Largest sphere inside a frustum
- by Will
How do you find the largest sphere that you can draw in perspective?
Viewed from the top, it'd be this:
Added: on the frustum on the right, I've marked four points I think we know something about. We can unproject all eight corners of the frusum, and the centres of the near and far ends. So we know point 1, 3 and 4. We also know that point 2 is the same distance from 3 as 4 is from 3. So then we can compute the nearest point on the line 1 to 4 to point 2 in order to get the centre? But the actual math and code escapes me.
I want to draw models (which are approximately spherical and which I have a miniball bounding sphere for) as large as possible.
Update: I've tried to implement the incircle-on-two-planes approach as suggested by
bobobobo and Nathan Reed :
function getFrustumsInsphere(viewport,invMvpMatrix) {
var midX = viewport[0]+viewport[2]/2,
midY = viewport[1]+viewport[3]/2,
centre = unproject(midX,midY,null,null,viewport,invMvpMatrix),
incircle = function(a,b) {
var c = ray_ray_closest_point_3(a,b);
a = a[1]; // far clip plane
b = b[1]; // far clip plane
c = c[1]; // camera
var A = vec3_length(vec3_sub(b,c)),
B = vec3_length(vec3_sub(a,c)),
C = vec3_length(vec3_sub(a,b)),
P = 1/(A+B+C),
x = ((A*a[0])+(B*a[1])+(C*a[2]))*P,
y = ((A*b[0])+(B*b[1])+(C*b[2]))*P,
z = ((A*c[0])+(B*c[1])+(C*c[2]))*P;
c = [x,y,z]; // now the centre of the incircle
c.push(vec3_length(vec3_sub(centre[1],c))); // add its radius
return c;
},
left = unproject(viewport[0],midY,null,null,viewport,invMvpMatrix),
right = unproject(viewport[2],midY,null,null,viewport,invMvpMatrix),
horiz = incircle(left,right),
top = unproject(midX,viewport[1],null,null,viewport,invMvpMatrix),
bottom = unproject(midX,viewport[3],null,null,viewport,invMvpMatrix),
vert = incircle(top,bottom);
return horiz[3]<vert[3]? horiz: vert;
}
I admit I'm winging it; I'm trying to adapt 2D code by extending it into 3 dimensions. It doesn't compute the insphere correctly; the centre-point of the sphere seems to be on the line between the camera and the top-left each time, and its too big (or too close). Is there any obvious mistakes in my code? Does the approach, if fixed, work?
