Approximate length of a cubic bezier curve

/* A point in two-dimensional space. */

typedef struct {
    double x, y;
} 
point;

/* The control points (0 and 1) and end point (2) of a cubic Bezier
   curve. */

typedef struct {
    point pt[3];
} 
curve;

/*
  Calculate parametric value of x or y given t and the four point
  coordinates of a cubic bezier curve. This is a separate function
  because we need it for both x and y values.
*/

double _bezier_point (double t, double start, double control_1,
			     double control_2, double end)
{
    /* Formula from Wikipedia article on Bezier curves. */
    return              start * (1.0 - t) * (1.0 - t)  * (1.0 - t) 
           + 3.0 *  control_1 * (1.0 - t) * (1.0 - t)  * t 
	   + 3.0 *  control_2 * (1.0 - t) * t          * t
  	   +              end * t         * t          * t;
}

#define STEPS 10

/* 
   Approximate length of the Bezier curve which starts at "start" and
   is defined by "c". According to Computing the Arc Length of Cubic Bezier Curves
   there is no closed form integral for it.
*/

double bezier_length (point start, curve * c)
{
    double t;
    int i;
    int steps;
    point dot;
    point previous_dot;
    double length = 0.0;
    steps = STEPS;
    for (i = 0; i <= steps; i++) {
	t = (double) i / (double) steps;
	dot.x = _bezier_point (t, start.x, c->pt[0].x, 
			       c->pt[1].x, c->pt[2].x);
	dot.y = _bezier_point (t, start.y, c->pt[0].y, 
			       c->pt[1].y, c->pt[2].y);
	if (i > 0) {
	    double x_diff = dot.x - previous_dot.x;
	    double y_diff = dot.y - previous_dot.y;
	    length += sqrt (x_diff * x_diff + y_diff * y_diff);
	}
	previous_dot = dot;
    }
    return length;
}

Web links

• Arc Length Parameterization of Spline Curves (PDF)