Affine Invariance of Bézier Curves on Digital Grid

Affine invariance is one of the most fundamental properties of free-form curves, ensuring that transformations such as translation, scaling, rotation, and shearing preserve the essential characteristics of the geometric shape. It is exploited by almost every software that uses such curves. However, this property only holds in a theoretical, mathematical sense. The transformation of a curve calculated and displayed on computers using finite precision arithmetic and representation may not be fully identical to the curve calculated from the transformed control points. This deviation, even pixel-level inaccuracy, can cause problems in various applications, such as Computer-Aided Geometric Design, medical image processing, numerical computations, and font design, where this level of error can have serious consequences. In this paper, we study and demonstrate the extent and nature of this deviation using geometric and statistical tools on a cubic Bézier curve. We provide practical methods to mitigate this inaccuracy and decrease the error level using fast and simple alternative computations of the curve, taking advantage of the symmetry of the basis functions, elevating the degree of the curve, and using reparametrization to evaluate the curve on integer values. The effectiveness of these alternatives is evaluated by statistical methods based on 500,000 transformations.