How to scale rotated matrix in a new coordinate?

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Abdulaziz Abutunis 2017-12-18

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评论： Abdulaziz Abutunis 2017-12-18

Hello All: I have rotated foil data (each point has x,y coordinate) and I want to scale the foil after the rotation. In other words, I want to apply the scaling factor in different coordinate ( say the new coordinate is rotated 15 deg).

Appreciate your suggestions

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Jan 2017-12-18

The question is not clear. You can rotate, translate, sheer and scale the coordinate systems. Scaling means to multiply the coordinates, so this seems to be trivial. Maybe it would help if you show us an example of the inputs and wanted outputs.

Abdulaziz Abutunis 2017-12-18

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Jan Thank you for your response. Yes, my question may be better addressed to algebra or other groups but I thought there may be some commands in Matlab ready to do that. What I meant was that I have an airfoil that I have rotated and then I did some other steps then I need to scale down the airfoil in the newly rotated x-direction only. It seems that even if I want to scale in newly rotated x-direction only but I also should consider the movement of data in y-coordinate to get what I want. I have simplified the what I need in the attached file. Thanks

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Answer 1

Jan 2017-12-18

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Your question is still vague, so I can post a vague answer only:

If you have a set of points, and want to scale the values "along a direction", rotate the points such, that the direction vector point to the x axis. Then divide the x component of the coordinates and rotate the point back. You can express this as 3 matrices:

T = Rot(alpha) * Scale * Rot(-alpha) =
  = [cos(alpha) -sin(alpha);  sin(alpha) cos(alpha)] * ...
    [0.5, 0; 0, 1] * ...
    [cos(alpha)  sin(alpha); -sin(alpha) cos(alpha)]

You can multiply this to get the matrix, which scales the points in the direction vector, which includes the signed angle alpha with the x axis.