what is the three parameter family of plane projective transformations which fix a unit circle at the origin(that is map the unit circle to itself)? I understand that one such transformation is a rotation but that accounts for just one parameter, what are the other two?
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Plane projective transformations can be viewed as linear operators in a 3D vector space V. A circle (more generally an ellipse) in the projective plane P(V) corresponds to a cone C in V. Such a cone is the vanishing locus of an indefinite quadratic form Q on V. Thus the operators we look for are operators preserving Q. In physical lingo, V is a 3D Minkowski space and the operators are Lorentz transformations. They can be described as follows. Choose a basis e1, e2, e3 for V in which Q takes canonical form -1 0 0 0 1 0 0 0 1 One family of Q-preserving operators is the rotations you mentioned: 1 0 0 0 cos(alpha) -sin(alpha) 0 sin(alpha) cos(alpha) Another family consists of hyperbolic rotations (in physical lingo, boosts): cosh(theta) sinh(theta) 0 sinh(theta) cosh(theta) 0 0 0 1 Analogously, we have hyperbolic rotations in another plane: cosh(theta) 0 sinh(theta) 0 1 0 sinh(thata) 0 cosh(theta) We can form a product of 3 matrices, with one of each family. This will yield a general operator of the desired form. This is completely analogous to decomposition of 3D rotations into rotations around separate axes (i.e. Euler angles). Note that 3D rotations are operators preserving a definite quadratic form |
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