Spline Representation

1. Introduction

In computer graphics, smooth curves are essential for modeling realistic objects. Spline representation provides flexibility and precision in defining such curves.

• Used for smooth curve generation
• Based on control points

2. What is a Spline?

A spline is a piecewise polynomial curve that passes through or near a set of control points, ensuring smoothness at the joins.

• Piecewise polynomial
• Smooth and continuous

3. Control Points

Control points define the shape of a spline curve. Moving a control point affects only a portion of the curve.

• Local control property
• Easy curve modification

4. Parametric Representation

Spline curves are represented parametrically using a parameter t that varies over a specific interval.

P(t) = (x(t), y(t), z(t)),   0 ≤ t ≤ 1

5. Properties of Splines

• Continuity (C0, C1, C2)
• Local control
• Smooth transitions

6. Types of Spline Curves

• Interpolating splines
• Approximating splines
• Parametric splines

7. Advantages of Spline Representation

• Smooth curve modeling
• Efficient data representation
• Flexible shape control

8. Disadvantages

• Complex mathematical formulation
• Computational overhead

9. Applications

• Curve and surface modeling
• Computer animation
• CAD/CAM systems

Practice Questions

1. What is a spline?
2. Explain control points in spline curves.
3. What are properties of spline curves?
4. List types of spline curves.
5. Where are splines used?

Practice Task