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

where...

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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