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In geometry, inversive geometry is the study of inversion, a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry become much more tractable when an inversion is applied.
A natural setting for problem of Apollonius is inversive geometry. [4] [12] The basic strategy of inversive methods is to transform a given Apollonius problem into another Apollonius problem that is simpler to solve; the solutions to the original problem are found from the solutions of the transformed problem by undoing the transformation ...
In inversive geometry, an inverse curve of a given curve C is the result of applying an inverse operation to C. Specifically, with respect to a fixed circle with center O and radius k the inverse of a point Q is the point P for which P lies on the ray OQ and OP·OQ = k 2. The inverse of the curve C is then the locus of P as Q runs over C.
Inversive geometry itself can be performed with the larger system known as Conformal Geometric Algebra(CGA), of which Plane-based GA is a subalgebra. CGA is also usually applied to 3D space, and is able to model general spheres, circles, and conformal (angle-preserving) transformations, which include the transformations seen on the Poincare ...
When we recently wrote about the toughest math problems that have been solved, we mentioned one of the greatest achievements in 20th-century math: the solution to Fermat’s Last Theorem. Sir ...
Example of a 2-dimensional figure with central symmetry, invariant under point reflection Dual tetrahedra that are centrally symmetric to each other In geometry , a point reflection (also called a point inversion or central inversion ) is a geometric transformation of affine space in which every point is reflected across a designated inversion ...
Inversive geometry#Circle inversion, a transformation of the Euclidean plane that maps generalized circles to generalized circles; Inversion in a point, or point reflection, a kind of isometric (distance-preserving) transformation in a Euclidean space; Inversion transformation, a conformal transformation (one which preserves angles of intersection)
An analogue of the Beckman–Quarles theorem holds true for the inversive distance: if a bijection of the set of circles in the inversive plane preserves the inversive distance between pairs of circles at some chosen fixed distance , then it must be a Möbius transformation that preserves all inversive distances. [3]
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