[Figure 39]
FIG. 39
*141. Introduction of infinite point; center of involution.* We connect
the projective theory of involution with the metrical, as usual, by the
introduction of the elements at infinity. In an involution of points on a
line the point which corresponds to the infinitely distant point is called
the _center_ of the involution. Since corresponding points in the
involution have been shown to be harmonic conjugates with respect to the
double points, the center is midway between the double points when they
exist. To construct the center (Fig. 39) we draw as usual through _A_ and
_A’_ any two rays and cut them by a line parallel to _AA’_ in the points
_K_ and _M_. Join these points to _B_ and _B’_, thus determining on _AK_
and _AN_ the points _L_ and _N_. _LN_ meets _AA’_ in the center _O_ of the
involution.
*142. Fundamental metrical theorem.* From the figure we see that the
triangles _OLB’_ and _PLM_ are similar, _P_ being the intersection of KM
and LN. Also the triangles _KPN_ and _BON_ are similar. We thus have
_OB : PK = ON : PN_
and
_OB’ : PM = OL : PL;_
whence
_OB · OB’ : PK · PM = ON · OL : PN · PL._
In the same way, from the similar triangles _OAL_ and _PKL_, and also
_OA’N_ and _PMN_, we obtain
_OA · OA’ : PK · PM = ON · OL : PN · PL,_
and this, with the preceding, gives at once the fundamental theorem, which
is sometimes taken also as the definition of involution:
_OA · OA’ = OB · OB’ = __constant__,_
or, in words,
_The product of the distances from the center to two corresponding points
in an involution of points is constant._
*143. Existence of double points.* Clearly, according as the constant is
positive or negative the involution will or will not have double points.
The constant is the square root of the distance from the center to the
double points. If _A_ and _A’_ lie both on the same side of the center,
the product _OA · OA’_ is positive; and if they lie on opposite sides, it
is negative. Take the case where they both lie on the same side of the
center, and take also the pair of corresponding points _BB’_. Then, since
_OA · OA’ = OB · OB’_, it cannot happen that _B_ and _B’_ are separated
from each other by _A_ and _A’_. This is evident enough if the points are
on opposite sides of the center. If the pairs are on the same side of the
center, and _B_ lies between _A_ and _A’_, so that _OB_ is greater, say,
than _OA_, but less than _OA’_, then, by the equation _OA · OA’ = OB ·
OB’_, we must have _OB’_ also less than _OA’_ and greater than _OA_. A
similar discussion may be made for the case where _A_ and _A’_ lie on
opposite sides of _O_. The results may be stated as follows, without any
reference to the center:
_Given two pairs of points in an involution of points, if the points of
one pair are separated from each other by the points of the other pair,
then the involution has no double points. If the points of one pair are
not separated from each other by the points of the other pair, then the
involution has two double points._
*144.* An entirely similar criterion decides whether an involution of
rays has or has not double rays, or whether an involution of planes has or
has not double planes.
[Figure 40]
FIG. 40
*145. Construction of an involution by means of circles.* The equation
just derived, _OA · OA’ = OB · OB’_, indicates another simple way in which
points of an involution of points may be constructed. Through _A_ and _A’_
draw any circle, and draw also any circle through _B_ and _B’_ to cut the
first in the two points _G_ and _G’_ (Fig. 40). Then any circle through
_G_ and _G’_ will meet the line in pairs of points in the involution
determined by _AA’_ and _BB’_. For if such a circle meets the line in the
points _CC’_, then, by the theorem in the geometry of the circle which
says that _if any chord is __ drawn through a fixed point within a circle,
the product of its segments is constant in whatever direction the chord is
drawn, and if a secant line be drawn from a fixed point without a circle,
the product of the secant and its external segment is constant in whatever
direction the secant line is drawn_, we have _OC · OC’ = OG · OG’ =_
constant. So that for all such points _OA · OA’ = OB · OB’ = OC · OC’_.
Further, the line _GG’_ meets _AA’_ in the center of the involution. To
find the double points, if they exist, we draw a tangent from _O_ to any
of the circles through _GG’_. Let _T_ be the point of contact. Then lay
off on the line _OA_ a line _OF_ equal to _OT_. Then, since by the above
theorem of elementary geometry _OA · OA’ = OT__2__ = OF__2_, we have one
double point _F_. The other is at an equal distance on the other side of
_O_. This simple and effective method of constructing an involution of
points is often taken as the basis for the theory of involution. In
projective geometry, however, the circle, which is not a figure that
remains unaltered by projection, and is essentially a metrical notion,
ought not to be used to build up the purely projective part of the theory.
*146.* It ought to be mentioned that the theory of analytic geometry
indicates that the circle is a special conic section that happens to pass
through two particular imaginary points on the line at infinity, called
the _circular points_ and usually denoted by _I_ and _J_. The above method
of obtaining a point-row in involution is, then, nothing but a special
case of the general theorem of the last chapter (§ 125), which asserted
that a system of conics through four points will cut any line in the plane
in a point-row in involution.
[Figure 41]
FIG. 41
*147. Pairs in an involution of rays which are at right angles. Circular
involution.* In an involution of rays there is no one ray which may be
distinguished from all the others as the point at infinity is
distinguished from all other points on a line. There is one pair of rays,
however, which does differ from all the others in that for this particular
pair the angle is a right angle. This is most easily shown by using the
construction that employs circles, as indicated above. The centers of all
the circles through _G_ and _G’_ lie on the perpendicular bisector of the
line _GG’_. Let this line meet the line _AA’_ in the point _C_ (Fig. 41),
and draw the circle with center _C_ which goes through _G_ and _G’_. This
circle cuts out two points _M_ and _M’_ in the involution. The rays _GM_
and _GM’_ are clearly at right angles, being inscribed in a semicircle.
If, therefore, the involution of points is projected to _G_, we have found
two corresponding rays which are at right angles to each other. Given now
any involution of rays with center _G_, we may cut across it by a straight
line and proceed to find the two points _M_ and _M’_. Clearly there will
be only one such pair unless the perpendicular bisector of _GG’_ coincides
with the line _AA’_. In this case every ray is at right angles to its
corresponding ray, and the involution is called _circular_.
*148. Axes of conics.* At the close of the last chapter (§ 140) we gave
the theorem: _A conic determines at every point in its plane an involution
of rays, corresponding rays __ being conjugate with respect to the conic.
The double rays, if any exist, are the tangents from the point to the
conic._ In particular, taking the point as the center of the conic, we
find that conjugate diameters form a system of rays in involution, of
which the asymptotes, if there are any, are the double rays. Also,
conjugate diameters are harmonic conjugates with respect to the
asymptotes. By the theorem of the last paragraph, there are two conjugate
diameters which are at right angles to each other. These are called axes.
In the case of the parabola, where the center is at infinity, and on the
curve, there are, properly speaking, no conjugate diameters. While the
line at infinity might be considered as conjugate to all the other
diameters, it is not possible to assign to it any particular direction,
and so it cannot be used for the purpose of defining an axis of a
parabola. There is one diameter, however, which is at right angles to its
conjugate system of chords, and this one is called the _axis_ of the
parabola. The circle also furnishes an exception in that every diameter is
an axis. The involution in this case is circular, every ray being at right
angles to its conjugate ray at the center.
*149. Points at which the involution determined by a conic is circular.*
It is an important problem to discover whether for any conic other than
the circle it is possible to find any point in the plane where the
involution determined as above by the conic is circular. We shall proceed
to the curious problem of proving the existence of such points and of
determining their number and situation. We shall then develop the
important properties of such points.
*150.* It is clear, in the first place, that such a point cannot be on
the outside of the conic, else the involution would have double rays and
such rays would have to be at right angles to themselves. In the second
place, if two such points exist, the line joining them must be a diameter
and, indeed, an axis. For if _F_ and _F’_ were two such points, then,
since the conjugate ray at _F_ to the line _FF’_ must be at right angles
to it, and also since the conjugate ray at _F’_ to the line _FF’_ must be
at right angles to it, the pole of _FF’_ must be at infinity in a
direction at right angles to _FF’_. The line _FF’_ is then a diameter, and
since it is at right angles to its conjugate diameter, it must be an axis.
From this it follows also that the points we are seeking must all lie on
one of the two axes, else we should have a diameter which does not go
through the intersection of all axes—the center of the conic. At least one
axis, therefore, must be free from any such points.
[Figure 42]
FIG. 42
*151.* Let now _P_ be a point on one of the axes (Fig. 42), and draw any
ray through it, such as _q_. As _q_ revolves about _P_, its pole _Q_ moves
along a line at right angles to the axis on which _P_ lies, describing a
point-row _p_ projective to the pencil of rays _q_. The point at infinity
in a direction at right angles to _q_ also describes a point-row
projective to _q_. The line joining corresponding points of these two
point-rows is always a conjugate line to _q_ and at right angles to _q_,
or, as we may call it, a _conjugate normal_ to _q_. These conjugate
normals to _q_, joining as they do corresponding points in two projective
point-rows, form a pencil of rays of the second order. But since the point
at infinity on the point-row _Q_ corresponds to the point at infinity in a
direction at right angles to _q_, these point-rows are in perspective
position and the normal conjugates of all the lines through _P_ meet in a
point. This point lies on the same axis with _P_, as is seen by taking _q_
at right angles to the axis on which _P_ lies. The center of this pencil
may be called _P’_, and thus we have paired the point _P_ with the point
_P’_. By moving the point _P_ along the axis, and by keeping the ray _q_
parallel to a fixed direction, we may see that the point-row _P_ and the
point-row _P’_ are projective. Also the correspondence is double, and by
starting from the point _P’_ we arrive at the point _P_. Therefore the
point-rows _P_ and _P’_ are in involution, and if only the involution has
double points, we shall have found in them the points we are seeking. For
it is clear that the rays through _P_ and the corresponding rays through
_P’_ are conjugate normals; and if _P_ and _P’_ coincide, we shall have a
point where all rays are at right angles to their conjugates. We shall now
show that the involution thus obtained on one of the two axes must have
double points.
[Figure 43]
FIG. 43
*152. Discovery of the foci of the conic.* We know that on one axis no
such points as we are seeking can lie (§ 150). The involution of points
_PP’_ on this axis can therefore have no double points. Nevertheless, let
_PP’_ and _RR’_ be two pairs of corresponding points on this axis (Fig.
43). Then we know that _P_ and _P’_ are separated from each other by _R_
and _R’_ (§ 143). Draw a circle on _PP’_ as a diameter, and one on _RR’_
as a diameter. These must intersect in two points, _F_ and _F’_, and since
the center of the conic is the center of the involution _PP’_, _RR’_, as
is easily seen, it follows that _F_ and _F’_ are on the other axis of the
conic. Moreover, _FR_ and _FR’_ are conjugate normal rays, since _RFR’_ is
inscribed in a semicircle, and the two rays go one through _R_ and the
other through _R’_. The involution of points _PP’_, _RR’_ therefore
projects to the two points _F_ and _F’_ in two pencils of rays in
involution which have for corresponding rays conjugate normals to the
conic. We may, then, say:
_There are two and only two points of the plane where the involution
determined by the conic is circular. These two points lie on one of the
axes, at equal distances from the center, on the inside of the conic.
These points are called the foci of the conic._
*153. The circle and the parabola.* The above discussion applies only to
the central conics, apart from the circle. In the circle the two foci fall
together at the center. In the case of the parabola, that part of the
investigation which proves the existence of two foci on one of the axes
will not hold, as we have but one axis. It is seen, however, that as _P_
moves to infinity, carrying the line _q_ with it, _q_ becomes the line at
infinity, which for the parabola is a tangent line. Its pole _Q_ is thus
at infinity and also the point _P’_, so that _P_ and _P’_ fall together at
infinity, and therefore one focus of the parabola is at infinity. There
must therefore be another, so that
_A parabola has one and only one focus in the finite part of the plane._
[Figure 44]
FIG. 44
*154. Focal properties of conics.* We proceed to develop some theorems
which will exhibit the importance of these points in the theory of the
conic section. Draw a tangent to the conic, and also the normal at the
point of contact _P_. These two lines are clearly conjugate normals. The
two points _T_ and _N_, therefore, where they meet the axis which contains
the foci, are corresponding points in the involution considered above, and
are therefore harmonic conjugates with respect to the foci (Fig. 44); and
if we join them to the point _P_, we shall obtain four harmonic lines. But
two of them are at right angles to each other, and so the others make
equal angles with them (Problem 4, Chapter II). Therefore
_The lines joining a point on the conic to the foci make equal angles with
the tangent._
It follows that rays from a source of light at one focus are reflected by
an ellipse to the other.
*155.* In the case of the parabola, where one of the foci must be
considered to be at infinity in the direction of the diameter, we have
[Figure 45]
FIG. 45
_A diameter makes the same angle with the tangent at its extremity as that
tangent does with the line from its point of contact to the focus (Fig.
45)._
*156.* This last theorem is the basis for the construction of the
parabolic reflector. A ray of light from the focus is reflected from such
a reflector in a direction parallel to the axis of the reflector.
*157. Directrix. Principal axis. Vertex.* The polar of the focus with
respect to the conic is called the _directrix_. The axis which contains
the foci is called the _principal axis_, and the intersection of the axis
with the curve is called the _vertex_ of the curve. The directrix is at
right angles to the principal axis. In a parabola the vertex is equally
distant from the focus and the directrix, these three points and the point
at infinity on the axis being four harmonic points. In the ellipse the
vertex is nearer to the focus than it is to the directrix, for the same
reason, and in the hyperbola it is farther from the focus than it is from
the directrix.
[Figure 46]
FIG. 46
*158. Another definition of a conic.* Let _P_ be any point on the
directrix through which a line is drawn meeting the conic in the points
_A_ and _B_ (Fig. 46). Let the tangents at _A_ and _B_ meet in _T_, and
call the focus _F_. Then _TF_ and _PF_ are conjugate lines, and as they
pass through a focus they must be at right angles to each other. Let _TF_
meet _AB_ in _C_. Then _P_, _A_, _C_, _B_ are four harmonic points.
Project these four points parallel to _TF_ upon the directrix, and we then
get the four harmonic points _P_, _M_, _Q_, _N_. Since, now, _TFP_ is a
right angle, the angles _MFQ_ and _NFQ_ are equal, as well as the angles
_AFC_ and _BFC_. Therefore the triangles _MAF_ and _NFB_ are similar, and
_FA : FM = FB : BN_. Dropping perpendiculars _AA_ and _BB’_ upon the
directrix, this becomes _FA : AA’ = FB : BB’_. We have thus the property
often taken as the definition of a conic:
_The ratio of the distances from a point on the conic to the focus and the
directrix is constant._
[Figure 47]
FIG. 47
*159. Eccentricity.* By taking the point at the vertex of the conic, we
note that this ratio is less than unity for the ellipse, greater than
unity for the hyperbola, and equal to unity for the parabola. This ratio
is called the _eccentricity_.
[Figure 48]
FIG. 48
*160. Sum or difference of focal distances.* The ellipse and the hyperbola
have two foci and two directrices. The eccentricity, of course, is the
same for one focus as for the other, since the curve is symmetrical with
respect to both. If the distances from a point on a conic to the two foci
are _r_ and _r’_, and the distances from the same point to the
corresponding directrices are _d_ and _d’_ (Fig. 47), we have _r : d = r’
: d’_; _(r ± r’) : (d ± d’)_. In the ellipse _(d + d’)_ is constant, being
the distance between the directrices. In the hyperbola this distance is
_(d - d’)_. It follows (Fig. 48) that
_In the ellipse the sum of the focal distances of any point on the curve
is constant, and in the hyperbola the difference between the focal
distances is constant._
PROBLEMS
1. Construct the axis of a parabola, given four tangents.
2. Given two conjugate lines at right angles to each other, and let them
meet the axis which has no foci on it in the points _A_ and _B_. The
circle on _AB_ as diameter will pass through the foci of the conic.
3. Given the axes of a conic in position, and also a tangent with its
point of contact, to construct the foci and determine the length of the
axes.
4. Given the tangent at the vertex of a parabola, and two other tangents,
to find the focus.
5. The locus of the center of a circle touching two given circles is a
conic with the centers of the given circles for its foci.
6. Given the axis of a parabola and a tangent, with its point of contact,
to find the focus.
7. The locus of the center of a circle which touches a given line and a
given circle consists of two parabolas.
8. Let _F_ and _F’_ be the foci of an ellipse, and _P_ any point on it.
Produce _PF_ to _G_, making _PG_ equal to _PF’_. Find the locus of _G_.
9. If the points _G_ of a circle be folded over upon a point _F_, the
creases will all be tangent to a conic. If _F_ is within the circle, the
conic will be an ellipse; if _F_ is without the circle, the conic will be
a hyperbola.
10. If the points _G_ in the last example be taken on a straight line, the
locus is a parabola.
11. Find the foci and the length of the principal axis of the conics in
problems 9 and 10.
12. In problem 10 a correspondence is set up between straight lines and
parabolas. As there is a fourfold infinity of parabolas in the plane, and
only a twofold infinity of straight lines, there must be some restriction
on the parabolas obtained by this method. Find and explain this
restriction.
13. State and explain the similar problem for problem 9.
14. The last four problems are a study of the consequences of the
following transformation: A point _O_ is fixed in the plane. Then to any
point _P_ is made to correspond the line _p_ at right angles to _OP_ and
bisecting it. In this correspondence, what happens to _p_ when _P_ moves
along a straight line? What corresponds to the theorem that two lines have
only one point in common? What to the theorem that the angle sum of a
triangle is two right angles? Etc.