SINUSOIDAL SPIRAL

 Curve studied by Maclaurin in 1718. Name given by Haton de la Goupillère in 1857. Other name in the case where n is a positive integer: lemniscate with n poles, or multifocal lemniscate.

 Polar equation of :   with n real. Complex equation: . Polar differential equation: . Curvilinear abscissa given by:  or . Polar tangential angle: . Pedal equation:  . Algebraic curve if and only if n is rational. In the case where n is a positive integer: Multipolar equation: , where  is a regular n-gon with radius a. Complex equation: . In the case where n = - m is a negative integer: Multipolar equation:  where  is a regular m-gon with radius a. Complex equation: . Length: ; area: .

The sinusoidal spirals are defined by their polar equation above.
When n is a positive integer, the sinusoidal spirals are the loci of the points for which the geometric mean of their distances to the vertices of a regular polygon is equal to the radius of this polygon; therefore, they are special cases of Cassinian curves with n poles.

When n is a negative integer, the sinusoidal spirals are the loci of the points M such that the mean of the angles formed by the lines joining the vertices of a regular polygon to M and a fixed direction is constant; therefore, they are special cases of stelloids.

and  are inverses of one another and the pedal of is .
The tangent can be easily constructed thanks to the relation: .
For n > 0, the curve is composed of a base pattern symmetrical about Ox obtained for : and transformed by all the rotations with angle for integral values of k; for n < 0, the base pattern has asymptotes.

When n is rational, we get the whole curve by p - 1 rotations of the base pattern for 1 £ k£ p - 1 where p is the numerator of n.

Examples for positive values of n:
 n = 1: circle n  = 2: lemniscate of Bernoulli n = 3: Kiepert curve n = 4 n = 5 n = 1/2: cardioid n = 3/2 n = 5/2 n = 7/2 n = 9/2 n  = 1/3: Cayley sextic n = 2/3 n = 4/3 n = 5/3 n = 7/3 n = 1/4 n = 3/4 n = 5/4 n = 7/4 n = 9/4 n = 1/5 n = 2/5 n = 3/5 n = 4/5 n = 6/5

Examples for negative values of n:
 n = -1: line x = a n  = -2: rectangular hyperbola n = -3: Humbert cubic n = -4 n = -5 n =- 1/2: parabola y2 = 4a (a - x) n = -3/2 Central negative pedal of the Kiepert curve n = -5/2 n = -7/2 n = -9/2 n  = -1/3: Tschirnhausen cubic n = -2/3 central negative pedal of the rectangular hyperbola n = -4/3 n = -5/3 n = -7/3 n = -1/4 n = -3/4 n = -5/4 n = -7/4 n = -9/4 n = -1/5 n = -2/5 n = -3/5 n = -4/5 n = -6/5

Compare to the roses.

The sinusoidal spiral of index n is the field line of the complex field (cf. the relation ):
 field lines of 1/z: hyperbolas of index -2 field lines of Öz: parabolas of index -1/2 field lines of z3/2 : cardioids of index 1/2 filed lines of z²: circles of index 1 field lines of z3: lemniscates of index 2 field lines of z4