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In mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Kochanek–Bartels spline or Kochanek–Bartels curve is a cubic Hermite spline
In numerical analysis, a cubic Hermite spline or cubic Hermite interpolator is a spline where each piece is a third-degree polynomial specified in Hermite form, that is, by its values and first derivatives at the end points of the correspondin ...
with tension, bias, and continuity parameters defined to change the behavior of the tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
s.
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
:
:
where...
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:
The code includes matrix summary needed to generate these splines in a BASIC
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dialect.
References
External links
* * {{DEFAULTSORT:Kochanek-Bartels spline Splines (mathematics) Interpolation