07 - Parametric Curves

Canonical form of a polynomial curve

f(u) = a0 + a1 u + a2 u^2 + a3 u^3 = sum ai u^i. The ai are vectors.

Canonical form in vector form

f(u) = u a, where u = [1 u u^2 u^3] and a = the column of coefficients.

Problem with the canonical form

Good for computing points, but it's hard to pick the ai to get the shape you want.

Blended form

f(u) = sum ci bi(u). ci are control points, bi(u) are blending (basis) functions.

Canonical form as a blended form

The special case bi(u) = u^i.

A point on a blended curve is…

A linear combination (weighted sum) of the control points.

Linear curve through endpoints p0, p1

f(u) = (1 - u) p0 + u p1. b0(u) = 1 - u, b1(u) = u.

Linear curve canonical coefficients

a0 = p0, a1 = p1 - p0.

Do control points have to be endpoints?

No: midpoint + endpoint, endpoint + vector, etc. Different blending functions, same canonical polynomial. You must know what each control point means.

Constraint matrix C

Writes each control point as an equation in the coefficients (substitute its u value or derivative): p = C a.

Blending (basis) matrix B

B = C^-1, so a = B p.

General curve formula

f(u) = u a = u B p. The entries of uB are the blending functions.

Midpoint/endpoint example: constraints

p0 = f(0.5) = a0 + 0.5 a1, p1 = f(1) = a0 + a1.

Midpoint/endpoint example: result

f(u) = (2 - 2u) p0 + (2u - 1) p1. Check u = 0.5 gives p0, u = 1 gives p1.

Why cubics?

High enough degree for smooth curves (C2 at joins), low enough to stay controllable.

Hermite curve control points

Position and derivative at both ends: [f(0), f'(0), f(1), f'(1)].

Derivative of a cubic

f'(u) = a1 + 2 a2 u + 3 a3 u^2

Hermite constraint matrix rows

f(0) = [1 0 0 0], f'(0) = [0 1 0 0], f(1) = [1 1 1 1], f'(1) = [0 1 2 3].

Hermite blending functions

f(u) = (1 - 3u^2 + 2u^3) f(0) + (u - 2u^2 + u^3) f'(0) + (3u^2 - 2u^3) f(1) + (-u^2 + u^3) f'(1).

Local vs global parameter

Each piece uses local u in [0, 1]; the whole curve uses a global parameter 0..n for n pieces.

Knots

Global parameter values where one piece ends and the next begins: 0, 1, 2, …, n. Parameter values, not positions.

Shared point scheme

Last control point of one piece is the first of the next. Assumes pieces defined by end values; too restrictive in general. Gives local control.

Dependency scheme

Evaluate the end of one piece and copy it to the start of the next. No control point needed at the end. Gives global control.

Explicit equation scheme

An explicit equation or constraint connecting the pieces (on control points or curve values).

Local control

Moving a control point changes only a few nearby pieces. Wanted for design.

Global control

Moving one control point changes the whole curve.

Interpolating curve

Passes through all its control points.

Approximating curve

Doesn't pass through all its control points; they just pull the shape.

The 3 out of 4 fact

We want cubic pieces, interpolation, local control, and C2, but no curve has all four. Each curve type picks 3.