Catmull-Rom splines - how do they work?

Question

From this site which seems to have the most detailed information about catmull-rom splines: http://www.mvps.org/directx/articles/catmull/ it makes mention of needing four points to create the spline. However it does not mention how the points p0 and p3 affect the values between p1 and p2.

Another question I have is how would you create continuous splines? Would it be as easy as defining the points p1, p2 to be continuous with p4, p5 by making p4 = p2 (that is assuming we have p0, p1, p2, p3, p4, p5, p6... pN).

A more general question is how would one calculate tangents on catmull rom splines? Would it have to involve taking two points on the spline (say at 0.01, 0.011) and getting the tangent based on pythagoras given the position coordinates those input values give?

Answer 1

The Wikipedia article goes into a little bit more depth. The general form of the spline takes as input 2 control points with associated tangent vectors. Additional spline segments can then be added provided that the tangent vectors at the common control points are equal, which preserves the C1 continuity.

In the specific Catmull-Rom form, the tangent vector at intermediate points is determined by the locations of neighboring control points. Thus, to create a C1 continuous spline through multiple points, it is sufficient to supply the set of control points and the tangent vectors at the first and last control point. I think the standard behavior is to use P1 - P0 for the tangent vector at P0 and PN - PN-1 at PN.

According to the Wikipedia article, to calculate the tangent at control point Pn, you use this equation:

T(n) = (P(n - 1) + P(n + 1)) / 2

This also answers your first question. For a set of 4 control points, P1, P2, P3, P4, interpolating values between P2 and P3 requires information form all 4 control points. P2 and P3 themselves define the endpoints through which the interpolating segment must pass. P1 and P3 determine the tangent vector the interpolating segment will have at point P2. P4 and P2 determine the tangent vector the segment will have at point P3. The tangent vectors at control points P2 and P3 influence the shape of the interpolating segment between them.

Answer 2

Take a look at equation 2 -- it describes how the control points affect the line. You can see points p0 and p3 go into the equation for plotting points along the curve from P1 to P2. You'll also see that the equation gives P1 when t == 0 and P2 when t == 1.

This example equation can be generalized. If you have points R0, R1, ... RN then you can plot the points between RK and RK+1 by using equation 2 with P0 = RK-1, P1 = RK, P2 = RK+1, P3 = RK+2.

You can't plot from R0 to R1 or from RN-1 to RN unless you add extra control points to stand in for R-1 and RN+1. The general idea is that you can pick whatever points you want to add to the head and tail of a sequence to give youself all the parameters to calculate the spline.

You can join two splines together by dropping one of the control points between them. Say you have R0, R1, ..., RN and S0, S1, ... SM they can be joined into R0, R1, ..., RN-1, S1, S2, ... SM.

To compute the tangent at any point just take the derivative of equation 2.