Bezier & B-Spline Surfaces

1. Introduction

Surface modeling is essential for representing complex 3D shapes. Bezier and B-Spline surfaces are parametric surfaces widely used for smooth and controllable surface generation.

• Used in CAD and animation
• Based on control points

2. Parametric Surfaces

A parametric surface is defined using two parameters (u, v) and expressed as a vector function.

P(u, v) = (x(u, v), y(u, v), z(u, v))

3. Bezier Surfaces

Bezier surfaces are defined using a rectangular grid of control points and Bernstein basis functions in both u and v directions.

• Control point grid (m × n)
• Surface lies within convex hull

4. Properties of Bezier Surfaces

• Global control (moving a point affects whole surface)
• Smooth and continuous
• Easy to implement

5. B-Spline Surfaces

B-Spline surfaces generalize Bezier surfaces by introducing knot vectors, providing better local control and flexibility.

• Local control of surface
• Defined by degree and knot vectors

6. Knot Vector

A knot vector is a non-decreasing sequence of parameter values that controls how basis functions influence the surface.

• Controls continuity
• Affects surface smoothness

7. Advantages of B-Spline Surfaces

• Local modification
• Better shape control
• Efficient for complex surfaces

8. Bezier vs B-Spline Surfaces

Bezier Surfaces          B-Spline Surfaces
---------------------    -----------------------------
Global control           Local control
Simple structure         More flexible
No knot vector           Uses knot vector
Less scalable            Scalable to complex models

9. Applications

• Automobile and aircraft design
• 3D animation and modeling
• Industrial CAD systems

Practice Questions

1. What is a Bezier surface?
2. Define B-Spline surface.
3. What is a knot vector?
4. Differentiate Bezier and B-Spline surfaces.
5. Where are parametric surfaces used?

Practice Task