Question

Suppose we have been given the coordinates of the centre of a solid rectangular box, the box’s length, breadth, height and a solid ball with a given centre and radius.

Is there a fast way to check if the box is a subset of the ball? The only simple method that comes to my mind is to check if each of the 8 corner vertices lies inside the sphere. If yes, the box is indeed a subset of the ball (by convexity property of the ball).

Answer 1

If you expect that most of the time the box will not be inside the sphere, you can do some quick tests:

if (sphere.center.x+sphere.radius<box.center.x-box.size.x) return false;
if (sphere.center.x-sphere.radius>box.center.x+box.size.x) return false;
etc.

If you expect that the box will usually be far inside the sphere, you can do other quick tests:

bsx = box.size.x;
bsy = box.size.y;
bsz = box.size.z;
box_radius = sqrt(bsx*bsx+bsy*bsy+bsz*bsz)/2;
brx = box.center.x-sphere.center.x;
bry = box.center.y-sphere.center.y;
brz = box.center.z-sphere.center.z;
box_dist = sqrt(brx*brx+bry*bry+brz*brz);
if (box_radius+box_dist<sphere.radius) return true;

You will still need more precise tests if the quick tests fail though.