Interpolating Cardinal and Catmull-Rom splines

Continuous curve with a kink in Fig.1 is called C ⁰ continuous. A curve is C ^k continuous if all k derivatives of the curve are continuous.

Interpolating piecewise Cardinal spline is composed of cubic Bezier splines joined with C¹ continuity (see Fig.2). The i-th Bezier segment goes through two neighbouring points P_i , P_i+1 . Derivatives at a point P_i are parallel to the P_i-1P_i+1 line, i.e.
P '_i = (P_i+1 - P_i-1)/α .
Derivatives at the terminal points P₀ , P_n go to the neighbouring nodes. Remember, that a cubic Bezier spline is determined by 4 vectors (e.g. four control points or two endpoints and two derivatives at the points). For the i -th segment we have
B₀ = P_i , B₃ = P_i+1 .
As for Bezier spline
B'(0) = 3(B₁ - B₀) , B'(1) = 3(B₃ - B₂) ,
therefore all control points B_0,1,2,3 are obtained easy for every segment. Control polygon for the left Bezier segment is shown in Fig.2. Derivatives at P₁ are parallel to the P₀P₂ line. Derivative at the endpoint P₀ goes to P₁.
One can take different α_i values for different points P_i and even different α_+,- for the "left" and "right" segments at a point too. For α = 2 we get the Catmul - Rom spline.

Interactive Cardinal spline Use finger or mouse to move the nearest control point. Use selector to move, add or delete control points. Drag up-down the small blue rectangle to the right to change α.
move
add
delete
set α

You can compare Cardinal and Bezier spline curves with the same control points.
subdivision Unfortunately the Cardinal curve cannot be subdivided in two sub-curves which coincide with the initial one. As since Fig.3 shows, that in common case the line P₀P₃ is not parallel to the P₀²P₁² one and the derivative at P(1/2) is not (P₃ - P₀)/α.

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updated 11 August 2001