/*

Java class to implement cubic and bicubic splines

      by Ken Perlin @ NYU, 1998, 2004.

You have my permission to use freely, as long as you keep the attribution. - Ken Perlin

What does the class do?

  1. Cubic spline: If you provide various geometric values for t, then this class creates an object that will interpolate a Cubic spline to give you the value within any value of t between 0 and 1.

    If you want to create a spline path, you can make a one dimensional array of such objects.

  2. Bicubic spline: If you provide a 4x4 grid of values for geometric quantities in u and v, this class creates an object that will interpolate a Bicubic spline to give you the value within any point of a unit tile in (u,v) space.

    If you want to create a spline surface, you can make a two dimensional array of such objects.

For a cubic spline the class provides a constructor and a method:

Cubic(double[] G)
Given four geometric values over t, calculate cubic coefficients.
double eval(double t)
Given a point in the interval t = [0 ... 1], return a value.
Algorithm:
f(t) = T M GT, where:
T = (t3 t2 t 1) ,
M is the basis matrix.
The constructor Cubic(G) calculates the matrix C = M GT

The method eval(t) calculates the value T C

For a bicubic spline the class provides a constructor and a method:

Cubic(double[][] G)
Given 4×4 geometric values over u×v, calculate bicubic coefficients.
double eval(double u, double v)
Given a point in the square [0 ... 1] × [0 ... 1], return a value.
Algorithm:
f(u,v) = U M G MT VT , where:
U = (u3 u2 u 1) ,
V = (v3 v2 v 1) ,
M is the basis matrix.
The constructor Cubic(G) calculates the matrix C = M G MT

The method eval(u,v) calculates the value U C VT 

*/

public class Cubic
{
   public static final double[][] BEZIER = {      // Bezier basis matrix
      {-1  ,  3  , -3  , 1  },
      { 3  , -6  ,  3  , 0  },
      {-3  ,  3  ,  0  , 0  },
      { 1  ,  0  ,  0  , 0  } 
   };
   public static final double[][] BSPLINE = {     // BSpline basis matrix
      {-1./6 ,  3./6 , -3./6 , 1./6 },
      { 3./6 , -6./6 ,  3./6 , 0.   },
      {-3./6 ,  0.   ,  3./6 , 0.   },
      { 1./6 ,  4./6 ,  1./6 , 0.   } 
   };
   public static final double[][] CATMULL_ROM = { // Catmull-Rom basis matrix
      {-0.5 ,  1.5 , -1.5 ,  0.5 },
      { 1   , -2.5 ,  2   , -0.5 },
      {-0.5 ,  0   ,  0.5 ,  0   },
      { 0   ,  1   ,  0   ,  0   } 
   };
   public static final double[][] HERMITE = {     // Hermite basis matrix
      { 2  , -2  ,  1  ,  1  },
      {-3  ,  3  , -2  , -1  },
      { 0  ,  0  ,  1  ,  0  },
      { 1  ,  0  ,  0  ,  0  } 
   };

   double a, b, c, d;                  // cubic coefficients vector

   Cubic(double[][] M, double[] G) {
      a = b = c = d;
      for (int k = 0 ; k < 4 ; k++) {  // (a,b,c,d) = M G
	 a += M[0][k] * G[k];
	 b += M[1][k] * G[k];
	 c += M[2][k] * G[k];
	 d += M[3][k] * G[k];
      }
   }

   public double eval(double t) {
      return t * (t * (t * a + b) + c) + d;
   }

   double[][] C = new double[4][4];    // bicubic coefficients matrix
   double[][] T = new double[4][4];    // scratch matrix

   Cubic(double[][] M, double[][] G) {
      for (int i = 0 ; i < 4 ; i++)    // T = G MT
      for (int j = 0 ; j < 4 ; j++)
      for (int k = 0 ; k < 4 ; k++)
	 T[i][j] += G[i][k] * M[j][k];
      
      for (int i = 0 ; i < 4 ; i++)    // C = M T
      for (int j = 0 ; j < 4 ; j++)
      for (int k = 0 ; k < 4 ; k++)
	 C[i][j] += M[i][k] * T[k][j];
   }

   double[] C3 = C[0], C2 = C[1], C1 = C[2], C0 = C[3];

   public double eval(double u, double v) {
      return u * (u * (u * (v * (v * (v * C3[0] + C3[1]) + C3[2]) + C3[3])
                         + (v * (v * (v * C2[0] + C2[1]) + C2[2]) + C2[3]))
                         + (v * (v * (v * C1[0] + C1[1]) + C1[2]) + C1[3]))
                         + (v * (v * (v * C0[0] + C0[1]) + C0[2]) + C0[3]);
   }
}