Piecewise Circular Approximation of Spirals and Polar Polynomials

 

Francesca Taponecco, Marc Alexa
Interactive Graphics System Group
Department of Computer Graphics
Technische Universitšt Darmstadt
Fraunhofer Str. 5, 64283 Darmstadt

Germany

{ftapone, alexa}@gris.informatik.tu-darmstadt.de

Keywords
Spirals, scan conversion, circular approximation

ABSTRACT

Spirals are surprisingly common in science, nature, physics, astronomy, flora and fauna, and the arts. Various types of spirals exist and most of them have simple descriptions in polar coordinates. However, in Cartesian coordinates they are typically transcendental functions, which makes the evaluation on Cartesian grids an inefficient process. We propose a construction scheme for piecewise circular approximations. The algorithm consists of generating center coordinates and radii for quarter circles given an arbitrary monotone polynomial, exponential, or logarithmic function in polar coordinates. Evaluating quarter circles as well as generating the parameters can be done incrementally with few integer operations, thus, the algorithm is fast and stable. We show that the approximation converges against the exact form with increasing winding number for any type of spiral.