The addition of a point at infinity to the space obviates the distinction between hyperplane and hypersphere; higher dimensional inversive geometry is frequently studied then in the presumed context of an M�bius transformations.
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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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In M�bius or inversive geometry, lines are viewed as circles through a point " at infinity " and for any line and any circle, there is a M�bius transformation which maps one to the other.
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Hyperbolic motions are often taken from inversive geometry : these are mappings composed of reflections in a line or a circle ( or in a hyperplane or a hypersphere for hyperbolic spaces of more than two dimensions ).
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More recently the mathematical structure of inversive geometry has been interpreted as an incidence structure where the generalized circles are called " blocks " : In incidence geometry, any affine plane together with a single point at infinity forms a M�bius plane, also known as an " inversive plane ".
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Again, " n "-dimensional anti-de Sitter space and ( " n " " 1 )-dimensional conformal space with " Lorentzian " signature ( in contrast with conformal space with " Euclidean " signature, which is identical to inversive geometry, for three dimensions or greater ) have isomorphic automorphism groups, but are distinct geometries.
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