Auto-Adaptive Step Straight-Line Algorithm

67 O ne of the main functions of graphic displays is drawing straight lines. To draw lines quickly, the speed of the applied algorithm is critical. The following three different approaches exist: I discrete differential analysis (DDA), introduced by Bresenham, 1 I combinatory analysis, 2 and I linguistic methods. 3,4 The most famous approach remains the DDA, since extended by N-step algorithms. Here, we focus on this class of algorithms. Since three N-step algorithms have been published (N = 2 5 , N = 3 6 , N = 4 7), we analyzed them, studying only their time complexity because they compute the same approximation of the continuous line. Our analysis shows that improvements are small and don't support our objectives for speed. We propose a new algorithm that uses other properties , some of them already presented. 8 We also compare the performances of these algorithms and present the theoretical analysis and benchmarks that prove the new algorithm is at least twice as fast as earlier ones. Before discussing these algorithms in more detail, we introduce additional notations used throughout the article: I " Line " denotes the discrete straight segment between two points P(xp, yp) and Q(xq, yq), and I (u, v) describes the slope of lines u = xq − xp and v = yq − yp. Since the coordinates of P and Q are integers, the segment is an exact translation of line (0, 0)(u, v). A line may be given by a set of points or by its first point and a set of moves. For slope (u, v), the set of moves is always composed using two different codes. In the first octant, the codes are I X, representing an axial move, and I D, representing a diagonal move. N-step algorithms use the DDA method. The main idea is to choose one pattern (a set of points) among different possible patterns. A succession of tests that compare predefined values and a term E realize this choice. Iterating this process determines the set of patterns composing the straight line. A simple operation—number of iterations × number of instructions per loop—gives the time complexity. Let N be the number of points of one pattern. Let C be the number of possible patterns. Inside an iteration , let t be the number of tests, a the number of additions, and i the number of increments of E. …