Projective Geometry/Classic/Projective Transformations

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This section gives a classical description of projective transformations of the projective line, plane, and 3-space.

A projective transformation is a transformation used in projective geometry: it is the composition of a pair of perspective projections. It describes what happens to the perceived positions of observed objects when the point of view of the observer changes. Projective transformations do not preserve sizes or angles but do preserve incidence and cross-ratio: two properties which are important in projective geometry. A projective transformation can also be called a projectivity. Projectivities form a group.[1]

As important special cases, a projective transformation can be in the (real) one-dimensional projective line RP1, the two-dimensional projective plane RP2, and the three-dimensional projective 3-space RP3; see:


  1. Richard Hartley and Andrew Zisserman (2003). Multiple View Geometry in computer vision. Cambridge University Press. ISBN 0-521-54051-8.