清华大学：《计算机图形学基础》课程教学资源（授课教案）参数曲线曲面、Bezier曲线、Bezier曲面

Bézier Curves and Surfaces • Parametric curves and surface: Some Concepts • Bézier Cuvres: Concept and properties • Bézier surfaces: Rectangular and Triangular • Conversion of Rectangular and Triangular Bézier surfaces

Parametric curves and surface • In Graphics, we usually design a scene and then generate realistic image by using rendering equation. • It’s necessary to introduce geometric modeling for scene design. • How to represent 3D shapes (models) in computer?

History of Geometric Modeling • Surface Modeling : – In 1962, Pierre Bézier, an engineer of French Renault Car company, propose a new kind of curve representation, and finally developed a system UNISURF for car surface design in 1972

• Pierre Etienne Bezier was born on September 1, 1910 in Paris. Son and grandson of engineers • He entered the Ecole Superieure d'Electricite and earnt a second degree in electrical engineering in 1931. In 1977, 46 years later, he received his DSc degree in mathematics from the University of Paris. • In 1933, aged 23, Bezier entered Renault and worked for this company for 42 years. • Bezier's academic career began in 1968 when he became Professor of Production Engineering at the Conservatoire National des Arts et Metiers. He held this position until 1979

• Solid Modeling : – In1973, Ian Braid of Cambridge University developed a solid modeling system for designing engineering parts. – Ian Braid presented in his dissertation “Designing with volumes”, this work being demonstrated with the BUILD-1 system

How to represent a curve? • There are three major types of object representation: – Explicit representation: the explicit form of a curve in 2D gives the value of one variable, the dependent variable, in terms of the other, the independent variable. In x, y space, we may write – For the line, we usually write y f x  ( ) y mx h  

– Implicit representation: In two dimensions, an implicit curve can be represented by the equation – For the line – For the circle 2 2 2 x y r    0 f x y ( , ) 0  ax by c    0

– Parametric form: The parametric form of a curve expresses the value of each spatial variable for points on the curve in terms of an independent variable , the parameter. – In 3D, we have three explicit functions t ( ) ( ) ( ) x x t y y t z z t   

– One of the advantages of the parametric form is that it is the same in two and three dimensions. In the former case, we simply drop the equation for z. – A useful representation of the parametric form is to visualize the locus of points ( ) being drawn as t varies. ( ) [ ( ), ( ), ( )]T P t x t y t z t 