Parametric Surface

Related Concepts

Algebraic Curve; Parametric Curve; Splines

Definition

A parametric surface is a surface in the Euclidean space ${\mathbb R}^3$ which is defined by a parametric equation with two parameters,

$$\begin{array}{lll} \mathbb{S}(\cdot): {\mathbb R}^2 &\rightarrow & {\mathbb R}^3 \nonumber\\ (u,v) &\mapsto & (x(u,v), y(u,v), z(u,v)), \end{array}$$

(1)

where u, v are the parameters and vary within a certain 2D domain in the parametric uv-plane. x(u,v), y(u,v), z(u,v) are the real-valued functions continuously mapping to the points on a surface.

For example, Bézier surface is one of the most commonly used parametric surfaces (patch) defined as

$$\begin{array}{lll} P(u,v)=\sum_{i=0}^m \sum_{j=0}^n B_i^m(u)B_j^n(v) {\bf p}_{ij}, \end{array}$$

(2)

where p _ij ($\in \mathbb{R}^3$) are the control points of Bézier spline surface. $B_i^n(\cdot):\mathbb{R}\rightarrow \mathbb{R}$ are the basis functions determined by Bernstein polynomials of degree n(see the detail in...