求贝赛尔 曲线的长度

Quadratic Bezier curves are defined by second order polynominal, andcan be written as
,
where t is real parameter with values in range [0,1]. P'sare respectively curve starting point, anchor point and the end point.Derivative of the quadratic Bezier curve can be written as
.

Length of any parametric (in general length of any well defined curve) curve can be computated using curve length integral.In case of 2nd order Bezier curve, using its derivatives, this integral can be written as
.
To simplify this integral we can make some substitutions. In this case it we will look like this
.
Next after doing some algebra and grouping elements in order to parameter t we will do another substitutions (to make this integral easier)
.

Finally we get simplified inegral, that can be written in form

This integral can be 'easily' simplified and calculated using relation
,
but we need to do some more substitutions
.

After doing elementary algebra we finaly get our expression. To calculate length of quadratic Bezier curve with thisexpression all we need arecoordinates of end points and control point. We dont need to use iterative methods anymore.
.

Accuracy of this evaluation

To check accuracy we will evalute length of quadric Bezier curve using previously calculated expression and approximation algorithm.Used algorithm approximates a Bezier curve with a set of line segments and calculates curve length as a sum over lengths of all that line segments.

In this case we will use a series of quadric Bezier curves. All curves have the same end points. For each curve control point is taken from a set ofpoints equally spaced on a circle, with given radius and centered in the middle between end points.

Presented plot shows comparison of curve length calculated using both (our expression and line approximation) methods for a set of Bezier quadriccurves.Green line shows results from approximation method, curve lengths calculated using our expression are drawn withred cross

Implementation

Implementation in c language can look as follows

float blen(v* p0, v* p1, v* p2)
{
v a,b;
a.x = p0->x - 2*p1->x + p2->x;
a.y = p0->y - 2*p1->y + p2->y;
b.x = 2*p1->x - 2*p0->x;
b.y = 2*p1->y - 2*p0->y;
float A = 4*(a.x*a.x + a.y*a.y);
float B = 4*(a.x*b.x + a.y*b.y);
float C = b.x*b.x + b.y*b.y;

float Sabc = 2*sqrt(A+B+C);
float A_2 = sqrt(A);
float A_32 = 2*A*A_2;
float C_2 = 2*sqrt(C);
float BA = B/A_2;

return ( A_32*Sabc + A_2*B*(Sabc-C_2) + (4*C*A-B*B)*log( (2*A_2+BA+Sabc)/(BA+C_2) ) )/(4*A_32);
};