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Kochanek–Bartels spline

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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 di+1 defined by

d i = ( 1 − t ) ( 1 + b ) ( 1 + c ) 2 ( p i − p i − 1 ) + ( 1 − t ) ( 1 − b ) ( 1 − c ) 2 ( p i + 1 − p i ) {\displaystyle \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})}
d i + 1 = ( 1 − t ) ( 1 + b ) ( 1 − c ) 2 ( p i + 1 − p i ) + ( 1 − t ) ( 1 − b ) ( 1 + c ) 2 ( p i + 2 − p i + 1 ) {\displaystyle \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

  1. http://news.povray.org/povray.binaries.tutorials/attachment/%3CXns91B880592482seed7@povray.org%3E/Splines.bas.txt
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