Fast exact digital differential analyzer for circle generation

Abstract

In the first part of the paper we present a short review of applications of digital differential analyzers (DDA) to generation of circles showing that they can be treated as one-step numerical schemes. In the second part we present and discuss a novel fast algorithm based on a two-step numerical scheme (explicit midpoint rule). Although our algorithm is as cheap as the simplest one-step DDA algoritm (and can be represented in terms of shifts and additions), it generates circles with maximal accuracy, i.e., it is exact up to round-off errors.

A Comparison of Standard One-Step DDA Circular Interpolators with a New Cheap Two-Step Algorithm

“…This algorithm was first proposed in [9]; see also [10]. Table 3 contains maximum values of absolute radial systematic error of proposed algorithm, given by (26), after full circle simulation in the case …”

“…This is an exact algorithm which can be derived as follows (for another approach, see [10]). We consider the rotation from the point , to +2 , +2 ; see Figure 2.…”

“…From these results we see that algorithm (11) is the most accurate. Therefore, in next sections we choose algorithm (11) in order to compare it with two new circular interpolators introduced in [10].…”

“…In paper [9] the idea of applying a Nystrom two-step scheme to circle generation appeared for the first time. Our next paper [10] developed theoretical and geometric aspects of this method. In the present paper we focus on practical and experimental issues.…”

“…In the present paper we focus on practical and experimental issues. Moreover, here we consider circles of arbitrary radius , while in [9,10] we assumed = 1.…”