For a homogeneous representation of 3D coordinates, the most general transformation is a projective transformation, represented by a 4 \ times 4 matrix \ mathbf { T }.
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However, a projective transformation is a bijection of a projective space, a property " not " shared with the " projections " of this article.
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Examples include projective spaces ( with " G " the group of projective transformations ) and spheres ( with " G " the group of conformal transformations ).
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Since the cross-ratio is invariant under projective transformations, it follows that the hyperbolic distance is invariant under the projective transformations that preserve the conic " C ".
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Since the cross-ratio is invariant under projective transformations, it follows that the hyperbolic distance is invariant under the projective transformations that preserve the conic " C ".
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It is sufficient to prove the theorem when the conic is a circle, because any ( non-degenerate ) conic can be reduced to a circle by a projective transformation.
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The projective general linear group of order " n " + 1, of projective transformations, is contained in the Cremona group of order " n ".
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In fact, proving they're all multiple of the SFF proves the fact that their non-degeneracy only depends on " M " up to projective transformation.
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The topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry ( the study of differential invariants of the projective transformations ).
30.
Any two polyhedra with the same face lattice and the same midsphere can be transformed into each other by a projective transformation of three-dimensional space that leaves the midsphere in the same position.
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