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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 ...
Triangle geometry, and circles associated with triangles, including the nine-point circle, Brocard circle, and Lemoine circle [1] [2] [3] The Problem of Apollonius on constructing a circle tangent to three given circles, and the Malfatti problem of constructing three mutually-tangent circles, each tangent to two sides of a given triangle [1] [3]
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 ...
In plane geometry there are two ways that two circles can be tangent: from the inside (one circle inside the other) or from the outside. In Moebius (or inversive) geometry, these two ways are equivalent, because an inversion, centered on a point which is outside one of the circles but inside the other, exchanges the two pictures.
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the density of the Earth from measurements of its gravity field. It is called an inverse problem because ...
A college student just solved a seemingly paradoxical math problem—and the answer came from an incredibly unlikely place. Skip to main content. 24/7 Help. For premium support please call: 800 ...
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.
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