next curve  previous curve  2D curves  3D curves  surfaces  fractals  polyhedra 
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.
next curve  previous curve  2D curves  3D curves  surfaces  fractals  polyhedra 
© Robert FERRÉOL
2017