       Re: how to calculate the 3D centre point of rotation given the angle of

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• Subject: [mg132128] Re: how to calculate the 3D centre point of rotation given the angle of
• From: carlson at wolfram.com
• Date: Fri, 20 Dec 2013 05:33:42 -0500 (EST)
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```On Dec 17, 2013, at 3:42 AM, parr.will at googlemail.com wrote:

> Hi,
>
> I've been looking at the Mathematica function: RotationTransform.
>
> in the mathematica help it says: =09
>
> RotationTransform[Theta,w,p]
> gives a 3D rotation (Theta in radians/degrees) around the axis w anchored at the point p.
>
> I have a problem where I have some 4 points in 3D space before and after a rotation is applied to them. I have the 3D rotation in radians/degrees and the axis (w), but want to find the point (p) around which the rotation occurs.
>
> eg:
> (*points before rotation*)
> pts1{{-21.365, -1.61273, 2.41973}, {-41.0366, -4.33682,
>  4.78811}, {-18.1104, -20.673, 7.53}, {-19.804, 3.79904, 21.6102}};
> (*points after rotation*)
> pts2={{-17.9409, -3.2446, -7.46078}, {-35.9907, -7.76684, =
-14.7927},{-14.3971, -22.658, -4.21113}, {-25.7926, -1.61099, 10.8609}};
>
> theta(*in degrees*)=29.3405
>
> axis of rotation(*normalised*)={0.347494, -0.904472, 0.247341}
>
> (*axis of rotation not normalised={19.9099, -51.8224, 14.1716}*)
>
> can anyone help me find the centre point of rotation (p) please?
>
> best wishes,
>
> Will

FindGeometricTransform will give you the transformation:

t = FindGeometricTransform[pts2, pts1][]

t /@ pts1

Then Solve will give you the fixed point, ie, the center of rotation:

{x, y, z} /. Solve[t[{x, y, z}] == {x, y, z}, {x, y, z}][]

Chris

```

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