A Geometric Method for Approximating Convex Arcs

Approximating a curve by a sequence of points along it is a problem of importance in such traditional fields as surveying and in such modern ones as computer graphics. The authors examine a simple geometric method for choosing points $p_1 , \cdots ,p_n $ along a planar convex curve; the point $p_{i + 1} $ is chosen so that the line segment $I(p_{i - 1,} p_{i + 1} )$ is parallel to a support line of the curve at $p_i $. This method tends to produce many points where the curvature is large and few points where the curvature is small, and thus it permits an accurate representation of the curve from limited information. The points of the approximating sequence can be generated in a mechanical manner from two succeeding terms, and mild regularity conditions on the curve insure that the sequence will be finite.