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Next, a translation of axes can reduce an equation of the form to an equation of the same form but with new variables (x', y') as coordinates, and with D and E both equal to zero (with certain exceptions—for example, parabolas). The principal tool in this process is "completing the square."
A point P has coordinates (x, y) with respect to the original system and coordinates (x′, y′) with respect to the new system. [1] In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle . A rotation of axes in more than two dimensions is defined similarly.
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is ...
C – translation vector. Contains the three translations along the coordinate axes; μ – scale factor, which is unitless; if it is given in ppm, it must be divided by 1,000,000 and added to 1. R – rotation matrix. Consists of three axes (small [clarification needed] rotations around each of the three coordinate axes) r x, r y, r z.
Although a translation is a non-linear transformation in a 2-D or 3-D Euclidean space described by Cartesian coordinates (i.e. it can't be combined with other transformations while preserving commutativity and other properties), it becomes, in a 3-D or 4-D projective space described by homogeneous coordinates, a simple linear transformation (a ...
Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as , see conventions in spherical coordinates). As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. θ has a range ...
In the passive transformation (right), point P stays fixed, while the coordinate system rotates counterclockwise by an angle θ about its origin. The coordinates of P ′ after the active transformation relative to the original coordinate system are the same as the coordinates of P relative to the rotated coordinate system.
In geodesy, geographic coordinate conversion is defined as translation among different coordinate formats or map projections all referenced to the same geodetic datum. [1] A geographic coordinate transformation is a translation among different geodetic datums. Both geographic coordinate conversion and transformation will be considered in this ...