PEDAL AND RECIPROCAL CURVES.

101

Pedal. The locus of the foot of the perpendicular from a fixed point on the tangent to a curve is called the pedal of the curve in regard to that point.

Let two tangents to the curve intersect in p, ct, ct be the perpendiculars on them. Because the angles ctp, ct'p are right angles, a circle on cp as diameter will pass through tt'. Now let the two tangents coalesce into one; then p will become a point on the curve, and t't will become tangent to the

pedal, and also to the circle on cp as diameter. Therefore the angle ctu = cpt, where tu is tangent to the pedal

at t.

Reciprocal. The inverse of the pedal of a curve, in regard to the same point, is called the reciprocal

curve.

Lets be the inverse point to t, and sn the tangent to the locus of s. We know that tu and sn make equal angles with cst; therefore

so that n, p are inverse points. Hence p is a point on the

n

reciprocal of the locus of s, or when one curve is reciprocal to a second, the second curve is reciprocal to the first. Hence the name, reciprocal.

We shall now shew that the reciprocal of a circle is always one of the conic sections. For this purpose it is necessary first to prove a certain property of these

curves.

Two points s and h in the major axis of an ellipse, such that sb = hbca, and consequently that cs2= ch2= ca2 — cb3, are called the foci of the curve. Draw pm perpendicular to the axis from any point p of the curve, and take

=

= (cs — cm)2 + ca2 — cs3- cm3 + cn3=-2cs. cm + ca2+ cn2

=

Similarly hpna. Therefore sp+hp aa', or the sum of the focal distances of any point on the ellipse is equal to the major axis. If we take

The ratio cs ca is called the eccentricity of the ellipse, and sometimes denoted by the letter e, so that spe.pl. The line dl is called the directrix.

Thus we see that the ellipse is the locus of a point whose distance from a fixed point (the focus) is in a constant ratio to its distance from a fixed line (the directrix). The distance from the focus is less than that from the directrix.

A precisely similar demonstration applies to the hyperbola; the points s and h being so taken that

= (cs - cm)2 + cn2 - cm2 - cs2 + ca2 = an

damsn

=

=

as before. So hp a'n, and hp-spaa'. On the other branch we should find sp - hp aa', or the difference of the focal distances of any point on the hyperbola is equal to the major axis. Taking cd: caca : cs, and drawing pl perpendicular to dl, we find as before that

sp plcs: ca.

Thus in the hyperbola also the distance from the focus is in a constant ratio to the distance from the directrix dl, but the ratio in this case is greater than unity.

In the parabola we know that pm* varies as am; take a point s on the axis so that pm2 = 4as. am. Then

if da

sp2 = sm2 + pm2 = sm2 + 4as. am

= as.

τ

d

α

=sm2+4as. sm + 4as2 = dm3,

Hence sp=pl, or the parabola is the locus of a point whose distance from the focus s is equal to its distance from the directrix.

t

We can now prove that the reciprocal of a circle is a conic section, of which the centre of reciprocation is a focus. Let s be the centre of reciprocation, st perpendicular to the tangent qt of the circle. Then the reciprocal curve of the circle is inverse to the locus of t; and the size of the circle of inversion will evidently affect only the size, not the shape, of the curve. Let d be the inverse point to c, then if

n

P

mc

a

p will be a point on the reciprocal curve. Now

୧

sc.sd = = sp. st = sp (sn + cq) = sm . sc+ sp .ca (since sp smsc : sn); or sp.ca: = sc (sd — sm) = md.sc. Therefore sp pl=sc: ca, or the locus of p is a conic section having s for focus, dl for corresponding directrix, and sc ca for eccentricity. Hence if s is within the circle this conic is an ellipse, if on the circumference a parabola, if outside the circle a hyperbola.

Since the reciprocal is the inverse of the pedal, and

W

the inverse of a circle is a circle except when it passes through the centre of inversion, it follows that the pedal of a conic section in regard to a focus is a circle in the case of the ellipse and hyperbola, and a straight line in the case of the parabola. We may prove this independently thus. The tangents to an ellipse or hyper

LAW OF CIRCULAR HODOGRAPH.

=

105

for

bola make equal angles with the focal distances r, since rr, is constant, Fr,; now is the component of velocity of p along sp, and r, along hp, and these being equal in magnitude, it follows that spy=hpz. Produce hp to w, making pw=ps, so that hwaa'. Then sw is perpendicular to py which bisects the angle spw. Hence sy is sw, and sc=sh, therefore cy=hw=ca, or the locus of y is the circle on aa' as diameter. This is called the auxiliary circle.

ACCELERATION INVERSELY AS SQUARE OF DISTANCE.

P

When the acceleration is directed to a fixed point, the hodograph is the reciprocal of the orbit turned through a right angle about the fixed point. Let py be tangent to the orbit, s the fixed point, su the velocity at p, sy perpendicular to py. Then we know that

ST =

su.sy=h,

which is constant. Hence if we mark off sr on sy, so that su, we shall have sr. sy = h, and therefore the locus of r is the reciprocal of the orbit. But the locus of u is the locus of r turned through a right angle.

When the acceleration is inversely as the square of the distance from the fixed point, the hodograph is a circle (Hamilton). Let the acceleration fμ: r2, so that fr2 = μ. We know that r2=h, therefore ƒ: 0=μ: h, or the acceleration is proportional to the angular velocity. Now the acceleration is the velocity in the hodograph, whose direction is that of the radius vector in the orbit; so that the angular velocity, which is the rate at which the radius vector turns round, is also the rate at which the tangent to the hodograph turns round. Since then the velocity in the hodograph is in a constant ratio to the rate at which its tangent turns round, the curvature of the hodograph is constant and equal to h: μ. Therefore the hodograph is a circle of radius μ : h