Efficient Subdivision of B\'{e}zier Curves/Surfaces via Blossoms

TL;DR

This paper introduces closed-form blossom evaluation formulas that enable efficient subdivision of Bézier curves and surfaces, significantly simplifying control point computation for dynamic shape refinement in CAD, animation, and design applications.

Contribution

It provides a novel, direct method for Bézier subdivision using blossoms, improving computational efficiency and simplicity over previous iterative approaches.

Findings

Closed-form formulas for blossom evaluation

Efficient subdivision of Bézier curves and surfaces

Simplified control point computation for dynamic refinement

Abstract

We consider the problem of B\'{e}zier curves/surfaces subdivision using blossoms. We propose closed-form formulae for blossoms evaluation, as needed for the calculation of control points. This approach leads to direct and efficient way to obtain subdivisions for B\'{e}zier curves and both tensor product and triangular B\'{e}zier surfaces. It simplifies considerably the computation of control points of subdivisions which is crucial in applications where curves/surfaces need to be refined or adapted dynamically. For instance, in CAD/CAM systems, architectural design, or animation, the ability to quickly and accurately determine new control points is essential for manipulation and rendering complex shapes. More efficient subdivision can facilitate complex operations like finding intersections between surfaces or smoothly blending multiple surfaces.