A Geometric structure for two-dimensional shapes and three-dimensional surfaces

Abstract

A geometric structure called slope change notation (SCN), which describes two-dimensional (2D) shapes and three-dimensional (3D) surfaces in a discrete representation, is presented. The SCN of a curve is obtained by placing constant-length straight-line segments around the curve (the endpoints of the straight-line segments always touching the curve), and calculating the slope changes between contiguous segments scaled to a continuous range from −1 to 1. The SCN is independent of translation and rotation (due to the fact that slope changes around the curve are used), and optionally, of size. The SCN for 2D shapes is 1D. This is an important characteristic, because shapes with particular characteristics are easily generated by numerical sequences; also, it is possible to perform arithmetic operations among shapes and surfaces. The SCN differs from other chain codes, for instance, Freeman chains ( Proc. Natn. Electron. Conf. 18, 312–324 (1961)), since the proposed notation does not use a grid (and so depends only on itself); its range of slope changes varies continuously from −1 to 1; its vertices always touch the curve, which produces a better description of the shape; and its discrete elements always have the same length. Using this geometric structure only slope changes are variable; the segments size of any shape is always constant. At the end of the paper a related theory “B”, that allows variable segment size as a function of slope changes, is introduced. These ideas are based on previous work ( Pattern Recognition 13, 123–137 (1981)) and the solutions to many problems which arose are presented in this paper.