Compose many Geometric Transformations for 3D Graphics

Question

I'm struggling to understand what would be a good way to compose several geometrical transformations to create a single TransformationFunction to be applied to a GeometricTransformation. One way could be to obtain a set of TransformationMatrix and multiply.

On this answer what I would have liked to achieve is to perform two rotations in different axis and then a translation in order to create a realistic trajectory of a falling cube, knowing height vs time and also the angles of elevation and azimuth vs time.

If I give a list of transformation functions these are applied to different copies of the original object instead of the same in sequence

Graphics3D[{
  Opacity[1]
  , Red
  , Arrow[{{0, 0, 0}, {1, 0, 0}}]
  , Green
  , Arrow[{{0, 0, 0}, {0, 1, 0}}]
  , Blue
  , Arrow[{{0, 0, 0}, {0, 0, 1}}]
  , Opacity[0.2]
  , GeometricTransformation[Cuboid[-{1, 1, 1}/4, {1, 1, 1}/4],
   {
    RotationTransform[Pi/4, {0, 0, 1}]
    , TranslationTransform[{1, 1, 1}]
    }
   ]
  }]

How do I properly compose several sequential geometrical transformations ?

I figure out the solution as soon as I posted the question and read one of the Related posts. I'm not sure if I should delete this or not. Anyhow I'm looking forward for other better solutions. — rhermans, Oct 15 '14 at 22:40
You can use Fold too, but matrix multiplication may be slightly faster: How to render a 3D ellipsoid with Graphics3D? — Kuba, Oct 15 '14 at 22:48
FYI: There is nothing wrong with answering your own question. In fact it is a recommended practice if you have something particularly nice you wish to share e.g. (54784). See: (834) — Mr.Wizard, Oct 15 '14 at 23:48

score 15 · Accepted Answer · answered Oct 15 '14 at 22:35

Using Composition I can apply RotationTransform, TranslationTransform , ShearingTransform one after the other.

Graphics3D[{
  Opacity[1]
  , Red
  , Arrow[{{0, 0, 0}, {1, 0, 0}}]
  , Green
  , Arrow[{{0, 0, 0}, {0, 1, 0}}]
  , Blue
  , Arrow[{{0, 0, 0}, {0, 0, 1}}]
  , Opacity[0.2]
  , GeometricTransformation[Cuboid[-{1, 1, 1}/4, {1, 1, 1}/4],
   Composition @@ {
     RotationTransform[Pi/4, {0, 0, 1}]
     , TranslationTransform[{1, 1, 1}]
     , ShearingTransform[Pi/8, {1, 0, 0}, {0, 0, 1}]
     }
   ]
  }]

I did not know that Composition would combine *Transform expressions in this manner so you just taught me something. Thanks and +1. — Mr.Wizard, Oct 15 '14 at 23:51

Compose many Geometric Transformations for 3D Graphics

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