The inverse of a conic with respect to a focus is a curve called
Pascal’s Limaçon. From the polar equation of a conic, the focus
being the pole, it is evident that the polar equation of the limaçon
may be written in the form:
e 1
r = ——— cos_x_ + ——— ;
p p
where e and p are constants, being respectively the eccentricity and
semi-latus rectum of the conic.
From the above equation it is readily seen that the curve may be traced
by drawing from a fixed point O on a circle any number of chords
and laying off a constant length on each of these lines, measured
from the circumference of the circle. The point O is the node of the
limaçon; and the fixed circle, which I shall call the base circle, is
the inverse of the directrix of the conic. This is readily shown as
follows:—the polar equation of the directrix is r = p/(e cos_x_). Hence
the equation of its inverse is r = (e cos_x_)/p, which is the equation of
the base circle of the limaçon.
If the conic which we invert be an ellipse, the point O will be an
acnode on the Limaçon; if the conic be a hyperbola, the point O is a
crunode. If the conic be a parabola, O is then a cusp and the inverse
curve is called the Cardioid.
The limaçon may also be traced as a roulette.
Let the circle A C have a diameter just twice that of the circle A B.
Then a given diameter of A C will always pass through a fixed point Q
on the circle A B, (Williamson’s Diff. Cal. Art. 286) and will have its
middle point on the circle A B. Now any point P on the diameter of A C
will always be at a fixed distance from C and will therefore describe a
limaçon of which A B will be the base circle.
The pedal of a circle with respect to any point is a limaçon. This may
be inferred from the general theorem that the pedal of a curve is the
inverse of its polar reciprocal, (Salmon’s H. P. C. Art. 122). For the
polar reciprocal of a conic from its focus is a circle and hence its
pedal is a limaçon.
The base circle is the locus of the instantaneous centre for all points
on the limaçon. Let B O P be a line cutting a circle in B and Q. Let
the line revolve about B, Q following the circle; the point P will
trace a limaçon.
Now, for any instant, the instantaneous center will be the same whether
Q be following the circle or the tangent at the point where the line
cuts the circle. Therefore the instantaneous center for the point P
is found by erecting a perpendicular to the line P B, through B, and
a normal to the circle at Q. (Williamson’s Diff. Cal. Art. 294). The
intersection (C) of these two lines is the instantaneous center for the
curve at the point P. But by elementary geometry C is on the circle.
Now as the line P B revolves through 360° around B, the line B C which
is always perpendicular to it also makes a complete revolution and the
instantaneous center C moves once round the base circle.
Below we give a list of theorems obtained by inverting the
corresponding theorems respecting a conic. In these theorems any circle
through the pole is called a nodal circle, any chord through the pole
is called a nodal chord, and the line through the pole perpendicular to
the axis of the curve is called the latus rectum. The letters _e_ and
_p_ signify respectively the eccentricity and half the latus rectum of
the inverted conic.
The locus of the point of intersection
of two tangents to a parabola which
cut one another at a constant angle is
a hyperbola having the same focus and
directrix as the original parabola.
The locus of the point of intersection
of two nodal tangent circles to a
cardioid which cut each other at a
constant angle is a limaçon having the
same double point and director circle.
The sum of the reciprocals of two focal
chords of a conic at right angles to
each other is constant.
The sum of any two nodal chords of a
limaçon at right angles to each other
is constant.
P Q is a chord of a conic which
subtends a right angle at the focus.
The locus of the pole of P Q and the
locus enveloped by P Q are each conics
whose latera recta are to that of the
original conic as √2 : 1 and 1 : √2
respectively.
If P and Q be two points on a limaçon
such that they intercept a right
angle at the node, then the locus
of the point of intersection of the
two nodal circles tangent at P and Q
respectively, is a limaçon whose latus
rectum is to that of the original
limaçon as ½√2 : 1. And the envelope
of the circle described on P Q as a
diameter is a limaçon, whose latus
rectum is to that of the original
limaçon as 1 : ½√2.
If two conics have a common focus,
two of their common chords will pass
through the point of intersection of
their directrices.
If two limaçons have a common node,
two nodal circles passing each through
two points of intersection of the
limaçons, will pass through the point
of intersection of their base circles.
Two conics have a common focus about
which one of them is turned; two of
their common chords will touch conics
having the fixed focus for focus.
Two limaçons have a common node about
which one of them is turned; two of
the nodal circles through two of their
points of intersection will envelope
limaçons having fixed node for node.
Two conics are described having the
same focus, and the distance of
this focus from the corresponding
directrix of each is the same; if the
conics touch one another, then twice
the sine of half the angle between
the transverse axes is equal to the
difference of the reciprocals of the
eccentricities.
If two limaçons are described having
the same node and base circles of the
same diameter, and if the limaçons
touch each other, then twice the sine
of half the angle between the axes of
the limaçons is equal to the difference
of the eccentricities.
If a circle of a given radius pass
through the focus (S) of a given conic
and cut the conic in the points A,
B, C, and D; then SA. SB. SC. SD
is constant.
If a circle of a given radius pass
through the node (S) of a given limaçon
and cut it in A, B, C, and D; then
1
——————————————— is constant.
(SA. SB. SC. SD)
A circle passes through the focus of
a conic whose latus rectum is 2l and
meets the conic in four points whose
distance from the focus are r₁, r₂,
r₃, r₄, then
1 1 1 1 2
——— + ——— + ——— + ——— = ——— .
r₁ r₂ r₃ r₄ l
A circle passes through the node of
a limaçon whose latus rectum is 2l,
meeting the curve in four points whose
distances from the node are r₁, r₂, r₃,
r₄, then
r₁ + r₂ + r₃ + r₄ = 2l.
Two points P and Q are taken, one on
each of two conics which have a common
focus and their axes in the same
direction, such that PS and QS are at
right angles, S being the common focus.
Then the tangents at P and Q meet on a
conic the square of whose eccentricity
is equal to the sum of the squares of
the eccentricities of the original
conics.
Two points P and Q are taken one on
each of two limaçons which have a
common node and their axes in the same
direction, such that PS and QS are at
right angles, S being the common node.
Then the nodal tangent circles at P and
Q intersect on a limaçon the square of
whose eccentricity is equal to the sum
of the squares of the eccentricities of
the original limaçons.
A series of conics are described with
a common latus rectum; the locus
of points upon them at which the
perpendicular from the focus on the
tangent is equal to the semi-latus
rectum is given by the equation
p = -r cos 2_x_
If a series of limaçons are described
with the same latus rectum, the locus
of points upon them at which the
diameter of the nodal tangent circle
is equal to the semi-latus rectum, is
given by the equation
pr = -cos 2_x_
If POP₁ be a chord of a conic through a
fixed point O, then will tan ½P₁SO tan ½PSO
be a constant, S being the focus of the conic.
If POP₁ be a nodal circle of a limaçon
passing through a fixed point O, then
will tan ½ P₁SO tan ½ PSO be a constant,
S being the node.
Conics are described with equal latera
recta and a common focus. Also the
corresponding directrices envelop
a fixed confocal conic. Then these
conics all touch two fixed conics, the
reciprocals of whose latera recta are
the sum and difference respectively of
those of the variable conic and their
fixed confocal, and which have the same
directrix as the fixed confocal.
Limaçons are described with equal
latera recta and a common node. Also
the director circles envelop a fixed
limaçon having a common node. Then
these limaçons all touch two fixed
limaçons whose latera recta are the
sum and difference respectively of the
reciprocals of the variable limaçon
and of the fixed limaçon, and which
have the same base circle as the fixed
limaçon.
Every focal chord of a conic is cut
harmonically by the curve, the focus,
and the directrix.
Every nodal chord of a limaçon is
bisected by the base circle.
The envelope of circles on the focal
radii of a conic as diameters is the
auxiliary circle.
The envelope of the perpendiculars at
the extremities of the nodal radii of
a limaçon is a circle having for the
diameter the axis of the limaçon.
Below we give a number of theorems respecting the cardioid obtained by
inverting the corresponding theorems concerning the parabola.
The straight line which bisects the
angle contained by two lines drawn
from the same point in a parabola,
the one to the focus, the other
perpendicular to the directrix, is a
tangent to the parabola at that point.
The nodal circle which bisects the
angle between the line drawn from any
point on a cardioid to the cusp and the
nodal circle through the point which
cuts the director circle orthogonally,
is a tangent circle at that point.
The latus rectum of a parabola is equal
to four times the distance from the
focus to the vertex.
The latus rectum of a cardioid is equal
to its length on the axis.
If a tangent to a parabola cut the axis
produced, the points of contact and of
intersection are equally distant from
the focus.
If a nodal tangent circle cut the
axis of a cardioid, the points of
intersection and of tangency are
equally distant from the cusp.
If a perpendicular be drawn from the
focus to any tangent to a parabola, the
point of intersection will be on the
vertical tangent.
If a nodal circle be drawn tangent to a
cardioid, the diameter of such circle
passing through the cusp will be a
common chord of this circle and another
described on the axis of the cardioid
as diameter.
The directrix of a parabola is the
locus of the intersection of tangents
that cut at right angles.
The base circle is the locus of the
intersection of nodal circles tangent
to a cardioid, which cut orthogonally.
The circle described on any focal chord
of a parabola as diameter will touch
the directrix.
The circle described an any nodal chord
of a cardioid as diameter will be
tangent to the base circle.
The locus of a point from which two
normals to a parabola can be drawn
making complementary angles with the
axis, is a parabola.
The locus of the point through which
two nodal circles, cutting a cardioid
orthogonally, and making complementary
angles with the axis, can be drawn is a
cardioid.
Two tangents to a parabola which
make equal angles with the axis and
directrix respectively, but are not at
right angles, meet on the latus rectum.
Two nodal circles tangent to a cardioid
which make equal angles with the axis
and latus rectum, respectively but do
not cut orthogonally intersect on the
latus rectum.
The circle which circumscribes the
triangle formed by three tangents to a
parabola passes through the focus.
If three nodal circles be drawn tangent
to a cardioid, the three points of
intersection of these three circles are
on a straight line.
If the two normals drawn to a parabola
from a point P make equal angles with
a straight line, the focus of P is a
parabola.
If the two nodal circles cutting a
cardioid orthogonally and pass through
the point P, make equal angles with a
fixed nodal circle, the locus of P is a
cardioid.
Any two parabolas which have a common
focus and their axes in opposite
directions intersect at right angles.
Any two cardioids which have a common
cusp and their axes in opposite
directions intersect at right angles.
A number of other theorems on the limaçon and cardioid are given in
Professor Newson’s article in this number of the QUARTERLY,
and these need not be repeated here.
Dialect Word-List.