Introduction to splines II - Bezier splines, etc.

Bezier splines:

A very popular interpolating spline is the Bezier spline. Unlike the Hermite spline, Bezier splines allow the user to do all manipulation by dragging around points. Here we have our two end points P₁ and P₄, which the curve goes through (which is why it's called an interpolating spline) but also two other "in-between" points P₂ and P₃, that guide the direction of the curve.

As we said in class, the Bezier spline is made from successively nesting linear interpolations.

The coefficients that you get when you do successive nested linear transformations are called the Bernstein polynomials, named for the mathematician who discovered them. Bezier cubic spline functions are simply the third order Bernstein polynomials.

Let's go over the math: We can implement linear interpolation by:

lerp(t, P₁, P₂) = (1-t) P₁ + t P₂

We can define a parabolic (ie: 2nd order) parametric curve, given three points P₁, P₂, P₃, by nested linear interpolations:

lerp(t, lerp(t, P₁, P₂), lerp(t, P₂, P₃)) =
(1-t) ((1-t) P₁ + t P₂) + t ((1-t) P₂ + t P₃)

Multiplying this out gives: (1-t)² P₁ + 2(1-t) t P₂ + t² P₃

Similarly, we can define a cubic (ie: 3rd order) parametric curve given four points P₁, P₂, P₃, P₄, by nested linear interpolations:

lerp(t, lerp(t, lerp(t, P₁, P₂), lerp(t, P₂, P₃)), lerp(t, lerp(t, P₂, P₃), lerp(t, P₃, P₄))) =
(1-t)³ P₁ + 3(1-t)² t P₂ + 3(1-t) t² P₃ + t³ P₄

Conveniently, the above equation contains four polynomials, each of which multiplies by only one of P₁, P₂, P₃ or P₄. These are in fact the third-order Bernstein polynomials. I've colored each polynomial differently so you can track them more easily in the following discussion.

These third-order Bernstein polynomials describe the cubic curves modulated by the geometric coefficients P₁,P₂,P₃,P₄, respectively, in terms of t³,t²,t and 1.

Now we just need to convert from one function basis to another:

We want cubic functions		We have geometric functions
t³		-t³+3t²-3t+1
t²	←	3t³-6t²+3t
t		-3t³+3t²
1		t³

The coefficients of the functions on the right give us the columns of the Bezier matrix:

a		-1	3	-3	1	P₁
b	←	3	-6	3	0	P₂
c	←	-3	3	0	0	P₃
d	←	1	0	0	0	P₄

Homework due Wednesday April 30:

Add smooth movement to your animations, using spline curves.

Optional things you can also try for extra credit:

Use cubic splines to make drawings with smooth curves.
Create more interesting mesh shapes for your z-buffer render by incorporating cubis splines into your mesh creation methods.
Surprise me - try something else interesting that makes use of cubic splines.