Kochanek–Bartels spline

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Kochanek bartels spline.svg

In mathematics, a Kochanek–Bartels spline or Kochanek–Bartels curve is a cubic Hermite spline with tension, bias, and continuity parameters defined to change the behavior of the tangents.

Given n + 1 knots,

p0, ..., pn,

to be interpolated with n cubic Hermite curve segments, for each curve we have a starting point pi and an ending point pi+1 with starting tangent di and ending tangent di+1 defined by

\mathbf{d}_i = \frac{(1-t)(1+b)(1+c)}{2}(\mathbf{p}_i-\mathbf{p}_{i-1}) + \frac{(1-t)(1-b)(1-c)}{2}(\mathbf{p}_{i+1}-\mathbf{p}_i)
\mathbf{d}_{i+1} = \frac{(1-t)(1+b)(1-c)}{2}(\mathbf{p}_{i+1}-\mathbf{p}_{i}) + \frac{(1-t)(1-b)(1+c)}{2}(\mathbf{p}_{i+2}-\mathbf{p}_{i+1})


t tension Changes the length of the tangent vector
b bias Primarily changes the direction of the tangent vector
c continuity Changes the sharpness in change between tangents

Setting each parameter to zero would give a Catmull–Rom spline.

The source code found here of Steve Noskowicz in 1996 actually describes the impact that each of these values has on the drawn curve:

Tension T = +1→ Tight T = −1→ Round
Bias B = +1→ Post Shoot B = −1→ Pre shoot
Continuity C = +1→ Inverted corners C = −1→ Box corners

The code includes matrix summary needed to generate these splines in a BASIC dialect.

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