Shape Analysis of Cubic Bézier Curves – Correspondence to Four Primitive Cubics

Abstract

In this paper, we show that any planar polynomial cubic Bezier curve can be described as an affine transformation of a part of four primitive cubics (x3+x2−3y2=0, x3−3y2=0, x3−x2−3y2=0, and x3−y=0), and propose an algorithm to derive the transformation matrix.For a given cubic Bezier curve, we derive the linear moving line that follows the curve, and find the double point D of the curve. If D is not a point at infinity, we derive the quadratic parallel moving line that follows the curve. By testing whether the quadratic moving line crosses D or not, we classify the curve into three cases (crunode, cusp, or acnode) which correspond to three primitive cubics. If D is a point at infinity, the curve is classified as fourth case (explicit cubic), which requires exceptional process. For each case, the affine transformation matrix between the primitive cubic and given Bezier curve can be derived. We confirm that the proposed algorithm never fails unless the cubic Bezier curve is degree reducible or consists of f...