Cube sphere intersection test?

Question

What's the easiest way of doing this? I fail at math, and i found pretty complicate formulaes over the internet... im hoping if theres some simpler one?

I just need to know if a sphere is overlapping a cube, i dont care about which point it does that etc.

I'm also hoping it would take advantage of the fact that both shapes are symmetric.

Edit: the cube is aligned straight in the x,y,z axises

Answer 1

Looking at half-spaces is not enough, you have to consider also the point of closest approach:

Borrowing Adam's notation:

Assuming an axis-aligned cube and letting C1 and C2 be opposing corners, S the center of the sphere, and R the radius of the sphere, and that both objects are solid:

inline float squared(float v) { return v * v; }
bool doesCubeIntersectSphere(vec3 C1, vec3 C2, vec3 S, float R)
{
    float dist_squared = R * R;
    /* assume C1 and C2 are element-wise sorted, if not, do that now */
    if (S.X < C1.X) dist_squared -= squared(S.X - C1.X);
    else if (S.X > C2.X) dist_squared -= squared(S.X - C2.X);
    if (S.Y < C1.Y) dist_squared -= squared(S.Y - C1.Y);
    else if (S.Y > C2.Y) dist_squared -= squared(S.Y - C2.Y);
    if (S.Z < C1.Z) dist_squared -= squared(S.Z - C1.Z);
    else if (S.Z > C2.Z) dist_squared -= squared(S.Z - C2.Z);
    return dist_squared > 0;
}

Answer 2

Jim Arvo has an algorithm for this in Graphics Gems 2 which works in N-Dimensions. I believe you want "case 3" at the bottom of this page: http://www.ics.uci.edu/~arvo/code/BoxSphereIntersect.c which cleaned up for your case is:

bool BoxIntersectsSphere(Vec3 Bmin, Vec3 Bmax, Vec3 C, float r) {
  float r2 = r * r;
  dmin = 0;
  for( i = 0; i < 3; i++ ) {
    if( C[i] < Bmin[i] ) dmin += SQR( C[i] - Bmin[i] );
    else if( C[i] > Bmax[i] ) dmin += SQR( C[i] - Bmax[i] );     
  }
  return dmin <= r2;
}