Kochanek–Bartels spline
# 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,

    $p_0,$ ..., $p_n,$

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 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.

