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In geometry, an inversion is a transformation that maps all circles into circles, where by a circle one may also mean a line (a circle with infinite radius). Table of Geometry, from the 1728 Cyclopaedia. ...
In mathematics, a transformation in elementary terms is any of a variety of different operations from geometry, such as rotations, reflections and translations. ...
Partial plot of a function f. ...
In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ...
A line, or straight line, can be described as an (infinitely) thin, (infinitely) long, perfectly straight curve (the term curve in mathematics includes straight curves). In Euclidean geometry, exactly one line can be found that passes through any two points. ...
The infinity symbol ∞ in several typefaces The word infinity comes from the Latin infinitas or unboundedness. ...
Remote Authentication Dial In User Service (RADIUS) is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. ...
Circle inversion
Inverse of a point
 P ' is the inverse of P with respect to the circle. In the plane, the inverse of a point P in respect to a circle of center O and radius R is a point P' such that P and P' are on the same ray going from O, and OP times OP' equals the radius squared, Image File history File links Inversion_illustration1. ...
Image File history File links Inversion_illustration1. ...
 This circle in respect to which inversion is performed will be called the reference circle.
 The inverse in respect to the red circle of a circle going through O (blue), is a line not going through O (green), and vice-versa.
 The inverse in respect to the red circle of a circle not going through O (blue), is a circle not going through O (green), and vice-versa.
 A procedure to construct the inverse P' of a point P outside a circle C. Let r be the radius of C. Since the triangles OPN and OP'N are similar, OP is to r as r is to OP'. One can check that the inverse of a point inside the reference circle is outside the reference circle and vice-versa. A point on the circle stays in the same place under inversion. The center of the circle gets transformed to infinity, and the infinity gets transformed to the circle center. This all can be summarized by saying that the closer a point is to the center, the further it goes when inverted, and the other way around, with the points on the circle staying where they are. Image File history File links Inversion_illustration2. ...
Image File history File links Inversion_illustration2. ...
Image File history File links Inversion_illustration3. ...
Image File history File links Inversion_illustration3. ...
Image File history File links InvertedCircle. ...
Image File history File links InvertedCircle. ...
Properties One may invert a set of points in respect to a circle by inverting each of the points which make it up. The following properties is what makes circle inversion important. - A line not passing through the center of the reference circle is inverted into a circle passing through the center of the reference circle, and vice versa; whereas a line passing through the center of the reference circle is inverted into itself.
- A circle not passing through the center of the reference circle is inverted into a circle not passing through the center of the reference circle. The circle (or line) after inversion stays as before if and only if it is orthogonal to the reference circle at their points of intersection.
Application Note that the center of a circle being inverted and the center of the circle as result of inversion are collinear with the center of the reference circle. This fact could be useful in proving the Euler line of the intouch triangle of a triangle coincides with its OI line. The proof roughly goes as below: In geometry, Eulers line (red line in the image), named after Leonhard Euler, is the line passing through the orthocenter (blue), the circumcenter (green), the centroid (yellow), and the center of the nine-point circle (red point) of any triangle. ...
Invert with respect to the incircle of triangle ABC. The medial triangle of the intouch triangle is inverted into triangle ABC, meaning the circumcenter of the medial triangle, that is, the nine-point center of the intouch triangle, the incenter and circumcenter of triangle ABC are collinear. It has been suggested that this article or section be merged into Stanford University. ...
The red triangle is the medial triangle of the black. ...
A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ...
In addition, two dimensional inversion can be extended to 3-dimensional by making use of a sphere instead.
Inversions in three dimensions Circle inversion is generalizable to sphere inversion in three dimensions. The inversion of a point P in 3D with respect to a reference sphere centered at a point O with radius R is a point P' such that  and the points P and P' are on the same ray going from O.
As with the 2D version, a sphere inverts to a sphere, except that if a sphere passes through the center O of the reference sphere, then it inverts to a plane. Any plane not passing through O, inverts to a sphere touching at O.
Stereographic projection is a special case of sphere inversion. Indeed, consider a sphere B of radius 1 and a plane P touching B at the South Pole S of B. Then P is the stereographic projection of B in respect to the North Pole N of B. Consider a sphere B2 of radius 2 centered at N. The inversion in respect to B2 transforms B into its stereographic projection P. Stereographic projection of a circle of radius R onto the x axis. ...
The Erlangen program 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. Remote Authentication Dial In User Service (RADIUS) is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. ...
In 2 dimensions, 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 in a plane at a fixed distance, called the radius, from a fixed point, the centre. ...
A hyperplane is a concept in geometry. ...
In mathematics, a hypersphere is a sphere which has dimension 3 or higher. ...
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. ...
Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. ...
Inversion of an algebraic curve We may invert a plane algebraic curve given by a single polynomial equation f(x, y) = 0 by setting In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ...
 Clearing denominators, we have the polynomial equations ux2 + uy2 − x = 0,vx2 + vy2 − y = 0, and eliminating x and y from the system of three equations in four unknowns consisting of these two equations and f (for instance, by using resultants) we can readily find the equation of the curve inverted in the unit circle. Now  and applying the transformation again leads back to the original curve. In mathematics, the resultant of two monic polynomials and over a field is defined as the product of the differences of their roots, where and take on values in the algebraic closure of . ...
For example, applying the above transformation to the lemniscate A lemniscate In mathematics, a lemniscate is a type of curve described by a Cartesian equation of the form: Graphing this equation produces a curve similar to . ...
- (x2 + y2)2 = a2(x2 − y2)
gives us - a2(u2 − v2) = 1,
the equation of a hyperbola; since inversion is a birational transformation and the hyperbola is a rational curve, this shows the lemniscate is also a rational curve, which is to say a curve of genus zero. If we apply it to the Fermat curve xn + yn = 1, where n is odd, we obtain In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ...
In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (X:Y:Z) by the Fermat equation Xn + Yn = Zn. ...
- (u2 + v2)n = un + vn.
Any rational point on the Fermat curve has a corresponding rational point on this curve, giving an equivalent formulation of Fermat's Last Theorem. Pierre de Fermat Fermats Last Theorem is one of the most famous theorems in the history of mathematics. ...
Conformal mapping property A transformation is conformal, or angle-preserving, if at every point the Jacobian is a scalar times an orthogonal matrix. This means that if J is the Jacobian, then JJT = kI. Computing the Jacobian in the case zi = xi/||x||2, where ||x||2 = x12 + ... + xn2 gives JJT = kI, with k = 1/||x||4; hence the inversive map is conformal. In mathematics, a conformal map is a function which preserves angles. ...
In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...
Inversive geometry and hyperbolic geometry The (n − 1)-sphere with equation  will have a positive radius so long as a12 + ... + an2 is greater than c, and on inversion gives the sphere  Hence, it will be invariant under inversion if and only if c = 1. But this is the condition of being orthogonal to the unit sphere. Hence we are led to consider the (n − 1)-spheres with equation  which are invariant under inversion, orthogonal to the unit sphere, and have centers outside of the sphere. These together with the subspace hyperplanes separating hemispheres are the hypersurfaces of the Poincaré disc model of hyperbolic geometry. In geometry, the Poincaré disk model, also called the conformal disk model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, and the lines of the geometry are segments of circles contained in the disk orthogonal to the...
Since inversion in the unit sphere leaves the spheres orthogonal to it invariant, the inversion maps the points inside the unit sphere to the outside and vice-versa. This is therefore true in general of orthogonal spheres, and in particular inversion in one of the spheres orthogonal to the unit sphere maps the unit sphere to itself. It also maps the interior of the unit sphere to itself, with points outside the orthogonal sphere mapping inside, and vice-versa; this defines the reflections of the Poincaré disc model if we also include with them the reflections through the diameters separating hemispheres of the unit sphere. These reflections generate the group of isometries of the model, which tells us that the isometries are conformal. Hence, the angle between two curves in the model is the same as the angle between two curves in the hyperbolic space.
External links - Inversion: Reflection in a Circle at cut-the-knot
- Wilson Stother's inversive geometry page
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