Calculating an AABB for a transformed sphere

Question

I have a sphere represented in object space by a center point and a radius. The sphere is transformed into world space with a transformation matrix that may include scales, rota

Answer 1

In general, a transformed sphere will be an ellipsoid of some sort. It's not too hard to get an exact bounding box for it; if you don't want go through all the math:

• note that M is your transformation matrix (scales, rotations, translations, etc.)
• read the definition of S below
• compute R as described towards the end
• compute the x, y, and z bounds based on R as described last.

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A general conic (which includes spheres and their transforms) can be represented as a symmetric 4x4 matrix: a homogeneous point p is inside the conic S when p^t S p < 0. Transforming your space by the matrix M transforms the S matrix as follows (the convention below is that points are column vectors):

A unit-radius sphere about the origin is represented by:
S = [ 1  0  0  0 ]
    [ 0  1  0  0 ]
    [ 0  0  1  0 ]
    [ 0  0  0 -1 ]

point p is on the conic surface when:
0 = p^t S p
  = p^t M^t M^t^-1 S M^-1 M p
  = (M p)^t (M^-1^t S M^-1) (M p)

transformed point (M p) should preserve incidence
-> conic S transformed by matrix M is:  (M^-1^t S M^-1)

The dual of the conic, which applies to planes instead of points, is represented by the inverse of S; for plane q (represented as a row vector):

plane q is tangent to the conic when:
0 = q S^-1 q^t
  = q M^-1 M S^-1 M^t M^t^-1 q^t
  = (q M^-1) (M S^-1 M^t) (q M^-1)^t

transformed plane (q M^-1) should preserve incidence
-> dual conic transformed by matrix M is:  (M S^-1 M^t)

So, you're looking for axis-aligned planes that are tangent to the transformed conic:

let (M S^-1 M^t) = R = [ r11 r12 r13 r14 ]  (note that R is symmetric: R=R^t)
                       [ r12 r22 r23 r24 ]
                       [ r13 r23 r33 r34 ]
                       [ r14 r24 r34 r44 ]

axis-aligned planes are:
  xy planes:  [ 0 0 1 -z ]
  xz planes:  [ 0 1 0 -y ]
  yz planes:  [ 1 0 0 -x ]

To find xy-aligned planes tangent to R:

  [0 0 1 -z] R [ 0 ] = r33 - 2 r34 z + r44 z^2 = 0
               [ 0 ]
               [ 1 ]
               [-z ]

  so, z = ( 2 r34 +/- sqrt(4 r34^2 - 4 r44 r33) ) / ( 2 r44 )
        = (r34 +/- sqrt(r34^2 - r44 r33) ) / r44

Similarly, for xz-aligned planes:

      y = (r24 +/- sqrt(r24^2 - r44 r22) ) / r44

and yz-aligned planes:

      x = (r14 +/- sqrt(r14^2 - r44 r11) ) / r44

This gives you an exact bounding box for the transformed sphere.