Sinusoidal spiral

A plane curve whose equation in polar coordinates has the form

When is rational, this is an algebraic curve. In particular, when it is a circle, when — an equilateral hyperbola, when — a cardioid, and when — a parabola.

For general the sinusoidal spiral passes through the pole, and is entirely contained within a circle of radius . When is negative, the radius vector of the curve can take arbitrarily large values and the curve does not pass through the pole. The sinusoidal spiral is symmetric with respect to the polar axis, and when is rational (where and are relatively prime numbers), it has axes of symmetry passing through the pole. When is a positive integer, the radius vector of the curve is a periodic function of period . As goes from 0 to , the curve consists of branches, each contained in an angle of . In this case the pole is a multiple point (see Fig.). When is a positive rational number, the curve consists of intersecting branches. When is a negative integer, the curve consists of infinite branches, which can be obtained by inverting the spiral with .

Figure: s085650a

References

[1]	A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)
[2]	J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)

References

[a1]	E.H. Lockwood, "A book of curves" , Cambridge Univ. Press (1967)