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Efficient Subdivision of Bézier Curves/Surfaces via Blossoms

Krassimira Vlachkova

TL;DR

Close-form formulae for blossoms evaluation are proposed, as needed for the calculation of control points of subdivisions, which simplifies considerably the computation of control points of subdivisions which is crucial in applications where curves/surfaces need to be refined or adapted dynamically.

Abstract

We consider the problem of Bézier 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 Bézier curves and both tensor product and triangular Bézier 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.

Efficient Subdivision of Bézier Curves/Surfaces via Blossoms

TL;DR

Close-form formulae for blossoms evaluation are proposed, as needed for the calculation of control points of subdivisions, which simplifies considerably the computation of control points of subdivisions which is crucial in applications where curves/surfaces need to be refined or adapted dynamically.

Abstract

We consider the problem of Bézier 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 Bézier curves and both tensor product and triangular Bézier 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.
Paper Structure (7 sections, 3 theorems, 39 equations, 2 figures, 2 tables)

This paper contains 7 sections, 3 theorems, 39 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Main Results
  3. Subdivision of a Bézier Curve
  4. Subdivision of a Tensor Product Bézier Surface
  5. Subdivision of a Triangular Bézier Surface
  6. Examples and Results
  7. Conclusions

Key Result

Theorem 1

The Bézier control points ${\bf w}_{\nu}$, $\nu=0,\dots ,n$, defined by (ec11) are

Figures (2)

  • Figure 1: Subdivision of TPB surfaces Example \ref{['example1']} using formula (\ref{['eq1']}): (a) The TPB surface $S_1(u,v)$ defined for $0\leq u\leq 1$, $0\leq v\leq 1$. (b) $S_1$ and the TPB surface $S_2(u,v)$ defined for $\frac{1}{3}\leq u\leq \frac{2}{3}$, $\frac{1}{4}\leq v\leq \frac{3}{4}$.
  • Figure 2: Subdivision of TB surfaces from Example \ref{['example2']} using formula (\ref{['eq2']}): (a) The TB surface $S_3$ of total degree 5 defined in the triangle with vertices ${\bf a}=(0,0)$, ${\bf b}=(1,0)$, ${\bf c}=(0,1)$. (b) The TPB surface $S_1$ and the TB surface $S_3$ with its control polygon. (c) The TB surface $S_4$ of total degree 5 defined in the triangle with vertices ${\bf a}=(0,\frac{1}{2})$, ${\bf b}=(\frac{1}{2},0)$, ${\bf c}=(\frac{1}{2},\frac{1}{2})$ and its control polygon. (d) The TB surfaces $S_3$ and $S_4$ with its control polygon.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3