Kochanek–Bartels spline

File:Kochanek bartels spline.svg

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>\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>\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...

Template:Mvar	tension	Changes the length of the tangent vector
Template:Mvar	bias	Primarily changes the direction of the tangent vector
Template:Mvar	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:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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

• {{#invoke:citation/CS1|citation

|CitationClass=web }}

• Template:Cite book