The Compendium Geometry is an eBook providing facts, formulas and explanations about geometry.

Coordinate Transform


In order to translate ("shift") a point P1 to the new position P1' one has to add the translation vector t [tx, ty]:

P1' = P1 + t

In terms of individual coordinate values, the new position [x', y'] is calculated as follows:

x' = x + tx
y' = y + ty


The 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 counter-clockwise rotation:

x' = rcos(φ1-β)
y' = rsin(φ1-β)


A 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. P6 in the figure below which is invariant to the reflection about the y-axis). A reflection changes the sense of the figures in the plane.

There are several special cases of 2D reflections:

  • Reflection about the x-axis: the x-coordinates are kept constant, the y-coordinates change their sign (P1 and P2, P3 and P4)
  • Reflection about the y-axis: the y-coordinates are kept constant, the x-coordinates change their sign (P1 and P4, P2 and P3)
  • Reflection about the angle bisector of the first quadrant: the x- and y-coordinates are exchanged (P1 and P5)
  • Inversion about the origin O: the signs of the both the x- and y-coordinate change (P1 and P3, P2 and P4)


Each individual coordinate of each point in the plane is contracted or expanded by the scaling factors Sx and Sy.

In the figure below the scaling factors are Sx=2 and Sy=0.5, thus the x-coordinates are multiplied by 2 (expanded), and the y-coordinates are divided by 2 (contracted). Consequently the point P1, for example, is transformed into point P1' by the following equations:

x1' = 2x1
y1' = 0.5y1

Last Update: 2011-01-11