STURM SPIRAL

 Curve studied by Sturm in 1857 and Masurel in 2014. Other name, in this article: Mannheim curve.

The Sturm spirals are the curves such that the radius of curvature is, at all points, proportional to the distance to a fixed point: .
The special case e = 1 is studied on the page dedicated to the Norwich spiral.

 Differential equation:  (p is the pedal radius). First integral: , hence the polar equation: .

 If a = 0, then , which is none other than a logarithmic spiral, with the limit case of the circle for e = 1 (no solution when e < 1). Elliptic case, e<1  Cartesian parametrization:  where . Complex parametrization  (it is therefore a tritrochoid). Curvilinear abscissa: . Cartesian tangential angle: . Radius of curvature:  When q is rational, the order of the rotation symmetry is equal to the denominator of q minus 1. Hyperbolic case, e >1  Cartesian parametrization:  where . Curvilinear abscissa: .  Cartesian tangential angle: . Radius of curvature: .

Remarkable properties (in the case e = 1 as well):
- the roulette with linear base of a Sturm spiral is a Duporcq curve with equal parameter e (hence the used of this letter e, associated to the eccentricity of a conic).
- the evolute of the Sturm spiral in the elliptic case is an epicycloid
- the evolute of the Sturm spiral in the hyperbolic case is a para- or hypercycloid.

 Consider now the curves such that the curvature is proportional to the distance to a fixed point; the differential equation gives the first integral , hence the polar equation . The case c = 0, gives , which is none other than a lemniscate of Bernoulli. The curvature of a lemniscate is proportional to the distance to its centre. Finally, one of the solution to is the cardioid.

Compare to the elastic curve, a curve such that the curvature is proportional to the distance to a fixed line, and a certain kind of radioid.

© Robert FERRÉOL  2017