Splines: Piecewise Polynomials
A springy wire minimizes its bending energy (subject to constraints)
Bending energy approximated by the integral of squared curvature
- minimize this and the curve looks real
- 2nd derivative approximates curvature
Intuitively: try to make curvature zero everywhere
- If you can't, distribute bends as uniformly as possible
A spline is a piecewise polynomial - many low degree polynomials are used to interpolate (pass through) the control points
Cubic polynomials are the most common:
- minimize the 2nd derivative
- lowest order polynomials that interpolate two points and allow the gradient at each point to be defined - C1 continuity is possible
- Higher or lower degrees are possible, of course