I want to solve an exercise regarding the connection between the projective model of möbius geometry and pencils of circles.
Consider the projective model of two-dimensional Möbius geometry P($R^{3,1}$) where points outside the möbius quadric correspond to circles. Show that a projective line given by a two-dimensional subspace
of signature $(+-)$ corresponds to a one-parameter family of circles all passing through the same two points (elliptic pencil).
of of signature $(++)$ corresponds to a one-parameter family of circles that do not intersect each other at any point (hyberbolic pencil).
of signature $(+0)$ corresponds to a one-parameter family of circles all tangent to each other at a single common point (parabolic pencil).
Of which type is the family of orthogonal circles in each case?
I do know that circles in the extended euclidean plane $R^2 \cup$ ${\infty}$ correspond to hyperplanes in the projective space P($R^{3,1}$) intersecting the unit sphere $S^2$. Each hyperplane corresponds to a point outside $S^2$ where the point is orthogonal to the plane w.r.t the Lorentz inner product $\langle\cdot,\cdot\rangle$.
Given a two-dimensional subspace U of $R^{3,1}$ there are three possible signatures that can be obtained by restricting $\langle\cdot,\cdot\rangle$ to U.
So each point [p] in the projective line P(U) indeed corresponds to a circle.
But I have some difficulty proving the claims since I do not see how the signature affects the pencil.
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