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 points,

p₀, ..., p_n,

to be interpolated with n cubic Hermite curves, 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 s_i+1 defined by

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

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

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.