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Combining rotations (in 2D space)


Browseruk

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The following image shows a set of points (A,,B,C,p) rotated (in this example 360/21°) anticlockwise around the origin o. The resultant set of points (A', B', C') are then rotated clockwise (18°) around the resultant point p'. My goal is to derive a single centre of rotation that will rotate the set of points [A,B,C] to [A' ',B' ', C' '], given only the position of p, and the two rotations: 

blob.thumb.png.c1c1563ba7a9581375b11bd90023dc7c.png

Graphically, by projecting normals to the centres of the lines A-A' ', B-B' ', C-C' ', (also p-p' '), where they cross at point r, gives me the center of a single rotation I seek. 

Question: how to do that mathematically given only: p = (7.160299318411282, 0) rotation1 = -360/21° & rotation2 = 18°?

Caveat: The above procedure does not work for all combinations of two rotations. Eg. In the next image p = (50,0) and the rotations are (-45° & +45°); which results in the normals to the bisectors all being parallel!

blob.thumb.png.42ef0db4bff7f83280f3645a8875ce9d.png 

I know that affine transformation using homogeneous coordinates can be composed [https://en.wikipedia.org/wiki/Transformation_matrix#Composing_and_inverting_transformations], but I am stuck for how to utilise that here as in the environment in which I am doing this (LUA embedded in a FEA package), I only have two mechanisms available: rotation about a point and translation in the XY plane.

Question2: Assuming that I get a solution to Q1 above, is my only option to deal with the Caveat case, to compare the angles of rotation and do something different if they are equal?

 

Thanks.

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