The Compendium Geometry is an eBook providing facts, formulas and explanations about geometry. 
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Coordinate Transform
TranslationIn order to translate ("shift") a point P_{1} to the new position P_{1}' one has to add the translation vector t [t_{x}, t_{y}]:
P_{1}' = P_{1} + t In terms of individual coordinate values, the new position [x', y'] is calculated as follows:
x' = x + t_{x} RotationThe rotation of a point around the origin O of the coordinate system does not change the length of the vector OP. Note that positive rotation angles correspond to counterclockwise rotation:x' = rcos(φ_{1}β) ReflectionA reflection flips all the points in the plane over a line, which is called the mirror. Points lying on the mirror line are called invariant points (points that map onto themselves, i.e. P_{6} in the figure below which is invariant to the reflection about the yaxis). A reflection changes the sense of the figures in the plane.There are several special cases of 2D reflections:
ScalingEach individual coordinate of each point in the plane is contracted or expanded by the scaling factors S_{x} and S_{y}.In the figure below the scaling factors are S_{x}=2 and S_{y}=0.5, thus the xcoordinates are multiplied by 2 (expanded), and the ycoordinates are divided by 2 (contracted). Consequently the point P_{1}, for example, is transformed into point P_{1}' by the following equations:
x_{1}' = 2x_{1}


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