There is a much easier and more efficient way to implement this which only requires you to compute your tangents using a different formula, without the need to implement the recursive evaluation algorithm of Barry and Goldman.
If you take the Barry-Goldman parametrization (referenced in Ted's answer) C(t) for the knots (t0,t1,t2,t3) and the control points (P0,P1,P2,P3), its closed form is pretty complicated, but in the end it's still a cubic polynomial in t when you constrain it to the interval (t1,t2). So all we need to describe it fully are the values and tangents at the two end points t1 and t2. If we work out these values (I did this in Mathematica), we find
C(t1) = P1
C(t2) = P2
C'(t1) = (P1 - P0) / (t1 - t0) - (P2 - P0) / (t2 - t0) + (P2 - P1) / (t2 - t1)
C'(t2) = (P2 - P1) / (t2 - t1) - (P3 - P1) / (t3 - t1) + (P3 - P2) / (t3 - t2)
We can simply plug this into the standard formula for computing a cubic spline with given values and tangents at the end points and we have our nonuniform Catmull-Rom spline. One caveat is that the above tangents are computed for the interval (t1,t2), so if you want to evaluate the curve in the standard interval (0,1), simply rescale the tangents by multiplying them with the factor (t2-t1).
I put a working C++ example on Ideone: http://ideone.com/NoEbVM
I'll also paste the code below.
#include <iostream>
#include <cmath>
using namespace std;
struct CubicPoly
{
float c0, c1, c2, c3;
float eval(float t)
{
float t2 = t*t;
float t3 = t2 * t;
return c0 + c1*t + c2*t2 + c3*t3;
}
};
/*
* Compute coefficients for a cubic polynomial
* p(s) = c0 + c1*s + c2*s^2 + c3*s^3
* such that
* p(0) = x0, p(1) = x1
* and
* p'(0) = t0, p'(1) = t1.
*/
void InitCubicPoly(float x0, float x1, float t0, float t1, CubicPoly &p)
{
p.c0 = x0;
p.c1 = t0;
p.c2 = -3*x0 + 3*x1 - 2*t0 - t1;
p.c3 = 2*x0 - 2*x1 + t0 + t1;
}
// standard Catmull-Rom spline: interpolate between x1 and x2 with previous/following points x0/x3
// (we don't need this here, but it's for illustration)
void InitCatmullRom(float x0, float x1, float x2, float x3, CubicPoly &p)
{
// Catmull-Rom with tension 0.5
InitCubicPoly(x1, x2, 0.5f*(x2-x0), 0.5f*(x3-x1), p);
}
// compute coefficients for a nonuniform Catmull-Rom spline
void InitNonuniformCatmullRom(float x0, float x1, float x2, float x3, float dt0, float dt1, float dt2, CubicPoly &p)
{
// compute tangents when parameterized in [t1,t2]
float t1 = (x1 - x0) / dt0 - (x2 - x0) / (dt0 + dt1) + (x2 - x1) / dt1;
float t2 = (x2 - x1) / dt1 - (x3 - x1) / (dt1 + dt2) + (x3 - x2) / dt2;
// rescale tangents for parametrization in [0,1]
t1 *= dt1;
t2 *= dt1;
InitCubicPoly(x1, x2, t1, t2, p);
}
struct Vec2D
{
Vec2D(float _x, float _y) : x(_x), y(_y) {}
float x, y;
};
float VecDistSquared(const Vec2D& p, const Vec2D& q)
{
float dx = q.x - p.x;
float dy = q.y - p.y;
return dx*dx + dy*dy;
}
void InitCentripetalCR(const Vec2D& p0, const Vec2D& p1, const Vec2D& p2, const Vec2D& p3,
CubicPoly &px, CubicPoly &py)
{
float dt0 = powf(VecDistSquared(p0, p1), 0.25f);
float dt1 = powf(VecDistSquared(p1, p2), 0.25f);
float dt2 = powf(VecDistSquared(p2, p3), 0.25f);
// safety check for repeated points
if (dt1 < 1e-4f) dt1 = 1.0f;
if (dt0 < 1e-4f) dt0 = dt1;
if (dt2 < 1e-4f) dt2 = dt1;
InitNonuniformCatmullRom(p0.x, p1.x, p2.x, p3.x, dt0, dt1, dt2, px);
InitNonuniformCatmullRom(p0.y, p1.y, p2.y, p3.y, dt0, dt1, dt2, py);
}
int main()
{
Vec2D p0(0,0), p1(1,1), p2(1.1,1), p3(2,0);
CubicPoly px, py;
InitCentripetalCR(p0, p1, p2, p3, px, py);
for (int i = 0; i <= 10; ++i)
cout << px.eval(0.1f*i) << " " << py.eval(0.1f*i) << endl;
}
