Appendix

A parametric cubic polynomial is the lowest-order formulation that can insure continuity of position, slope, and curvature where two such curves meet. A curve section is specified witha set of four `guide coordinates' and a `curve basis' defining how the guide points influence the shape of the curve. A B-spline curve is constrained to have continuous first and second derivatives where adjacent curve sections meet. The B-spline curves do not, in general, pass through their guide coordinates.

The matrix algebra follows the IRIS manual and Clark (1981). A 4 x 4 matrix, M, is computed from guide coordinates, the curve basis, and the desired sampling as follows: M = SBG.
If N line segments are desired to approximate the curve section, the sampling matrix is given by:

        S =  {  6/N**3    0        0     0  }
             {  6/N**3    2/N**2   0     0  }
             {  1/N**3    1/N**2   1/N   0  }
             {  0         0        0     1  }

The basis matrix for a cubic B-spline is given by:

        B = 1/6 {  -1    3   -3   1  }
                {   3   -6    3   0  }
                {  -3    0    3   0  }
                {   1    4    1   0  }

The geometry matrix is constructed from the guide coordinates as:

       G =   {  x_1   y_1   z_1   1.0  }
             {  x_2   y_2   z_2   1.0  }
             {  x_3   y_3   z_3   1.0  }
             {  x_4   y_4   z_4   1.0  }

The line segments are now generated with the `forward difference algorithm' given in the C language below:

       /*  Comments:
       *
       *   M[4][4] is the current matrix  ( indexed 0..3 )
       *   N is the number of line segments to be drawn
       *   MoveTo() and DrawTo() represent generic graphics calls
       */
       MoveTo( M[3][0]/M[3][3], M[3][1]/M[3][3], M[3][2]/M[3][3] );
       for( k=0; k<N; k=k+1 ) {
           for( i=3; i>0; i=i-1 ) {
               for( j=0; j<4; j=j+1 ) {
                   M[i][j] = M[i][j] + M[i-1][j];
               }
           }
           DrawTo( M[3][0]/M[3][3], M[3][1]/M[3][3], M[3][2]/M[3][3] );
       }

Thus if guide coordinates x_1,x_2,x_3,x_4 are used to create the first section of the curve, the coordinates x_2,x_3,x_4,x_5 will create the next curve section, smoothly joining the first.