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Encyclopedia > Inversive geometry

In mathematics, inversive geometry is the geometry of circles and the set of transformations that map all circles into circles. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. ... In Euclidean geometry, a circle is the set of all points at a fixed distance, called the radius, from a fixed point, called the centre (center). ... Transformation may refer to: In molecular biology: In genetics transformation involves the genetic alteration of a cell resulting from the introduction, uptake and expression of foreign DNA. In cell division, the transformation process converts normal cells into cells that will continue to divide without limit. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...


In the plane, the inverse of a point P inside a circle with center O is collinear with P and lying on the same side of O as P such that the radius of the circle is the geometric mean between the distances of P and its inverse to O, i.e., let P′ be the inverse of P then OP times OP′ equals the radius squared. The geometric mean of a set of positive data is defined as the product of all the members of the set, raised to a power equal to the reciprocal of the number of members. ...


For two dimensions, conformal geometry with reflections aka inversive geometry is the Riemann sphere with meromorphic functions. In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ... In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles fore the function. ...


In the spirit of the Erlangen program, inversive geometry is the study of transformations generated by the Euclidean transformations together with inversions, which in coordinate form, basically are conjugate to An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. ... In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ... In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure. ...

where r is the radius of the inversion. In classical geometry, a radius of a circle or sphere is any line segment with one endpoint on the circle (i. ...


In 2D with r=1 this is circle inversion with respect to the unit circle. In the complex plane this corresponds to taking the reciprocal of the conjugate.


As said, in inversive geometry there is no distinction made between a straight line and a circle (or hyperplane and hypersphere): a line is just nothing more and nothing less than a circle in its particular embedding in a Euclidean geometry (with a point added at infinity) and one can always be transformed into another. A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ... In Euclidean geometry, a circle is the set of all points at a fixed distance, called the radius, from a fixed point, called the centre (center). ... In geometry, a hyperplane is a linear, affine, or projective subspace of codimension 1. ... A hypersphere is a higher-dimensional analogue of a sphere. ... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ...


Duality of inversive geometry and hyperbolic geometry

This is usually referred to as the (Euclidean) conformal geometry/hyperbolic geometry duality. An (n − 1)-dimensional inversive geometry is dual to an n-dimensional hyperbolic geometry. There is now a one-to-one correspondence between (n − 2)-dimensional circles (or hyperspheres if you wish) and (n − 1)-dimensional hyperplanes in hyperbolic geometry. An inversion about this hypersphere corresponds to a reflection about the hyperplane. This can be seen in the Poincaré disc model and/or the Poincaré half-plane model where the (n − 2)-dimensional hypersphere is the boundary of the (n − 1)-dimensional hyperplane. A triangle immersed in a saddle-shape plane (an hyperbolic paraboloid), as well as two diverging parallel lines. ... A dual is a pair – a grouping of two. ... A triangle immersed in a saddle-shape plane (an hyperbolic paraboloid), as well as two diverging parallel lines. ... In non-Euclidean geometry, the Poincaré model is a model of two-dimensional hyperbolic geometry as a homogeneous space for the group of Möbius transformations. ...


This is the Wick-rotated version of the AdS/CFT duality. In fact, since most calculations are performed in the Wick-rotated model, this is the duality which is really being used. In physics, a Wick rotation is the process by which a theory in Euclidean space is analytically continued into one in Minkowski space and vice versa. ... In physics, the AdS/CFT correspondence is the equivalence between a string theory or supergravity defined on some sort of Anti de Sitter space and a conformal field theory defined on its boundary whose dimension is lower by one. ...


See also AdS/CFT. In physics, the AdS/CFT correspondence is the equivalence between a string theory or supergravity defined on some sort of Anti de Sitter space and a conformal field theory defined on its boundary whose dimension is lower by one. ...


Computer program

The following QBasic program can be used to interactively explore the nature of inversive transformations: The opening screen of QBasic 1. ...

 'PROGRAM INVGEOM.BAS pi = 4 * ATN(1) SCREEN 13 WINDOW (0, 0)-(300, 3 / 4 * 300) 'diatessaron newx = 143 newy = 98 newrad = 14 cx = 150 cy = 100 R = 30 main: CLS 'given circle FOR thet = 0 TO 2 * pi STEP .01 PSET (newx + newrad * COS(thet), newy - newrad * SIN(thet)) NEXT COLOR 2 'identity circle which defines inversive transformation FOR thet = 0 TO 2 * pi STEP .01 PSET (cx + R * COS(thet), cy - R * SIN(thet)) NEXT COLOR 15 'transformed version of given circle FOR thet = 0 TO 2 * pi STEP .01 x = newx + newrad * COS(thet) y = newy - newrad * SIN(thet) xdif = x - cx ydif = y - cy rho = SQR(xdif ^ 2 + ydif ^ 2) rhop = R ^ 2 / rho xdifp = xdif / rho * rhop 'xdifp = xdif / rho^2 * R^2 ydifp = ydif / rho * rhop xp = cx + xdifp yp = cy + ydifp PSET (xp, yp) NEXT waiting: k$ = INKEY$ IF k$ = "" THEN GOTO waiting 'move given circle one step to the left IF LCASE$(k$) = "q" THEN newx = newx - 1 'move given circle one step to the right IF LCASE$(k$) = "w" THEN newx = newx + 1 'move given circle one step up IF LCASE$(k$) = "a" THEN newy = newy - 1 'move given circle one step down IF LCASE$(k$) = "s" THEN newy = newy + 1 'decrease radius of given circle by one unit IF LCASE$(k$) = "z" THEN newrad = newrad - 1 'increase radius of given circle by one unit IF LCASE$(k$) = "x" THEN newrad = newrad + 1 'move identity circle one step to the left IF LCASE$(k$) = "e" THEN cx = cx - 1 'move identity circle one step to the right IF LCASE$(k$) = "r" THEN cx = cx + 1 'move identity circle one step up IF LCASE$(k$) = "d" THEN cy = cy - 1 'move identity circle one step down IF LCASE$(k$) = "f" THEN cy = cy + 1 'decrease the radius of the identity circle by one unit IF LCASE$(k$) = "c" THEN R = R - 1 'increase the radius of the identity circle by one unit IF LCASE$(k$) = "v" THEN R = R + 1 GOTO main 

The inversive transformation is defined by an "identity circle" which is shown in green. A pair of white circles represent the "given circle" and the "transformed version of the given circle". While the program is running, the given circle can be moved with the keys: Q (left), W (right), A (up), D (down), Z (decrease radius), X (increase radius). The identity circle can be moved with the keys: E (left), R (right), D (up), F (down), C (decrease radius), V (increase radius).


External link

  • Inversion: Reflection in a Circle

  Results from FactBites:
 
Inversive geometry - definition of Inversive geometry in Encyclopedia (404 words)
In mathematics, inversive geometry is the geometry of circles and the set of transformations that map all circles into circles.
Note that in inversive geometry, there is no distinction made between a straight line and a circle (or hyperplane and hypersphere): a line is just nothing more and nothing less than a circle in its particular embedding in a Euclidean geometry (with a point added at infinity) and one can always be transformed into another.
An (n − 1)-dimensional inversive geometry is dual to an n-dimensional hyperbolic geometry.
  More results at FactBites »

 
 

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