3D Maths for CG

Interpolation - Catmull-Rom Spline

A cubic spline. In the current context we're using it for interpolation purposes in 1-D. The Catmull-Rom spline takes a set of keyframe values to describe a smooth piecewise cubic curve that passes through all the keys. In order to use this routine we need four keyframe values. So we have v0,v1,v2, and v3, plus a parameter that specifies a key, x between v1 and v2 to interpolate. The return value, f(x) is the value associated with the key x.

    	f(x) = [1, x, x^2, x^3] * M * [v0, v1, v2, v3]

where M is defined as:

The image bellow shows an example of a curve between v1 and v2. The curve was obtained calling the function bellow varying x between 0.0 and 1.0.

    
	for (x=0;x<=1;x+=0.01)
		printf("%f\n",catmullRomSpline(x,v0,v1,v2,v3));

The following C code performs the required operation. The return value is the value for the parameter key x. The remaining parameters are the four values as mentione above. Note that x is assumed to be between v1 and v2 . The value for x must be between 0.0 and 1.0. If x = 0.0 then the return value is v1. For x = 1.0, the return value is v2.

    
/* Coefficients for Matrix M */
#define M11	 0.0	
#define M12	 1.0
#define M13	 0.0
#define M14	 0.0
#define M21	-0.5
#define M22	 0.0
#define M23	 0.5
#define M24	 0.0
#define M31	 1.0
#define M32	-2.5
#define M33	 2.0
#define M34	-0.5
#define M41	-0.5
#define M42	 1.5
#define M43	-1.5
#define M44	 0.5

double catmullRomSpline(float x, float v0,float v1,
				float v2,float v3) {

	double c1,c2,c3,c4;

	c1 =  	      M12*v1;
	c2 = M21*v0          + M23*v2;
	c3 = M31*v0 + M32*v1 + M33*v2 + M34*v3;
	c4 = M41*v0 + M42*v1 + M43*v2 + M44*v3;

	return(((c4*x + c3)*x +c2)*x + c1);
}

Further information about the formulas can be found at On-Line Geometric Modeling Notes.

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