Bezier Curves and Surfaces
Bezier curves and surfaces are fundamental tools in computer graphics, animation, and design, named after Pierre Bézier, an engineer at Renault. Here's a detailed look into these mathematical constructs:
History and Development
Pierre Bézier introduced Bézier curves in the 1960s while working on automobile design at Renault. His work was published in 1962, although similar concepts were being developed independently by Paul de Casteljau at Citroën. Bézier's approach was more accessible and became widely adopted in CAD (Computer-Aided Design) systems.
Mathematical Definition
A Bézier curve of degree n is defined by:
- Control Points: A set of points, P0, P1, ..., Pn, where P0 is the start point and Pn is the end point.
- Bernstein Polynomials: The curve is constructed using Bernstein basis polynomials, which ensure that the curve starts at P0, ends at Pn, and lies within the convex hull of the control points.
The equation for a Bézier curve is:
B(t) = Σ[ (n choose i) * (1-t)^(n-i) * t^i * Pi ]
Where t varies from 0 to 1, and (n choose i) is the binomial coefficient.
Properties
- Convex Hull Property: The curve lies entirely within the convex hull of its control points.
- Variation Diminishing Property: The number of intersections of the curve with any line is less than or equal to the number of control points.
- Affine Invariance: The curve retains its shape under affine transformations like rotation, scaling, and translation.
- Endpoint Tangent Property: The tangent to the curve at the endpoints is the line connecting the first two and the last two control points, respectively.
Applications
- Computer Graphics: Used for smooth interpolation between keyframes in animations, font design, and modeling.
- Engineering: In CAD/CAM systems for designing curves and surfaces of vehicle bodies, aircraft wings, and other complex shapes.
- Typography: Bézier curves are used to define the outlines of characters in digital fonts.
Bézier Surfaces
Extending from curves, Bézier surfaces are defined by a grid of control points. They are constructed similarly but use tensor product of Bernstein polynomials for both u and v parameters:
S(u, v) = Σ Σ [Bi,n(u) * Bj,m(v) * Pij]
References
Related Topics