Circle line-segment collision detection algorithm?

Question

I have a line from A to B and a circle positioned at C with the radius R.

What is a good alg

Answer 1

Circle is really a bad guy :) So a good way is to avoid true circle, if you can. If you are doing collision check for games you can go with some simplifications and have just 3 dot products, and a few comparisons.

I call this "fat point" or "thin circle". its kind of a ellipse with zero radius in a direction parallel to a segment. but full radius in a direction perpendicular to a segment

First, i would consider renaming and switching coordinate system to avoid excessive data:

s0s1 = B-A;
s0qp = C-A;
rSqr = r*r;

Second, index h in hvec2f means than vector must favor horisontal operations, like dot()/det(). Which means its components are to be placed in a separate xmm registers, to avoid shuffling/hadd'ing/hsub'ing. And here we go, with most performant version of simpliest collision detection for 2D game:

bool fat_point_collides_segment(const hvec2f& s0qp, const hvec2f& s0s1, const float& rSqr) {
    auto a = dot(s0s1, s0s1);
    //if( a != 0 ) // if you haven't zero-length segments omit this, as it would save you 1 _mm_comineq_ss() instruction and 1 memory fetch
    {
        auto b = dot(s0s1, s0qp);
        auto t = b / a; // length of projection of s0qp onto s0s1
        //std::cout << "t = " << t << "\n";
        if ((t >= 0) && (t <= 1)) // 
        {
            auto c = dot(s0qp, s0qp);
            auto r2 = c - a * t * t;
            return (r2 <= rSqr); // true if collides
        }
    }   
    return false;
}

I doubt you can optimize it any further. I am using it for neural-network driven car racing collision detection, to process millions of millions iteration steps.