Kochanek–Bartels spline

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, ..., p_n,

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

$\mathbf{d}_i = \frac{(1-t)(1%2Bb)(1%2Bc)}{2}(\mathbf{p}_{i}-\mathbf{p}_{i-1}) %2B \frac{(1-t)(1-b)(1-c)}{2}(\mathbf{p}_{i%2B1}-\mathbf{p}_{i})$

$\mathbf{d}_{i%2B1} = \frac{(1-t)(1%2Bb)(1-c)}{2}(\mathbf{p}_{i%2B1}-\mathbf{p}_{i}) %2B \frac{(1-t)(1-b)(1%2Bc)}{2}(\mathbf{p}_{i%2B2}-\mathbf{p}_{i%2B1})$

where t is the tension, b is the bias, and c is the continuity parameter.

The tension parameter, t, changes the length of the tangent vector. The bias parameter, b, primarily changes the direction of the tangent vector. The continuity parameter, c, 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.

External links

Shane Aherne. "Kochanek and Bartels Splines". Motion Capture — exploring the past, present and future. http://www.shaneaherne.com/research/splines.html. Retrieved 2009-04-15.