## Piecewise Circular Approximation of Spirals
and Polar Polynomials

Francesca
Taponecco, Marc Alexa

Interactive Graphics System Group

Department of Computer Graphics

Technische Universität Darmstadt

Fraunhofer Str. 5, 64283 Darmstadt

Germany

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

Keywords

Spirals, scan conversion, circular approximation

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**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.