Calculation example – Similarity Transformation in 2-D space

Calculate the coordinates of a known point P in an orthogonal coordinate system to a reference orthogonal coordinate system preserving shapes (angles) during transformation, in 2-D space.

Known values: Point P coordinates x, y on initial coordinate system, translation vector parameters ΔΧ , ΔΥ between origin centers of the two coordinate systems, scale factor s, rotation angle a.

Solution: Point P coordinates X, Y on reference coordinate system.

From the geometry of the scene applies:

X=s⋅(x⋅cosa-y⋅sina)+ΔΧ

Υ=s⋅(x⋅sina+y⋅cosa)+ΔΥ

*Rotation angle a is measured counterclockwise from reference coordinate system XY to initial coordinate system xy.