animate 3d plot with some further requirements in mathematica

I posted at this 开发者_运维百科post before, but I still could not solve the following problem completely. As an example:

{pA, pB, pC, pD} = {{0, 0, Sqrt[61/3]}, {Sqrt[7], 4*Sqrt[2/3], 0}, {0, -5*Sqrt[2/3], 0}, {-Sqrt[71], 4*Sqrt[2/3], 0}};
axis={1,0,0};pt={0,1,0};
plotPolygon[{a_, b_, c_}] := {Opacity[.4], Polygon[{a, b, c}]};
graph=Graphics3D[{plotPolygon[{pA, pB, pC}], plotPolygon[{pA, pB, pD}], 
            plotPolygon[{pB, pC, pD}], plotPolygon[{pA, pC, pD}]}, 
            Axes -> True, AxesOrigin->pt];
Animate[graph/.gg : Graphics3D[___] :> Rotate[gg, theta, axis], {theta, 0., 2.*Pi}]

I want to rotate along an axis axis={1,0,0} which passes the point pt={0,1,0}. But I don't know how to specify the point information. Also the rotation animation seems very chaotic in the sense that I would expect at least one point (in this case, the origin?) is not rotating.

You need to first change the origin of vertices of your polygon, rotate, and translate back. You can do this by hand

(RotationMatrix[theta,axis].(#-pt) + pt)& /@ {pA, pB, pC, pD}

Or, you can combine the transformations using Composition

Composition[
 AffineTransform[{RotationMatrix[theta,axis],pt}],TranslationTransform[-pt]
] /@ {pA, pB, pC, pD}

Or, you can take the previous composition and apply it directly to your Graphics object

GeometricTransformation[ <graphics>, Composition[ ... ]]

This documentation gives a thorough list of what can be done.

Edit: Here's a working animation script

Animate[
  graph /. Graphics3D[prims__, opts : OptionsPattern[]] :> 
    Graphics3D[
      GeometricTransformation[prims,
        Composition[
          AffineTransform[{RotationMatrix[theta, axis], pt}],
          TranslationTransform[-pt]
        ]
      ],
      opts
    ], 
  {theta, 0., 2.*Pi}
]

There's a couple of things to note here. First, GeometricTransformation only appears to work on the primitives themselves, so I had to split out the primitives from the options in Graphics3D via the rule Graphics3D[prims__, opts : OptionsPattern[]]. Also, the transformation itself needs to be within Animate to use the local version of theta.