that the centres of the projective pencils
generating the cubic are not distinguished from any other points on
the cubic. If we take any two points S, S' on the cubic, and draw the
secants through each of them, we obtain two quadric cones, which have
the line SS' in common, and which intersect besides along the cubic.
If we make these two pencils having S and S' as centres projective by
taking four rays on the one cone as corresponding to the four rays on
the other which meet the first on the cubic, the correspondence is
determined. These two pencils will generate a cubic, and the two cones
of secants having S and S' as centres will be identical with the above
cones, for each has five rays in common with one of the first, viz.
the line SS' and the four lines determined for the correspondence;
therefore these two cones intersect in the original cubic. This gives
the theorem:--
_On a twisted cubic any two points may be taken as centres of
projective pencils which generate the cubic, corresponding planes
being those which meet on the same secant._
Of the two projective pencils at S and S' we may keep the first fixed,
and move the centre of the other along the curve. The pencils will
hereby remain projective, and a plane [alpha] in S will be cut by its
corresponding plane [alpha]' always in the same secant a. Whilst S'
moves along the curve the plane [alpha]' will turn about a, describing
an axial pencil.
AUTHORITIES.--In this article we have given a purely geometrical
theory of conics, cones of the second order, quadric surfaces, &c. In
doing so we have followed, to a great extent, Reye's _Geometrie der
Lage_, and to this excellent work those readers are referred who wish
for a more exhaustive treatment of the subject. Other works especially
valuable as showing the development of the subject are: Monge,
_Geometrie descriptive_: Carnot, _Geometrie de position_ (1803),
containing a theory of transversals; Poncelet's great work _Traite des
proprietes projectives des figures_ (1822); Mobins, _Barycentrischer
Calcul_ (1826); Steiner, _Abhangigkeit geometrischer Gestalten_
(1832), containing the first full discussion of the projective
relations between rows, pencils, &c.; Von Staudt, _Geometrie der Lage_
(1847) and _Beitrage zur Geometrie der Lage_ (1856-1860), in which a
system of geometry is built up from the beginning without any
reference to number, so that ultimately a number itself gets a
geometrical definition, and in which imaginary elements are
systematically introduced into pure geometry; Chasles, _Apercu
historique_ (1837), in which the author gives a brilliant account of
the progress of modern geometrical methods, pointing out the
advantages of the different purely geometrical methods as compared
with the analytical ones, but without taking as much account of the
German as of the French authors; Id., _Rapport sur les progres de la
geometrie_ (1870), a continuation of the _Apercu_; Id., _Traite de
geometrie superieure_ (1852); Cremona, _Introduzione ad una teoria
geometrica delle curve piane_ (1862) and its continuation _Preliminari
di una teoria geometrica delle superficie_ (German translations by
Curtze). As more elementary books, we mention: Cremona, _Elements of
Projective Geometry_, translated from the Italian by C. Leudesdorf
(2nd ed., 1894); J.W. Russell, _Pure Geometry_ (2nd ed., 1905).
(O. H.)
III. DESCRIPTIVE GEOMETRY
This branch of geometry is concerned with the methods for representing
solids and other figures in three dimensions by drawings in one plane.
The most important method is that which was invented by Monge towards
the end of the 18th century. It is based on parallel projections to a
plane by rays perpendicular to the plane. Such a projection is called
orthographic (see PROJECTION, S 18). If the plane is horizontal the
projection is called the plan of the figure, and if the plane is
vertical the elevation. In Monge's method a figure is represented by its
plan and elevation. It is therefore often called drawing in plan and
elevation, and sometimes simply orthographic projection.
S 1. We suppose then that we have two planes, one horizontal, the
other vertical, and these we call the planes of plan and of elevation
respectively, or the horizontal and the vertical plane, and denote
them by the letters [pi]1 and [pi]2. Their line of intersection is
called the axis, and will be denoted by xy.
If the surface of the drawing paper is taken as the plane of the plan,
then the vertical plane will be the plane perpendicular to it through
the axis xy. To bring this also into the plane of the drawing paper we
turn it about the axis till it coincides with the horizontal plane.
This process of turning one plane down till it coincides with another
is called _rabatting_ one to the other. Of course there is no
necessity to have one of the two planes horizontal, but even when this
is not the case it is convenient to retain the above names.
[Illustration: FIG. 37.]
[Illustration: FIG. 38.]
The whole arrangement will be better understood by referring to fig.
37. A point A in space is there projected by the perpendicular AA1 and
AA2 to the planes [pi]1 and [pi]2 so that A1 and A2 are the horizontal
and vertical projections of A.
If we remember that a line is perpendicular to a plane that is
perpendicular to every line in the plane if only it is perpendicular
to any two intersecting lines in the plane, we see that the axis which
is perpendicular both to AA1 and to AA2 is also perpendicular to A1A0
and to A2A0 because these four lines are all in the same plane. Hence,
if the plane [pi]2 be turned about the axis till it coincides with the
plane [pi]1, then A2A0 will be the continuation of A1A0. This position
of the planes is represented in fig. 38, in which the line A1A2 is
perpendicular to the axis x.
Conversely any two points A1, A2 in a line perpendicular to the axis
will be the projections of some point in space when the plane [pi]2 is
turned about the axis till it is perpendicular to the plane [pi]1,
because in this position the two perpendiculars to the planes [pi]1
and [pi]2 through the points A1 and A2 will be in a plane and
therefore meet at some point A.
_Representation of Points._--We have thus the following method of
representing in a single plane the position of points in space:--_we
take in the plane a line xy as the axis, and then any pair of points
A1, A2 in the plane on a line perpendicular to the axis represent a
point A in space_. If the line A1A2 cuts the axis at A0, and if at A1
a perpendicular be erected to the plane, then the point A will be in
it at a height A1A = A0A2 above the plane. This gives the position of
the point A relative to the plane [pi]1. In the same way, if in a
perpendicular to [pi]2 through A2 a point A be taken such that A2A =
A0A1, then this will give the point A relative to the plane [pi]2.
[Illustration: FIG. 39.]
S 2. The two planes [pi]1, [pi]2 in their original position divide
space into four parts. These are called the four quadrants. We suppose
that the plane [pi]2 is turned as indicated in fig. 37, so that the
point P comes to Q and R to S, then the quadrant in which the point A
lies is called the first, and we say that in the first quadrant a
point lies above the horizontal and in front of the vertical plane.
Now we go round the axis in the sense in which the plane [pi]2 is
turned and come in succession to the second, third and fourth
quadrant. In the second a point lies above the plane of the plan and
behind the plane of elevation, and so on. In fig. 39, which represents
a side view of the planes in fig. 37 the quadrants are marked, and in
each a point with its projection is taken. Fig. 38 shows how these are
represented when the plane [pi]2 is turned down. We see that
_A point lies in the first quadrant if the plan lies below, the
elevation above the axis; in the second if plan and elevation both lie
above; in the third if the plan lies above, the elevation below; in
the fourth if plan and elevation both lie below the axis._
_If a point lies in the horizontal plane_, its elevation lies in the
axis and the plan coincides with the point itself. _If a point lies in
the vertical plane_, its plan lies in the axis and the elevation
coincides with the point itself. _If a point lies in the axis_, both
its plan and elevation lie in the axis and coincide with it.
Of each of these propositions, which will easily be seen to be true,
the converse holds also.
S 3. _Representation of a Plane._--As we are thus enabled to represent
points in a plane, we can represent any finite figure by representing
its separate points. It is, however, not possible to represent a plane
in this way, for the projections of its points completely cover the
planes [pi]1 and [pi]2, and no plane would appear different from any
other. But any plane [alpha] cuts each of the planes [pi]1, [pi]2 in a
line. These are called the traces of the plane. They cut each other in
the axis at the point where the latter cuts the plane [alpha].
_A plane is determined by its two traces, which are two lines that
meet on the axis_, and, conversely, _any two lines which meet on the
axis determine a plane_.
_If the plane is parallel to the axis its traces are parallel to the
axis._ Of these one may be at infinity; then the plane will cut one of
the planes of projection at infinity and will be parallel to it. Thus
a plane parallel to the horizontal plane of the plan has only one
finite trace, viz. that with the plane of elevation.
[Illustration: FIG. 40.]
_If the plane passes through the axis both its traces coincide with
the axis._ This is the only case in which the representation of the
plane by its two traces fails. A third plane of projection is
therefore introduced, which is best taken perpendicular to the other
two. We call it simply the third plane and denote it by [pi]3. As it
is perpendicular to [pi]1, it may be taken as the plane of elevation,
its line of intersection [gamma] with [pi]1 being the axis, and be
turned down to coincide with [pi]1. This is represented in fig. 40. OC
is the axis xy whilst OA and OB are the traces of the third plane.
They lie in one line [gamma]. The plane is rabatted about [gamma] to
the horizontal plane. A plane [alpha] through the axis xy will then
show in it a trace [alpha]3. In fig. 40 the lines OC and OP will thus
be the traces of a plane through the axis xy, which makes an angle POQ
with the horizontal plane.
We can also find the trace which any other plane makes with [pi]3. In
rabatting the plane [pi]3 its trace OB with the plane [pi]2 will come
to the position OD. Hence a plane [beta] having the traces CA and CB
will have with the third plane the trace [beta]3, or AD if OD = OB.
It also follows immediately that--
_If a plane [alpha] is perpendicular to the horizontal plane, then
every point in it has its horizontal projection in the horizontal
trace of the plane_, as all the rays projecting these points lie in
the plane itself.
_Any plane which is perpendicular to the horizontal plane has its
vertical trace perpendicular to the axis._
_Any plane which is perpendicular to the vertical plane has its
horizontal trace perpendicular to the axis and the vertical
projections of all points in the plane lie in this trace._
S 4. _Representation of a Line._--A line is determined either by two
points in it or by two planes through it. We get accordingly two
representations of it either by projections or by traces.
First.--_A line a is represented by its projections a1 and a2 on the
two planes [pi]1 and [pi]2._ These may be any two lines, for, bringing
the planes [pi]1, [pi]2 into their original position, the planes
through these lines perpendicular to [pi]1 and [pi]2 respectively will
intersect in some line a which has a1, a2 as its projections.
Secondly.--_A line a is represented by its traces--that is, by the
points in which it cuts the two planes [pi]1, [pi]2._ Any two points
may be taken as the traces of a line in space, for it is determined
when the planes are in their original position as the line joining the
two traces. This representation becomes undetermined if the two traces
coincide in the axis. In this case we again use a third plane, or else
the projections of the line.
The fact that there are different methods of representing points and
planes, and hence two methods of representing lines, suggests the
principle of duality (section ii., _Projective Geometry_, S 41). It is
worth while to keep this in mind. It is also worth remembering that
traces of planes or lines always lie in the planes or lines which they
represent. Projections do not as a rule do this excepting when the
point or line projected lies in one of the planes of projection.
Having now shown how to represent points, planes and lines, we have to
state the conditions which must hold in order that these elements may
lie one in the other, or else that the figure formed by them may
possess certain metrical properties. It will be found that the former
are very much simpler than the latter.
Before we do this, however, we shall explain the notation used; for it
is of great importance to have a systematic notation. We shall denote
points in space by capitals A, B, C; planes in space by Greek letters
[alpha], [beta], [gamma]; lines in space by small letters a, b, c;
horizontal projections by suffixes 1, like A1, a1; vertical
projections by suffixes 2, like A2, a2; traces by single and double
dashes [alpha]' [alpha]", a', a". Hence P1 will be the horizontal
projection of a point P in space; a line a will have the projections
a1, a2 and the traces a' and a"; a plane [alpha] has the traces
[alpha]' and [alpha]".
S 5. _If a point lies in a line, the projections of the point lie in
the projections of the line._
_If a line lies in a plane, the traces of the line lie in the traces
of the plane._
These propositions follow at once from the definitions of the
projections and of the traces.
If a point lies in two lines its projections must lie in the
projections of both. Hence
_If two lines, given by their projections, intersect, the intersection
of their planes and the intersection of their elevations must lie in a
line perpendicular to the axis_, because they must be the projections
of the point common to the two lines.
Similarly--_If two lines given by their traces lie in the same plane
or intersect, then the lines joining their horizontal and vertical
traces respectively must meet on the axis_, because they must be the
traces of the plane through them.
S 6. _To find the projections of a line which joins two points A, B
given by their projections A1, A2 and B1, B2_, we join A1, B1 and A2,
B2; these will be the projections required. For example, the traces of
a line are two points in the line whose projections are known or at
all events easily found. They are the traces themselves and the feet
of the perpendiculars from them to the axis.
Hence _if a' a" (fig. 41) are the traces of a line a, and if the
perpendiculars from them cut the axis in P and Q respectively, then
the line a'Q will be the horizontal and a"P the vertical projection of
the line_.
[Illustration: FIG. 41.]
Conversely, if the projections a1, a2 of a line are given, and if
these cut the axis in Q and P respectively, then _the perpendiculars
Pa' and Qa" to the axis drawn through these points cut the projections
a1 and a2 in the traces a' and a"_.
_To find the line of intersection of two planes_, we observe that this
line lies in both planes; its traces must therefore lie in the traces
of both. Hence the points where the horizontal traces of the given
planes meet will be the horizontal, and the point where the vertical
traces meet the vertical trace of the line required.
S 7. _To decide whether a point A, given by its projections, lies in a
plane [alpha], given by its traces_, we draw a line p by joining A to
some point in the plane [alpha] and determine its traces. If these lie
in the traces of the plane, then the line, and therefore the point A,
lies in the plane; otherwise not. This is conveniently done by joining
A1 to some point p' in the trace [alpha]'; this gives p1; and the
point where the perpendicular from p' to the axis cuts the latter we
join to A2; this gives p2. If the vertical trace of this line lies in
the vertical trace of the plane, then, and then only, does the line p,
and with it the point A, lie in the plane [alpha].
S 8. _Parallel planes have parallel traces_, because parallel planes
are cut by any plane, hence also by [pi]1 and by [pi]2, in parallel
lines.
_Parallel lines have parallel projections_, because points at infinity
are projected to infinity.
_If a line is parallel to a plane, then lines through the traces of
the line and parallel to the traces of the plane must meet on the
axis_, because these lines are the traces of a plane parallel to the
given plane.
S 9. _To draw a plane through two intersecting lines or through two
parallel lines_, we determine the traces of the lines; the lines
joining their horizontal and vertical traces respectively will be the
horizontal and vertical traces of the plane. They will meet, at a
finite point or at infinity, on the axis if the lines do intersect.
_To draw a plane through a line and a point without the line_, we join
the given point to any point in the line and determine the plane
through this and the given line.
_To draw a plane through three points which are not in a line_, we
draw two of the lines which each join two of the given points and draw
the plane through them. If the traces of all three lines AB, BC, CA be
found, these must lie in two lines which meet on the axis.
S 10. We have in the last example got more points, or can easily get
more points, than are necessary for the determination of the figure
required--in this case the traces of the plane. This will happen in a
great many constructions and is of considerable importance. It may
happen that some of the points or lines obtained are not convenient in
the actual construction. The horizontal traces of the lines AB and AC
may, for instance, fall very near together, in which case the line
joining them is not well defined. Or, one or both of them may fall
beyond the drawing paper, so that they are practically non-existent
for the construction. In this case the traces of the line BC may be
used. Or, if the vertical traces of AB and AC are both in convenient
position, so that the vertical trace of the required plane is found
and one of the horizontal traces is got, then we may join the latter
to the point where the vertical trace cuts the axis.
The draughtsman must remember that the lines which he draws are not
mathematical lines without thickness, and therefore every drawing is
affected by some errors. It is therefore very desirable to be able
constantly to check the latter. Such checks always present themselves
when the same result can be obtained by different constructions, or
when, as in the above case, some lines must meet on the axis, or if
three points must lie in a line. A careful draughtsman will always
avail himself of these checks.
S 11. _To draw a plane through a given point parallel to a given plane
[alpha]_, we draw through the point two lines which are parallel to
the plane [alpha], and determine the plane through them; or, as we
know that the traces of the required plane are parallel to those of
the given one (S 8), we need only draw one line l through the point
parallel to the plane and find one of its traces, say the vertical
trace l"; a line through this parallel to the vertical trace of
[alpha] will be the vertical trace [beta]" of the required plane
[beta], and a line parallel to the horizontal trace of [alpha] meeting
[beta]" on the axis will be the horizontal trace [beta]'.
[Illustration: FIG. 42.]
Let A1 A2 (fig. 42) be the given point, [alpha]' [alpha]" the given
plane, a line l1 through A1, parallel to [alpha]' and a horizontal
line l2 through A2 will be the projections of a line l through A
parallel to the plane, because the horizontal plane through this line
will cut the plane [alpha] in a line c which has its horizontal
projection c1 parallel to [alpha]'.
S 12. We now come to the metrical properties of figures.
_A line is perpendicular to a plane if the projections of the line are
perpendicular to the traces of the plane._ We prove it for the
horizontal projection. If a line p is perpendicular to a plane
[alpha], every plane through p is perpendicular to [alpha]; hence also
the vertical plane which projects the line p to p1. As this plane is
perpendicular both to the horizontal plane and to the plane [alpha],
it is also perpendicular to their intersection--that is, to the
horizontal trace of [alpha]. It follows that every line in this
projecting plane, therefore also p1, the plan of p, is perpendicular
to the horizontal trace of [alpha].
_To draw a plane through a given point A perpendicular to a given line
p_, we first draw through some point O in the axis lines [gamma]',
[gamma]" perpendicular respectively to the projections p1 and p2 of
the given line. These will be the traces of a plane [gamma] which is
perpendicular to the given line. We next draw through the given point
A a plane parallel to the plane [gamma]; this will be the plane
required.
Other metrical properties depend on the determination of the real size
or shape of a figure.
In general the projection of a figure differs both in size and shape
from the figure itself. But figures in a plane parallel to a plane of
projection will be identical with their projections, and will thus be
given in their true dimensions. In other cases there is the problem,
constantly recurring, either to find the true shape and size of a
plane figure when plan and elevation are given, or, conversely, to
find the latter from the known true shape of the figure itself. To do
this, the plane is turned about one of its traces till it is laid down
into that plane of projection to which the trace belongs. This is
technically called rabatting the plane respectively into the plane of
the plan or the elevation. As there is no difference in the treatment
of the two cases, we shall consider only the case of rabatting a plane
[alpha] into the plane of the plan. The plan of the figure is a
parallel (orthographic) projection of the figure itself. The results
of parallel projection (see PROJECTION, SS 17 and 18) may therefore
now be used. The trace [alpha]' will hereby take the place of what
formerly was called the axis of projection. Hence we see that
corresponding points in the plan and in the rabatted plane are joined
by lines which are perpendicular to the trace [alpha]' and that
corresponding lines meet on this trace. We also see that the
correspondence is completely determined if we know for one point or
one line in the plan the corresponding point or line in the rabatted
plane.
Before, however, we treat of this we consider some special cases.
S 13. _To determine the distance between two points A, B given by
their projections A1, B1 and A2, B2, or, in other words, to determine
the true length of a line the plan and elevation of which are given._
[Illustration: FIG. 43.]
_Solution._--The two points A, B in space lie vertically above their
plans A1, B1 (fig. 43) and A1A = A0A2, B1B = B0B2. The four points A,
B, A1, B1 therefore form a plane quadrilateral on the base A1B1 and
having right angles at the base. This plane we rabatt about A1B1 by
drawing A1A and B1B perpendicular to A1B1 and making A1A = A0A2, B1B =
B0B2. Then AB will give the length required.
The construction might have been performed in the elevation by making
A2A = A0A1 and B2B = B0B1 on lines perpendicular to A2B2. Of course AB
must have the same length in both cases.
This figure may be turned into a model. Cut the paper along A1A, AB
and BB1, and fold the piece A1ABB1 over along A1B1 till it stands
upright at right angles to the horizontal plane. The points A, B will
then be in their true position in space relative to [pi]1. Similarly
if B2BAA2 be cut out and turned along A2B2 through a right angle we
shall get AB in its true position relative to the plane [pi]2. Lastly
we fold the whole plane of the paper along the axis x till the plane
[pi]2 is at right angles to [pi]1. In this position the two sets of
points AB will coincide if the drawing has been accurate.
Models of this kind can be made in many cases and their construction
cannot be too highly recommended in order to realize orthographic
projection.
S 14. _To find the angle between two given lines a, b of which the
projections a1, b1 and a2, b2 are given._
[Illustration: FIG. 44.]
_Solution._--Let a1, b1 (fig. 44) meet in P1, a2, b2 in T, then if the
line P1T is not perpendicular to the axis the two lines will not meet.
In this case we draw a line parallel to b to meet the line a. This is
easiest done by drawing first the line P1P2 perpendicular to the axis
to meet a2 in P2, and then drawing through P2 a line c2 parallel to
b2; then b1, c2 will be the projections of a line c which is parallel
to b and meets a in P. The plane [alpha] which these two lines
determine we rabatt to the plan. We determine the traces a' and c' of
the lines a and c; then a'c' is the trace [alpha]' of their plane. On
rabatting the point P comes to a point S on the line P1Q perpendicular
to a'c', so that QS = QP. But QP is the hypotenuse of a triangle PP1Q
with a right angle P1. This we construct by making QR = P0P2; then P1R
= PQ. The lines a'S and c'S will therefore include angles equal to
those made by the given lines. It is to be remembered that two lines
include two angles which are supplementary. Which of these is to be
taken in any special case depends upon the circumstances.
_To determine the angle between a line and a plane_, we draw through
any point in the line a perpendicular to the plane (S 12) and
determine the angle between it and the given line. The complement of
this angle is the required one.
_To determine the angle between two planes_, we draw through any point
two lines perpendicular to the two planes and determine the angle
between the latter as above.
In special cases it is simpler to determine at once the angle between
the two planes by taking a plane section perpendicular to the
intersection of the two planes and rabatt this. This is especially the
case if one of the planes is the horizontal or vertical plane of
projection.
Thus in fig. 45 the angle P1QR is the angle which the plane [alpha]
makes with the horizontal plane.
S 15. We return to the general case of rabatting a plane [alpha] of
which the traces [alpha]' [alpha]" are given.
[Illustration: FIG. 45.]
Here it will be convenient to determine first the position which the
trace [alpha]"--which is a line in [alpha]--assumes when rabatted.
Points in this line coincide with their elevations. Hence it is given
in its true dimension, and we can measure off along it the true
distance between two points in it. If therefore (fig. 45) P is any
point in [alpha]" originally coincident with its elevation P2, and if
O is the point where [alpha]" cuts the axis xy, so that O is also in
[alpha]', then the point P will after rabatting the plane assume such
a position that OP = OP2. At the same time the plan is an orthographic
projection of the plane [alpha]. Hence the line joining P to the plan
P1 will after rabatting be perpendicular to [alpha]'. But P1 is known;
it is the foot of the perpendicular from P2 to the axis xy. We draw
therefore, to find P, from P1 a perpendicular P1Q to [alpha]' and find
on it a point P such that OP = OP2. Then the line OP will be the
position of [alpha]" when rabatted. This line corresponds therefore to
the plan of [alpha]"--that is, to the axis xy, corresponding points on
these lines being those which lie on a perpendicular to [alpha]'.
We have thus one pair of corresponding lines and can now find for any
point B1 in the plan the corresponding point B in the rabatted plane.
We draw a line through B1, say B1P1, cutting [alpha]' in C. To it
corresponds the line CP, and the point where this is cut by the
projecting ray through B1, perpendicular to [alpha]', is the required
point B.
Similarly any figure in the rabatted plane can be found when the plan
is known; but this is usually found in a different manner without any
reference to the general theory of parallel projection. As this method
and the reasoning employed for it have their peculiar advantages, we
give it also.
Supposing the planes [pi]1 and [pi]2 to be in their positions in space
perpendicular to each other, we take a section of the whole figure by
a plane perpendicular to the trace [alpha]' about which we are going
to rabatt the plane [alpha]. Let this section pass through the point Q
in [alpha]'. Its traces will then be the lines QP1 and P1P2 (fig. 9).
These will be at right angles, and will therefore, together with the
section QP2 of the plane [alpha], form a right-angled triangle QP1P2
with the right angle at P1, and having the sides P1Q and P1P2 which
both are given in their true lengths. This triangle we rabatt about
its base P1Q, making P1R = P1P2. The line QR will then give the true
length of the line QP in space. If now the plane [alpha] be turned
about [alpha]' the point P will describe a circle about Q as centre
with radius QP = QR, in a plane perpendicular to the trace [alpha]'.
Hence when the plane [alpha] has been rabatted into the horizontal
plane the point P will lie in the perpendicular P1Q to [alpha]', so
that QP = QR.
If A1 is the plan of a point A in the plane [alpha], and if A1 lies in
QP1, then the point A will lie vertically above A1 in the line QP. On
turning down the triangle QP1P2, the point A will come to A0, the line
A1A0 being perpendicular to QP1. Hence A will be a point in QP such
that QA = QA0.
If B1 is the plan of another point, but such that A1B1 is parallel to
[alpha]', then the corresponding line AB will also be parallel to
[alpha]'. Hence, if through A a line AB be drawn parallel to [alpha]',
and B1B perpendicular to [alpha]', then their intersection gives the
point B. Thus of any point given in plan the real position in the
plane [alpha], when rabatted, can be found by this second method. This
is the one most generally given in books on geometrical drawing. The
first method explained is, however, in most cases preferable as it
gives the draughtsman a greater variety of constructions. It requires
a somewhat greater amount of theoretical knowledge.
If instead of our knowing the plan of a figure the latter is itself
given, then the process of finding the plan is the reverse of the
above and needs little explanation. We give an example.
S 16. _It is required to draw the plan and elevation of a polygon of
which the real shape and position in a given plane [alpha] are known._
We first rabatt the plane [alpha] (fig. 46) as before so that P1 comes
to P, hence OP1 to OP. Let the given polygon in [alpha] be the figure
ABCDE. We project, not the vertices, but the sides. To project the
line AB, we produce it to cut [alpha]' in F and OP in G, and draw GG1
perpendicular to [alpha]'; then G1 corresponds to G, therefore FG1 to
FG. In the same manner we might project all the other sides, at least
those which cut OF and OP in convenient points. It will be best,
however, first to produce all the sides to cut OP and [alpha]' and
then to draw all the projecting rays through A, B, C ... perpendicular
to [alpha]', and in the same direction the lines G, G1, &c. By drawing
FG we get the points A1, B1 on the projecting ray through A and B. We
then join B to the point M where BC produced meets the trace [alpha]'.
This gives C1. So we go on till we have found E1. The line A1 E1 must
then meet AE in [alpha]', and this gives a check. If one of the sides
cuts [alpha]' or OP beyond the drawing paper this method fails, but
then we may easily find the projection of some other line, say of a
diagonal, or directly the projection of a point, by the former
methods. The diagonals may also serve to check the drawing, for two
corresponding diagonals must meet in the trace [alpha]'.
[Illustration: FIG. 46.]
Having got the plan we easily find the elevation. The elevation of G
is above G1 in [alpha]", and that of F is at F2 in the axis. This
gives the elevation F2G2 of FG and in it we get A2B2 in the verticals
through A1 and B1. As a check we have OG = OG2. Similarly the
elevation of the other sides and vertices are found.
S 17. We proceed to give some applications of the above principles to
the representation of solids and of the solution of problems connected
with them.
_Of a pyramid are given its base, the length of the perpendicular from
the vertex to the base, and the point where this perpendicular cuts
the base; it is required first to develop the whole surface of the
pyramid into one plane, and second to determine its section by a plane
which cuts the plane of the base in a given line and makes a given
angle with it._
1. As the planes of projection are not given we can take them as we
like, and we select them in such a manner that the solution becomes as
simple as possible. We take the plane of the base as the horizontal
plane and the vertical plane perpendicular to the plane of the
section. Let then (fig. 47) ABCD be the base of the pyramid, V1 the
plan of the vertex, then the elevations of A, B, C, D will be in the
axis at A2, B2, C2, D2, and the vertex at some point V2 above V1 at a
known distance from the axis. The lines V1A, V1B, &c., will be the
plans and the lines V2A2, V2B2, &c., the elevations of the edges of
the pyramid, of which thus plan and elevation are known.
We develop the surface into the plane of the base by turning each
lateral face about its lower edge into the horizontal plane by the
method used in S 14. If one face has been turned down, say ABV to ABP,
then the point Q to which the vertex of the next face BCV comes can be
got more simply by finding on the line V1Q perpendicular to BC the
point Q such that BQ = BP, for these lines represent the same edge BV
of the pyramid. Next R is found by making CR = CQ, and so on till we
have got the last vertex--in this case S. The fact that AS must equal
AP gives a convenient check.
2. The plane [alpha] whose section we have to determine has its
horizontal trace given perpendicular to the axis, and its vertical
trace makes the given angle with the axis. This determines it. To find
the section of the pyramid by this plane there are two methods
applicable: we find the sections of the plane either with the faces or
with the edges of the pyramid. We use the latter.
As the plane [alpha] is perpendicular to the vertical plane, the trace
[alpha]" contains the projection of every figure in it; the points
E2, F2, G2, H2 where this trace cuts the elevations of the edges will
therefore be the elevations of the points where the edges cut [alpha].
From these we find the plans E1, F1, G1, H1, and by joining them the
plan of the section. If from E1, F1 lines be drawn perpendicular to
AB, these will determine the points E, F on the developed face in
which the plane [alpha] cuts it; hence also the line EF. Similarly on
the other faces. Of course BF must be the same length on BP and on BQ.
If the plane [alpha] be rabatted to the plan, we get the real shape of
the section as shown in the figure in EFGH. This is done easily by
making F0F = OF2, &c. If the figure representing the development of
the pyramid, or better a copy of it, is cut out, and if the lateral
faces be bent along the lines AB, BC, &c., we get a model of the
pyramid with the section marked on its faces. This may be placed on
its plan ABCD and the plane of elevation bent about the axis x. The
pyramid stands then in front of its elevations. If next the plane
[alpha] with a hole cut out representing the true section be bent
along the trace [alpha]' till its edge coincides with [alpha]", the
edges of the hole ought to coincide with the lines EF, FG, &c., on the
faces.
S 18. Polyhedra like the pyramid in S 17 are represented by the
projections of their edges and vertices. But solids bounded by curved
surfaces, or surfaces themselves, cannot be thus represented.
For a surface we may use, as in case of the plane, its traces--that
is, the curves in which it cuts the planes of projection. We may also
project points and curves on the surface. A ray cuts the surface
generally in more than one point; hence it will happen that some of
the rays touch the surface, if two of these points coincide. The
points of contact of these rays will form some curve on the surface,
and this will appear from the centre of projection as the boundary of
the surface or of part of the surface. The outlines of all surfaces of
solids which we see about us are formed by the points at which rays
through our eye touch the surface. The projections of these contours
are therefore best adapted to give an idea of the shape of a surface.
[Illustration: FIG. 47.]
Thus the tangents drawn from any finite centre to a sphere form a
right circular cone, and this will be cut by any plane in a conic. It
is often called the projection of a sphere, but it is better called
the contour-line of the sphere, as it is the boundary of the
projections of all points on the sphere.
If the centre is at infinity the tangent cone becomes a right circular
cylinder touching the sphere along a great circle, and if the
projection is, as in our case, orthographic, then the section of this
cone by a plane of projection will be a circle equal to the great
circle of the sphere. We get such a circle in the plan and another in
the elevation, their centres being plan and elevation of the centre of
the sphere.
Similarly the rays touching a cone of the second order will lie in two
planes which pass through the vertex of the cone, the contour-line of
the projection of the cone consists therefore of two lines meeting in
the projection of the vertex. These may, however, be invisible if no
real tangent rays can be drawn from the centre of projection; and this
happens when the ray projecting the centre of the vertex lies within
the cone. In this case the traces of the cone are of importance. Thus
in representing a cone of revolution with a vertical axis we get in
the plan a circular trace of the surface whose centre is the plan of
the vertex of the cone, and in the elevation the contour, consisting
of a pair of lines intersecting in the elevation of the vertex of the
cone. The circle in the plan and the pair of lines in the elevation do
not determine the surface, for an infinite number of surfaces might be
conceived which pass through the circular trace and touch two planes
through the contour lines in the vertical plane. The surface becomes
only completely defined if we write down to the figure that it shall
represent a cone. The same holds for all surfaces. Even a plane is
fully represented by its traces only under the silent understanding
that the traces are those of a plane.
S 19. Some of the simpler problems connected with the representation
of surfaces are the determination of plane sections and of the curves
of intersection of two such surfaces. The former is constantly used in
nearly all problems concerning surfaces. Its solution depends of
course on the nature of the surface.
To determine the curve of intersection of two surfaces, we take a
plane and determine its section with each of the two surfaces,
rabatting this plane if necessary. This gives two curves which lie in
the same plane and whose intersections will give us points on both
surfaces. It must here be remembered that two curves in space do not
necessarily intersect, hence that the points in which their
projections intersect are not necessarily the projections of points
common to the two curves. This will, however, be the case if the two
curves lie in a common plane. By taking then a number of plane
sections of the surfaces we can get as many points on their curve of
intersection as we like. These planes have, of course, to be selected
in such a way that the sections are curves as simple as the case
permits of, and such that they can be easily and accurately drawn.
Thus when possible the sections should be straight lines or circles.
This not only saves time in drawing but determines all points on the
sections, and therefore also the points where the two curves meet,
with equal accuracy.
S 20. We give a few examples how these sections have to be selected. A
cone is cut by every plane through the vertex in lines, and if it is a
cone of revolution by planes perpendicular to the axis in circles.
A cylinder is cut by every plane parallel to the axis in lines, and if
it is a cylinder of revolution by planes perpendicular to the axis in
circles.
A sphere is cut by every plane in a circle.
Hence in case of two cones situated anywhere in space we take sections
through both vertices. These will cut both cones in lines. Similarly
in case of two cylinders we may take sections parallel to the axis of
both. In case of a sphere and a cone of revolution with vertical axis,
horizontal sections will cut both surfaces in circles whose plans are
circles and whose elevations are lines, whilst vertical sections
through the vertex of the cone cut the latter in lines and the sphere
in circles. To avoid drawing the projections of these circles, which
would in general be ellipses, we rabatt the plane and then draw the
circles in their real shape. And so on in other cases.
Special attention should in all cases be paid to those points in which
the tangents to the projection of the curve of intersection are
parallel or perpendicular to the axis x, or where these projections
touch the contour of one of the surfaces. (O. H.)
IV. ANALYTICAL GEOMETRY
1. In the name _geometry_ there is a lasting record that the science had
its origin in the knowledge that two distances may be compared by
measurement, and in the idea that measurement must be effectual in the
dissociation of different directions as well as in the comparison of
distances in the same direction. The distance from an observer's eye of
an object seen would be specified as soon as it was ascertained that a
rod, straight to the eye and of length taken as known, could be given
the direction of the line of vision, and had to be moved along it a
certain number of times through lengths equal to its own in order to
reach the object from the eye. Moreover, if a field had for two of its
boundaries lines straight to the eye, one running from south to north
and the other from west to east, the position of a point in the field
would be specified if the rod, when directed west, had to be shifted
from the point one observed number of times westward to meet the former
boundary, and also, when directed south, had to be shifted another
observed number of times southward to meet the latter. Comparison by
measurement, the beginning of geometry, involved counting, the basis of
arithmetic; and the science of number was marked out from the first as
of geometrical importance.
But the arithmetic of the ancients was inadequate as a science of
number. Though a length might be recognized as known when measurement
certified that it was so many times a standard length, it was not every
length which could be thus specified in terms of the same standard
length, even by an arithmetic enriched with the notion of fractional
number. The idea of possible incommensurability of lengths was
introduced into Europe by Pythagoras; and the corresponding idea of
irrationality of number was absent from a crude arithmetic, while there
were great practical difficulties in the way of its introduction. Hence
perhaps it arose that, till comparatively modern times, appeal to
arithmetical aid in geometrical reasoning was in all possible ways
restrained. Geometry figured rather as the helper of the more difficult
science of arithmetic.
2. It was reserved for algebra to remove the disabilities of arithmetic,
and to restore the earliest ideas of the land-measurer to the position
of controlling ideas in geometrical investigation. This unified science
of pure number made comparatively little headway in the hands of the
ancients, but began to receive due attention shortly after the revival
of learning. It expresses whole classes of arithmetical facts in single
statements, gives to arithmetical laws the form of equations involving
symbols which may mean any known or sought numbers, and provides
processes which enable us to analyse the information given by an
equation and derive from that equation other equations, which express
laws that are in effect consequences or causes of a law started from,
but differ greatly from it in form. Above all, for present purposes, it
deals not only with integral and fractional number, but with number
regarded as capable of continuous growth, just as distance is capable of
continuous growth. The difficulty of the arithmetical expression of
irrational number, a difficulty considered by the modern school of
analysts to have been at length surmounted (see FUNCTION), is not vital
to it. It can call the ratio of the diagonal of a square to a side, for
instance, or that of the circumference of a circle to a diameter, a
number, and let a or x denote that number, just as properly as it may
allow either letter to denote any rational number which may be greater
or less than the ratio in question by a difference less than any minute
one we choose to assign.
Counting only, and not the counting of objects, is of the essence of
arithmetic, and of algebra. But it is lawful to count objects, and in
particular to count equal lengths by measure. The widened idea is that
even when a or x is an irrational number we may speak of a or x unit
lengths by measure. We may give concrete interpretation to an
algebraical equation by allowing its terms all to mean numbers of times
the same unit length, or the same unit area, or &c. and in any equation
lawfully derived from the first by algebraical processes we may do the
same. Descartes in his _Geometrie_ (1637) was the first to systematize
the application of this principle to the inherent first notions of
geometry; and the methods which he instituted have become the most
potent methods of all in geometrical research. It is hardly too much to
say that, when known facts as to a geometrical figure have once been
expressed in algebraical terms, all strictly consequential facts as to
the figure can be deduced by almost mechanical processes. Some may well
be unexpected consequences; and in obtaining those of which there has
been suggestion beforehand the often bewildering labour of constant
attention to the figure is obviated. These are the methods of what is
now called _analytical_, or sometimes _algebraical_, _geometry_.
3. The modern use of the term "analytical" in geometry has obscured, but
not made obsolete, an earlier use, one as old as Plato. There is nothing
algebraical in this analysis, as distinguished from synthesis, of the
Greeks, and of the expositors of pure geometry. It has reference to an
order of ideas in demonstration, or, more frequently, in discovering
means to effect the geometrical construction of a figure with an
assigned special property. We have to suppose hypothetically that the
construction has been performed, drawing a rough figure which exhibits
it as nearly as is practicable. We then analyse or critically examine
the figure, treated as correct, and ascertain other properties which it
can only possess in association with the one in question. Presently one
of these properties will often be found which is of such a character
that the construction of a figure possessing it is simple. The means of
effecting synthetically a construction such as was desired is thus
brought to light by what Plato called _analysis_. Or again, being asked
to prove a theorem A, we ascertain that it must be true if another
theorem B is, that B must be if C is, and so on, thus eventually finding
that the theorem A is the consequence, through a chain of
intermediaries, of a theorem Z of which the establishment is easy. This
geometrical analysis is not the subject of the present article; but in
the reasoning from form to form of an equation or system of equations,
with the object of basing the algebraical proof of a geometrical fact on
other facts of a more obvious character, the same logic is utilized, and
the name "analytical geometry" is thus in part explained.