Cubic Form

A parametric cubic curve segment is P(u) = a + bu + cu^2 + d*u^3, u in [0, 1], where a, b, c, d are 3D vectors. With four vector coefficients we have 12 scalar degrees of freedom - enough to control endpoint position, endpoint tangent, and curvature.

Why Cubic?

• Linear: just a line.
• Quadratic: cannot have an inflection point; cannot represent S-curves naturally.
• Cubic: minimum degree allowing one inflection; smallest derivatives for C2 continuity.
• Higher: oscillates (Runge phenomenon), expensive to evaluate.

Parametric Continuity (C-class)

Two curve segments P1(u) and P2(u) joined at u=1 of P1 and u=0 of P2.

• C0: P1(1) = P2(0) - positions match (no gap).
• C1: also P1'(1) = P2'(0) - tangent vectors equal in magnitude and direction.
• C2: also P1''(1) = P2''(0) - same curvature direction and rate.

Geometric Continuity (G-class)

Looser - direction matches but magnitude can differ.

• G0: same as C0.
• G1: P1'(1) = k * P2'(0) for some k > 0 - tangent direction continuous, magnitudes may differ. Visually smooth, animations may have varying speed.
• G2: matching curvature direction (osculating circle).

Practical Distinction

• For shape, G-continuity is usually enough (the curve looks smooth).
• For animation along a path, you often want C1 or higher so a constant-u sweep produces smooth motion (no velocity discontinuity).

Matrix Form

For any cubic curve family we can write P(u) = U M G, where U = [u^3, u^2, u, 1], M is the 4x4 basis matrix (Hermite, Bezier, B-Spline differ here), and G is the geometry vector of control data (4 points / tangents). This unification is why graphics libraries can render any of these with the same evaluator just by swapping M and G.

Numerical Quick Sanity Check

For Hermite with endpoints P0, P1 and tangents R0, R1: P(0)=P0, P(1)=P1, P'(0)=R0, P'(1)=R1. Verify by plugging u=0,1 into the basis - we will derive this in the next lesson.

Effect on Modeling

Spline editors expose handles to enforce the desired continuity. Pixar's animation pipeline famously prefers G-continuity for character motion paths because animators care about path shape, not parameter speed.