Cardinal spline

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A cardinal spline is a cubic Hermite spline whose tangents are defined by the points and a tension parameter. This spline creates a curve from one waypoint to another taking into account the waypoints before and after. By taking into account the way points before and after the current curve, the curves appear to join together making one seamless curve.

Given n+1 points

p₀, ..., p_n,

to be interpolated with n cubic Hermite curve segments, for each curve we have a starting point p_i and an ending point p_i+1 with starting tangent m_i and ending tangent m_i+1 with the tangents defined by

where the first and last tangent m₀ and m_n are given and c is a constant that modifies the length of the tangent (the tension parameter).

[edit] Code Example

This is a sample of a perl subroutine to render a cardinal spline. A reference to an array containing 4 sets of coordinates is fed to the formula as well as a tension and a point along the curve.

X and Y coordinates of the point are returned.

sub EvaluateCardinal2D
 {
 use Math::Matrix;
 my($ra_coords,$T,$u)    = @_;
 my $s                   = (1-$T)/2;
 my $u_matrix            = new Math::Matrix    # 4 x 1
     (                   # Matrix based off the point in the curve
      [($u ** 3),    ($u ** 2),    ($u),        (1)        ],
     );
 my $cardinal_matrix     = new Math::Matrix    # 4 x 4
     (                   # Guts of the Cardinal Spline formula
      [(-1 * $s),    (2 - $s),    ($s - 2),        ($s)        ],
      [(2 * $s),     ($s - 3),    (3-(2 * $s)),    (-1 * $s)   ],
      [(-1 * $s),    (0),         ($s),            (0)         ],
      [(0),          (1),         (0),             (0)         ],
     );
 my $x_matrix            = new Math::Matrix    # 1 x 4
     (                   # X coords for point:
      [${${$ra_coords}[0]}[0]],    # 1
      [${${$ra_coords}[1]}[0]],    # 2
      [${${$ra_coords}[2]}[0]],    # 3
      [${${$ra_coords}[3]}[0]],    # 4
     );
 my $y_matrix            = new Math::Matrix    # 1 x 4
     (                   # Y coords
      [${${$ra_coords}[0]}[1]],    # 1
      [${${$ra_coords}[1]}[1]],    # 2
      [${${$ra_coords}[2]}[1]],    # 3
      [${${$ra_coords}[3]}[1]],    # 4
     );
 my $xt    = int ($u_matrix * $cardinal_matrix * $x_matrix); # Compute for X
 my $yt    = int ($u_matrix * $cardinal_matrix * $y_matrix); # Compute for Y
 return($xt,$yt);
 }

[edit] What a cardinal spline looks like

This is a graphical plotting of a cardinal spline based off of 10 points on a canvas of 720x480. The line represents the curve, and the squares represent the points. Notice that the curve does not reach the first and last points, they do however affect the shape of the curve.

The points are:

023x024, 123x064, 167x200, 212x285, 293x297, 552x205, 537x071, 358x122, 262x130, 238x024

The tension is set to .1. To see the perl source code used to create this image, go to the image's page