Defining Data

A Hermite cubic segment is defined by:

• Two endpoints P0 and P1.
• Two tangent vectors R0 at P0 and R1 at P1.

This is intuitive: "where it starts, where it ends, how it leaves, how it arrives."

Hermite Basis Functions

P(u) = h0(u) P0 + h1(u) P1 + h2(u) R0 + h3(u) R1, where:

• h0(u) = 2u^3 - 3u^2 + 1
• h1(u) = -2u^3 + 3u^2
• h2(u) = u^3 - 2u^2 + u
• h3(u) = u^3 - u^2

Check:

• h0(0)=1, h0(1)=0; h1(0)=0, h1(1)=1 -> endpoints match.
• h2'(0)=1; others give 0 at u=0,1 -> tangents match.

Matrix Form

P(u) = U M_h G_h, U = [u^3 u^2 u 1], M_h = [ 2 -2 1 1 ] [ -3 3 -2 -1 ] [ 0 0 1 0 ] [ 1 0 0 0 ], G_h = [P0; P1; R0; R1].

Joining Hermites

Two segments S1 and S2:

• For C1 at the join: end tangent of S1 = start tangent of S2.
• For C2: also second derivatives match. Cubic has only 4 DOF per axis -> imposing C2 leaves only 1 free DOF per join.

Pros

• Geometrically intuitive: position + velocity at each end.
• Common in animation tweening (key positions and velocities).

Cons

• Tangent magnitudes affect shape strongly; small numerical changes flip the curve.
• Tangents are not visually displayed in CAD tools, so designers prefer Bezier.

Worked Mini-Example

P0 = (0,0), P1 = (4,0), R0 = (0,4), R1 = (0,-4). At u=0.5:

• h0(0.5) = 20.125 - 30.25 + 1 = 0.25 - 0.75 + 1 = 0.5
• h1(0.5) = -20.125 + 30.25 = -0.25 + 0.75 = 0.5
• h2(0.5) = 0.125 - 0.5 + 0.5 = 0.125
• h3(0.5) = 0.125 - 0.25 = -0.125

P(0.5) = 0.5(0,0) + 0.5(4,0) + 0.125(0,4) + (-0.125)(0,-4) = (2, 0.5 + 0.5) = (2, 1).

Conversion

Hermite <-> Bezier is a linear remap: a Bezier with control points (P0, P0+R0/3, P1-R1/3, P1) traces the same Hermite segment.