CatmullRomsplines

Catmull-Rom splines are a type of interpolating parametric curve. They are a specific instance of cubic Hermite splines, meaning that at each control point, the curve passes through the point and its tangent at that point is determined by the positions of the neighboring control points. The name Catmull-Rom comes from Edwin Catmull and Robert Rom, who developed the algorithm.

Unlike B-splines, Catmull-Rom splines are guaranteed to pass through all of their control points. This makes them useful for applications where precise point interpolation is desired, such as in computer graphics for defining smooth paths or animation curves. The curve's shape is influenced by four points at a time: the point the curve is currently passing through, the next point, and the two points surrounding them.

The mathematical formulation involves a parameter t, typically ranging from 0 to 1, which defines the position along the curve segment between two control points. The tangent at a control point is calculated as a scaled difference between the preceding and succeeding points. This calculation ensures that the curve is C1 continuous, meaning it has a continuous first derivative, resulting in a smooth transition between curve segments without any sharp corners.

While they offer intuitive control and guaranteed interpolation, Catmull-Rom splines can sometimes exhibit unwanted oscillations between closely spaced control points. However, their simplicity and direct interpolation make them a popular choice in various graphics and animation software.