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

[math]\displaystyle{ \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) }[/math]

[math]\displaystyle{ \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}) }[/math]

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.

External links

• Shane Aherne. "Kochanek and Bartels Splines". Motion Capture — exploring the past, present and future. Archived from the original on 2007-07-05. https://web.archive.org/web/20070705165751/http://www.shaneaherne.com/research/splines.html. Retrieved 2009-04-15. 

• Doris H. U. Kochanek, Richard H. Bartels. "Interpolating splines with local tension, continuity, and bias control". SIGGRAPH '84 Proceedings of the 11th annual conference on Computer graphics and interactive techniques. ACM. pp. 33–41. http://dl.acm.org/citation.cfm?id=808575. Retrieved 2014-09-23.