Let's say we define the orientation of the ellipsoid from its major axis (the largest axis of the ellipsoid). Assuming the 3 axes of the ellipsoid to be on the three coordinates with lengths of a, b and c along each axis (with a ≥ b and a ≥ c), then only a single affine transformation (to a sphere) is necessary: [ a 0 0 0 0 b 0 0 0 0 c 0 0 0 0 1 ] Now, if we first rotate the major axis by θ from the first axis towards the second axis, and then rotate it by ϕ from the (rotated) first axis towards the third axis, the combined affine transformation becomes: [ a 0 0 0 0 b 0 0 0 0 c 0 0 0 0 1 ] [ c o s θ − s i n θ 0 0 s i n θ c o s θ 0 0 0 0 1 0 0 0 0 1 ] [ c o s ϕ 0 − s i n ϕ 0 0 1 0 0 s i n ϕ 0 c o s ϕ 0 0 0 0 1 ] Is the multiplied matrix (from left to right) the correct affine transformation that must go into the equations in the links above?

Question

Let&#039;s say we define the orientation of the ellipsoid from its major axis (the largest axis of the ellipsoid). Assuming the 3 axes of the ellipsoid to be on the three coordinates with lengths of   a,   b and   c along each axis (with   a  ≥  b and   a  ≥  c), then only a single affine transformation (to a sphere) is necessary:      [                            a                          0                          0                          0                                      0                          b                          0                          0                                      0                          0                          c                          0                                      0                          0                          0                          1                      ]  Now, if we first rotate the major axis by   θ from the first axis towards the second axis, and then rotate it by   ϕ from the (rotated) first axis towards the third axis, the combined affine transformation becomes:      [                            a                          0                          0                          0                                      0                          b                          0                          0                                      0                          0                          c                          0                                      0                          0                          0                          1                      ]        [                            c          o          s          θ                          −          s          i          n          θ                          0                          0                                      s          i          n          θ                          c          o          s          θ                          0                          0                                      0                          0                          1                          0                                      0                          0                          0                          1                      ]        [                            c          o          s          ϕ                          0                          −          s          i          n          ϕ                          0                                      0                          1                          0                          0                                      s          i          n          ϕ                          0                          c          o          s          ϕ                          0                                      0                          0                          0                          1                      ]  Is the multiplied matrix (from left to right) the correct affine transformation that must go into the equations in the links above?

Christina Ward · Accepted Answer

Let&#039;s say you have an ellipsoid centered at       c    →    with semi-principal axes               r      →        1  ,               r      →        2  , and               r      →        3  :              r      →        1    =      [                                        r                      11                                                            r                      21                                                            r                      31                                ]    ,                r      →        2    =      [                                        r                      12                                                            r                      22                                                            r                      32                                ]    ,                r      →        3    =      [                                        r                      13                                                            r                      23                                                            r                      33                                ]  The surface of the ellipsoid passes through points       c    →    ±              r      →        1  ,       c    →    ±              r      →        2   and       c    →    ±              r      →        3  .The semi-principal axes are all perpendicular:               r      →        1    ⋅              r      →        2    =  0,               r      →        1    ⋅              r      →        3    =  0, and               r      →        2    ⋅              r      →        3    =  0.The axis-aligned bounding box for the above ellipsoid is defined by                    x        =                  c          x                ±                              r                      11            2                    +                      r                      12            2                    +                      r                      13            2                                              y        =                  c          y                ±                              r                      21            2                    +                      r                      22            2                    +                      r                      23            2                                              z        =                  c          z                ±                              r                      31            2                    +                      r                      32            2                    +                      r                      33            2                              Note that this applies also to the case where the homogenous transformation matrix M that transforms the unit sphere to the ellipsoid is  M  =      [                                        r                      11                                                r                      12                                                r                      13                                                c            x                                                            r                      21                                                r                      22                                                r                      23                                                c            y                                                            r                      31                                                r                      32                                                r                      33                                                c            z                                                0                          0                          0                          1                      ]

Brenden Tran · Accepted Answer

Here is full soluion:We can describe the transformation of the unit sphere to the above ellipsoid using                    (1)                                    p          →                (        u        ,        v        ,        w        )        =                        p          →                (                        t          →                )        =                        c          →                +        u                                      r            →                    1                +        v                                      r            →                    2                +        w                                      r            →                    3                =                        c          →                +              R                        t          →                    where       t    →    =  (  u  ,  v  ,  w  ) fulfills      ‖            t      →        ‖    =  1      ⟺          u    2    +      v    2    +      w    2    =  1on the surface of the ellipsoid, and      ‖            t      →        ‖    &amp;lt;  1      ⟺          u    2    +      v    2    +      w    2    &amp;lt;  1within the ellipsoid, and matrix R has the semi-principal axes as column vectors,  R  =      [                                                                r              →                        1                                                                        r              →                        2                                                                        r              →                        3                                ]    =      [                                        r                      11                                                r                      12                                                r                      13                                                            r                      21                                                r                      22                                                r                      23                                                            r                      31                                                r                      32                                                r                      33                                ]  Essentially, we scale the unit sphere by the semi-principal axes, rotate it to the new orientation, and translate the center.The above operation is invertible, if the ellipsoid is not degenerate; that is, if       ‖                      r        →            1        ‖    &amp;gt;  0,       ‖                      r        →            2        ‖    &amp;gt;  0, and       ‖                      r        →            3        ‖    &amp;gt;  0.Since rotation matrices are orthonormal, its inverse is its transpose. To counteract the scaling, we scale by the inverses of the semi-principal axes.The scaling vectors are now row vectors of the transformation matrix, with lengths inverse to the corresponding semi-principal axis length; i.e.              s      →        n    =            (                                            r            →                    n                                      ‖                                                      r                →                            n                        ‖                    2                    )          T    ,    n  =  1  ,  2  ,  3so that      ‖                      s        →            n        ‖    =            ‖                              r          →                n            ‖              −      1        ,    n  =  1  ,  2  ,  3i.e.              s      →        1    =                        r        →            1                      ‖                                      r            →                    1                ‖            2        =      [                                        s                      11                                                s                      12                                                s                      13                                ]                s      →        2    =                        r        →            2                      ‖                                      r            →                    2                ‖            2        =      [                                        s                      21                                                s                      22                                                s                      23                                ]                s      →        3    =                        r        →            3                      ‖                                      r            →                    3                ‖            2        =      [                                        s                      31                                                s                      32                                                s                      33                                ]        s          i      j        =            r              j        i                            r                  1          i                2            +              r                  2          i                2            +              r                  3          i                2              ,    i  =  1  ,  2  ,  3  ;    j  =  1  ,  2  ,  3  S  =      [                                        s                      11                                                s                      12                                                s                      13                                                            s                      21                                                s                      22                                                s                      23                                                            s                      31                                                s                      32                                                s                      33                                ]  Now,                    (2)                                    t          →                (                        p          →                )        =                                      s            →                    1                ⋅        (                        p          →                −                        c          →                )        +                                      s            →                    2                ⋅        (                        p          →                −                        c          →                )        +                                      s            →                    3                ⋅        (                        p          →                −                        c          →                )        =              S                  (                              p            →                    −                              c            →                    )                    Essentially, (1) and (2) are the inverse of each other.If you insist on using Euler or Tait-Bryan angles or other similar angular convention, I recommend you use them to calculate the semi-principal axes               r      →        1  ,               r      →        2  , and               r      →        3   only, and then apply the above vector solution.

Let's say we define the orientation of the ellipsoid from its major axis (the largest axis of the el

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