I am writing a game in Python (with pygame) that requires me to generate random but nice-looking “sea” for each new game. After a long search I settled on an algorithm that involves Bezier curves as defined in padlib.py. I now need to figure out when the curves generated by padlib intersect a line segment.
The brute force method would be to just use the set of approximating line segments produced by padlib to find the answer. However, I suspect that a better answer can be found analytically. I only have a few dozen spline segments – searching them should be faster than thousand of line segments.
A little search took me down this road: Bezier Curve -> Kochanek-Bartels Spline -> Cubic Hermite spline
On the last page, I found this function:
p(t) = h00(t)p0 + h10(t)m0 + h01(t)p1 + h11(t)m1
where p(t) is a actually a point (2-dimensional vector), hij(t) functions are cubic polynomials, p0, p1, m0 and m1 are points I can get from padlib code.
Now, I can see that the solution to my problem is p(t) = u + v * t1, where u and v are the end of my line segment.
However, working out the analytical solution is beyond me. Does anyone here know of an existing solution? Or can help me with solving the equations?
As a rough outline, rotate and translate the system so that the line segment lies on the X axis. Now the y coordinate is a cubic function of the parameter t. Find the ‘zeros’ (the analytic formulae will be found in good math texts or wikipedia). Now evaluate the x coordinates corresponding to those zero points and test against your line segment.