CURVES REFERRED TO POLAR COORDINATES.

60. It is frequently advantageous to refer curves to polar coordinates instead of rectangular ones, especially in the investigations of physical astronomy, and we shall now advert to the method of drawing tangents to spiral and other curves by means of their polar equations.

x = — r cos 0, y = r sin 0, x2 + y2 = p2;

and differentiating these, we get

dx

=

-cos Odr+r sine de; dy = sine dr+r cos e do ; x dx+y dy = r dr.

Now tan TPO tan (POM-PT'M) =

VOL. I.

tan POM tan PT' M 1+tan P O M tan PT' M 2 B

Draw the normal PR perpendicular to the tangent, PT meeting TO produced in R; then we have

If OY be drawn perpendicular to the tangent PT; then if O Y = p, we shall have, by the similar triangles POT and TOY,

OY=OT cos TOY OT cos TPO

=

от
sec TPO'

1. If the straight line OA revolve uniformly round O as a centre, the point P which moves uniformly from O along O A will trace out a curve called the spiral of Archimedes.

Let the radius vector OP = r, and angle A OP the value of r when the revolving line has made a complete revolution; then a'r :: 2 : 0; which gives 2 r = a' 0; but if we put a' 2 a, then the equation of the curve will be simply r = a 0.

=

To draw a tangent to this spiral, we have from the equation to the curve,

2. Find the polar equation to the ellipse, the focus being the pole, and draw a tangent to the curve.

Let FP=r, angle A FP=0, and O F

=

√ (a2-b*) = a

2 ae b2r cos 0+ a2 e2 r2 cos 20 = (b2 — aer cos 0)2;

b2 - aer cos 0, and r =

This equation is of the planetary

polar equation to the ellipse, the focus being the pole.
much used in physical astronomy, since the orbits
bodies are ellipses, and the sun is situated in one of their foci.
Differentiating this equation, we get

p3:

==

(1 + e cos 0)2

=

(e sin 20+ 1+2 e cos 0 + e2 cos 20)

r2 d 02 { 2 (1 + e cos 0) − (1 − e3) } ̧
(1 + e cos 0)2

Now substituting for 1+ e cos 0 its value

a (1 − e2)

we get

In a similar manner, the square of the perpendicular from the other focus upon the tangent is found to be (2 a - being the radius vector)

the product of the perpendiculars from the foci to the tangent at any point of an ellipse is equal to the square of the semi-minor axis.

3. Draw a tangent to the logarithmic or equiangular spiral, its equation being r =

4. Find the polar subtangent in the curve whose equation is r=

5. Find the polar subtangent in the hyperbolic spiral whose equation

is r =

a

Ans. Subtangent = a.

ASYMPTOTES TO POLAR CURVES.

61. Since an asymptote is a tangent to a curve, at a point infinitely distant, which passes at a finite distance from the origin, we have only

to assume r = ∞, and find the value of 0 from the equation to the curve

=

rf0; then if the polar subtangent r2

d Ꮎ dr

be finite, there will be an

asymptote corresponding to the value assigned to 0.

a

1. To find whether the curver = has an asymptote. Assume

0

62. Let APB be a curve, and let the arc AP OM = x, ON = x', MP = y, NQ = y', and the chord PQ = c; then the arc PQ=s's, and by common algebra, we have

(1).

X -x

H

Now when Q approaches P, the limiting values of

'y' - y`
x' -x

N

dy

; hence dx

or ds

= d x2 + dy2

Hence the square of the differential of an arc is equal to the sum of the squares of the differentials of its coordinates.

63. If x = r cos 0 and y = r sin ; then will de r sin o de, dy

dx2 + dy2 = dr2 + r2 do2 ·.... (3),

which is the formula for the differential of an arc of a curve referred to polar coordinates.

THE DIRECTION OF CURVATURE OF A CURVE.

64. From the theory of tangents to curves, the direction of their curvature may be easily determined. For since (50)

d2 y
dx

de

dx

is always the same as that of and if be positive, O increases when

x increases, and therefore the concavity of the curve must be turned

d Ꮎ dx

upwards as in Art. 50 (fig. 2); but if be negative, diminishes when x increases, and therefore the concavity is turned downwards, as in (fig. 1); consequently the curve will be convex towards the axis of x if the second differential coefficient of y be positive, and concave if it be negative. We have supposed that y is positive in the preceding remarks, and if y be negative, the conclusion in respect of the signs will be reversed. The following enunciation of the principle will apply in all cases :

A curve at any point (x y) is convex to the axis of x, if y and dy have the same sign, but concave if the signs be different.

RADIUS OF CURVATURE.

=

65. If a circle touch a straight line A B in any point P, it is evident that in the immediate vicinity of P the circle will tend to coincide with A B, as the radius OP p, of the circle increases; therefore the greater the radius is, the curvature of the circle is the less, and the less the radius is, the greater is the curvature; consequently the reciprocal of the radius.

A

B

QS be normals, and PT, QT' tangents at these points. Let the arc

=

s, the arc A Q = s', angle PTX

=

, angle QT' X = ', and

=

AP
the chord of the arc PQ = c; then arc PQ s'
TCT' = 4 — '. Now in the triangle P Q S we have
sin S sin SQ P