Discrete circles, rings and spheres

Abstract

This paper presents a new approach to discrete circles, rings, and an immediate extension to spheres. The circle, called arithmetical circle is defined by diophantine equations. The integer radius circles with the same centre pave the plane. It is easy to determine if a point is on, inside, or outside a circle. This was not easy to do with previous definitions of circles, like Bresenham's. We show that the arithmetical circle extends Bresenham's circle. We give an efficient incremental generation algorithm. The arithmetical circle has many extensions. We present briefly half-integer centered circles with a generation algorithm, 4-connected circles, and a general ring definition. We finish with the arithmetical sphere, an immediate 3D extension of arithmetical circle. We give elements to build a algorithm for generating the sphere.