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. It was invented by Doris Kochanek of National Film Board of Canada and Richard Bartels of University of Waterloo in Canada to automate the process of creating the effect desired by the animator for interpolated motion between key frames in computer animation, reducing the need to input additional information.

Definition The key positions (data points) of each keyframe are p i {\displaystyle \mathbf {p} _{i}} , p i + 1 , … {\displaystyle \mathbf {p} _{i+1},\ldots } , and are interpolated using a cubic Hermite spline.

For each p i {\displaystyle \mathbf {p} _{i}} , define the incoming tangent vector D S {\displaystyle \mathbf {DS} } and the outgoing tangent vector D D {\displaystyle \mathbf {DD} } as follows:

D S 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 {DS} _{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 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 {DD} _{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})}

For the interval between the start point p i {\displaystyle \mathbf {p} _{i}} and the end point p i + 1 {\displaystyle \mathbf {p} _{i+1}} , Kochanek-Bartels spline is obtained by applying the starting tangent vector m i = D D i {\displaystyle \mathbf {m} _{i}=\mathbf {DD} _{i}} and the ending tangent vector m i + 1 = D S i + 1 {\displaystyle \mathbf {m} _{i+1}=\mathbf {DS} _{i+1}} to the definition formula of cubic Hermite spline.

Parameters and Effects

Example of implementation The source code of Steve Noskowicz in 1996 actually describes the impact that each of these values has on the drawn curve:

The code includes matrix summary needed to generate these splines in a BASIC dialect.

References

External links Shane Aherne. "Kochanek and Bartels Splines". Motion Capture — exploring the past, present and future. Archived from the original on 2007-07-05. Retrieved 2009-04-15.