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

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

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

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 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. Retrieved 2009-04-15.

• Doris H. U. Kochanek, Richard H. Bartels (1984). Interpolating splines with local tension, continuity, and bias control. ACM. pp. 33–41. doi:10.1145/800031.808575. ISBN 0-89791-138-5. Retrieved 2014-09-23. {{cite book}}: |work= ignored (help)