![]() This can be illustrated by the figure below:įigure above: A line projectivity. However, the composition is a projectivity because a perspectivity isĪ projectivity, and projectivities form a group (closed), so that the composition of two The composition of two perspectivities is not in generalĪ perspectivity. The difference between a perspectivity and projectivity is made clear by considering the composition of two perspectivities. The distinctive property of a perspectivity is thatlines joining corresponding points are concurrent. What is the main difference between perspectivity and general projectivity? The transformation is not defined on the line L, where P intersects the plane parallel to Q and going throught O. Strictly speaking it gives a transformation from one plane to another, but if we identify the two planes by (for example) fixing a cartesian system in each, we get a projective transformation from the plane to itself, as shown in the figure below.įigure above: A perspective transformation with center O, mapping the plane P to the plane Q. It can be seen as a common example of projective transformation. Perspective transformation projects a 3D geometric object into a 2D plane. The term perspecive transformation is also commonly seen. Projective transformations (if not affine) are not defined on all of the plane, but only on the complement of a line (the missing line is “mapped to infinity”).įigure above: In projective transformations (if not affine), a vanishing line in infinity can be warped to be a finite line. Any plane projective transformation can be expressed by an invertible 3×3 matrix in homogeneous coordinates:Ĭonversely, any invertible 3×3 matrix defines a projective transformation of the plane. ![]() Projective TransformationĪ transformation that maps lines to lines (but does not necessarily preserve parallelism) is a projective transformation. The corresponding matrix in homogeneous coordinates isĮvery affine transformation is obtained by composing a scaling transformation with an isometry, or a shear with a homothety and an isometry. Where r is the shearing factor (see Figure 1). The corresponding matrix in homogeneous coordinates isĪ shear preserving horizontal lines has the form Where a,b != 0 are the scaling factors (real numbers). There are two important particular cases of such transformations:Ī nonproportional scaling transformation centered at the origin has the form Projective: lines mapped to lines, but parallelism may not be kept Īffine: collinearity and parallelism are both keptĪ transformation that preserves lines and parallelism (maps parallel lines to parallel lines) is an affine transformation. Several researchers crack the genetic code. Animation 22: DNA words are three letters long. Francis Crick describes RNA and its role and Paul Zamecnick explains protein synthesis. Animation 21: RNA is an intermediary between DNA and protein. Key differences about projective and affine transformations: 3D animation of translation: RNA to protein. Typical Types of Transformation of 2D Planes
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