Surfaces

Surfaces
An Introduction

Prof. David Bernstein
James Madison University

Computer Science Department

bernstdh@jmu.edu

Review of our Discussion of Curves

Parametric Polynomial Curves:
- \(\bs{p}(\lambda) = \sum_{k=0}^{k=n} \bs{c}_k \lambda^k \)
Bézier Curves:
- \( \bs{p}(\lambda) = \left[ \begin{array}{l l l l} q_{01} & q_{11} & q_{21} & q_{31} \\ q_{02} & q_{12} & q_{22} & q_{32} \\ q_{03} & q_{13} & q_{23} & q_{33} \\ \end{array} \right] \left[ \begin{array}{r r r r} -1 & 3 & -3 & 1 \\ 3 & -6 & 3 & 0 \\ -3 & 3 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right] \left[ \begin{array}{l} \lambda^3 \\ \lambda^2 \\ \lambda \\ 1 \\ \end{array} \right] \)

Bézier Surfaces/Patches:

The Concept:
- A cubic Bézier surface, \(\bs{s}(u,v)\), can be constructed from 16 points, \(\bs{p}_{0,0}, \ldots, \bs{p}_{3,3}\), by successively calculating cubic Bézier curves in the \(u\) and \(v\) directions
Visualization:

The Points

The "\(x\)" Coordinates:
- \( \bs{P}_1 = \left[ \begin{array}{l l l l} p_{0,0,1} & p_{0,1,1} & p_{0,2,1} & p_{0,3,1} \\ p_{1,0,1} & p_{1,1,1} & p_{1,2,1} & p_{1,3,1} \\ p_{2,0,1} & p_{2,1,1} & p_{2,2,1} & p_{2,3,1} \\ p_{3,0,1} & p_{3,1,1} & p_{3,2,1} & p_{3,3,1} \\ \end{array} \right] \)
The "\(y\)" Coordinates:
- \( \bs{P}_2 = \left[ \begin{array}{l l l l} p_{0,0,2} & p_{0,1,2} & p_{0,2,2} & p_{0,3,2} \\ p_{1,0,2} & p_{1,1,2} & p_{1,2,2} & p_{1,3,2} \\ p_{2,0,2} & p_{2,1,2} & p_{2,2,2} & p_{2,3,2} \\ p_{3,0,2} & p_{3,1,2} & p_{3,2,2} & p_{3,3,2} \\ \end{array} \right] \)
The "\(z\)" Coordinates:
- \( \bs{P}_3 = \left[ \begin{array}{l l l l} p_{0,0,3} & p_{0,1,3} & p_{0,2,3} & p_{0,3,3} \\ p_{1,0,3} & p_{1,1,3} & p_{1,2,3} & p_{1,3,3} \\ p_{2,0,3} & p_{2,1,3} & p_{2,2,3} & p_{2,3,3} \\ p_{3,0,3} & p_{3,1,3} & p_{3,2,3} & p_{3,3,3} \\ \end{array} \right] \)

From the Bernstein Polynomials

The Geometry Matrix:
- \( \bs{G} = \left[ \begin{array}{r r r r} -1 & 3 & -3 & 1 \\ 3 & -6 & 3 & 0 \\ -3 & 3 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right] \)
The Parameter Vectors:
- \( \bs{u} = \left[ \begin{array}{c} u^3 \\ u^2 \\ u \\ 1 \\ \end{array} \right] \;\;\;\;\; \bs{v} = \left[ \begin{array}{c} v^3 \\ v^2 \\ v \\ 1 \\ \end{array} \right] \)

Putting it all Together

\( \bs{s}(u,v) = \left[ \begin{array}{c} \bs{v}^T (\bs{G}^T \bs{P}_1 \bs{G}) \bs{u} \\ \bs{v}^T (\bs{G}^T \bs{P}_2 \bs{G}) \bs{u} \\ \bs{v}^T (\bs{G}^T \bs{P}_3 \bs{G}) \bs{u} \\ \end{array} \right] \)