The above-mentioned augmented matrix is called " affine transformation matrix ", or " projective transformation matrix " ( as it can also be used to perform projective transformations ).
32.
The above-mentioned augmented matrix is called " affine transformation matrix ", or " projective transformation matrix " ( as it can also be used to perform projective transformations ).
33.
For example, projective geometry is about the group of projective transformations that act on the real projective plane, whereas inversive geometry is concerned with the group of inversive transformations acting on the extended complex plane.
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An alternative explanation for the invariance of the cross-ratio is based on the fact that the group of projective transformations of a line is generated by the translations, the homotheties, and the multiplicative inversion.
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We will briefly outline and depict the results for a simple approach to find a " H " and " H "'projective transformation that rectify the image pair from the example scene.
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The choice of a projective embedding of " X ", " modulo " projective transformations is likewise equivalent to the choice of a very ample line bundle on " X ".
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A third equivalence relation, introduced by Gaspard Monge in 1795, occurs in projective geometry : not only ellipses, but also parabolas and hyperbolas, turn into circles under appropriate projective transformations; they all are projectively equivalent figures.
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He noted that every Euclidean congruence is an affine mapping, and each of these is a projective transformation; therefore the group of projectivities contains the group of affine maps, which in turn contains the group of Euclidean congruencies.
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A "'homography "'( or " projective transformation " ) of PG ( 2, " K " ) is a collineation of this type of projective plane which is a linear transformation of the underlying vector space.
40.
Also keep in mind that all conics are basically the same curve in projective since the equations can all be transformed under a projective transformation to x 2 + y 2 = z 2, which just happens to be the equation of the cone in space.
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