Bezier surfaces with prescribed diagonals

TL;DR

The paper addresses the problem of constructing tensor-product Bezier surfaces of degree $n\times n$ with prescribed main diagonal curves, potentially together with prescribed boundary or boundary tangent data. It derives explicit necessary and sufficient linear conditions on the diagonal control points, expressing the diagonals via $Q_k=\frac{D^n_k({\mathcal{P}})}{\binom{2n}{k}}$ and $R_k=\frac{E^n_k({\mathcal{P}})}{\binom{2n}{k}}$, and reveals parity-dependent relations for even and odd $n$ that characterize admissible diagonal pairs; the diagonals-equivalence class of surfaces is shown to be an affine space of dimension $(n-1)^2$. When a prescribed boundary is also imposed, the solution space tightens to an affine subspace of dimension $(n-3)^2$ (boundary+diagonals), and with additional $C^1$-boundary data the dimension reduces to $(n-5)^2$, with the smallest degrees yielding unique surfaces for $n\le 3$ or $n\le 5$ respectively. The results provide a rigorous framework for controlled surface design in geometric modeling and multi-patch CAD, and they point to functional minimization approaches to select canonical surfaces within the resulting affine families.

Abstract

The affine space of all tensor product Bézier patches of degree nxn with prescribed main diagonal curves is determined. First, the pair of Bézier curves which can be diagonals of a Bézier patch is characterized. Besides prescribing the diagonal curves, other related problems are considered, those where boundary curves or tangent planes along boundary curves are also prescribed.

Figures (8)

• Figure 1: The two main diagonals of a tensor product Bézier surface. Notice that both have the same midpoint.
• Figure 2: Control polygon (red) of the diagonal curve of a biquadratic Bézier surface obtained by repeated degree elevation. The corresponding degree elevation control points are also in red.
• Figure 3: If a designer gave two curves that do not intersect (dashed line), one could modify them slightly by means of Corollary \ref{['corollary-two-necessary-conditions']} in order to get two admissible diagonal curves.
• Figure 4: $n=3$. Three Bézier surfaces with the same prescribed main diagonals.
• Figure 5: $n=3$. Prescribed diagonal curves and boundary (properly related).

Theorems & Definitions (24)

• Lemma 1
• proof
• Lemma 2
• Lemma 3
• proof
• Theorem 1
• proof
• Remark 1
• Corollary 1
• Definition 1