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, ..., 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 si+1 defined by
- <math>\mathbf{s}_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>\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> 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.