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.
The concept of inversion can be generalized to higher-dimensional spaces.
To invert a number in arithmetic usually means to take its reciprocal. A closely related idea in geometry is that of "inverting" a point. In the plane, the inverse of a point P with respect to a reference circle (Ø) with center O and radius r is a point P', lying on the ray from O through P such that
This is called circle inversion or plane inversion. The inversion taking any point P (other than O) to its image P' also takes P' back to P, so the result of applying the same inversion twice is the identity transformation on all the points of the plane other than O (self-inversion). To make inversion an involution it is necessary to introduce a point at infinity, a single point placed on all the lines, and extend the inversion, by definition, to interchange the center O and this point at infinity.
It follows from the definition that the inversion of any point inside the reference circle must lie outside it, and vice versa, with the center and the point at infinity changing positions, whilst any point on the circle is unaffected (is invariant under inversion). In summary, the nearer a point to the center, the further away its transformation, and vice versa.
To construct the inverse P' of a point P outside a circle Ø:
To construct the inverse P of a point P' inside a circle Ø:
There is a construction of the inverse point to A with respect to a circle P that is independent of whether A is inside or outside P.
Consider a circle P with center O and a point A which may lie inside or outside the circle P.
The inversion of a set of points in the plane with respect to a circle is the set of inverses of these points. The following properties make circle inversion useful.
Additional properties include:
For a circle not passing through the center of inversion, the center of the circle being inverted and the center of its image under inversion are collinear with the center of the reference circle. This fact can be used to prove that the Euler line of the intouch triangle of a triangle coincides with its OI line. The proof roughly goes as below:
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.
Any two non-intersecting circles may be inverted into concentric circles. Then the inversive distance (usually denoted δ) is defined as the natural logarithm of the ratio of the radii of the two concentric circles.
In addition, any two non-intersecting circles may be inverted into congruent circles, using circle of inversion centered at a point on the circle of antisimilitude.
The Peaucellier–Lipkin linkage is a mechanical implementation of inversion in a circle. It provides an exact solution to the important problem of converting between linear and circular motion.
If point R is the inverse of point P then the lines perpendicular to the line PR through one of the points is the polar of the other point (the pole).
Poles and polars have several useful properties:
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 starting at 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. A circle, that is, the intersection of a sphere with a secant plane, inverts into a circle, except that if the circle passes through O it inverts into a line. This reduces to the 2D case when the secant plane passes through O, but is a true 3D phenomenon if the secant plane does not pass through O.
The simplest surface (besides a plane) is the sphere. The first picture shows a non trivial inversion (the center of the sphere is not the center of inversion) of a sphere together with two orthogonal intersecting pencils of circles.
The inversion of a cylinder, cone, or torus results in a Dupin cyclide.
A spheroid is a surface of revolution and contains a pencil of circles which is mapped onto a pencil of circles (see picture). The inverse image of a spheroid is a surface of degree 4.
A hyperboloid of one sheet, which is a surface of revolution contains a pencil of circles which is mapped onto a pencil of circles. A hyperboloid of one sheet contains additional two pencils of lines, which are mapped onto pencils of circles. The picture shows one such line (blue) and its inversion.
A stereographic projection usually projects a sphere from a point (north pole) of the sphere onto the tangent plane at the opposite point (south pole). This mapping can be performed by an inversion of the sphere onto its tangent plane. If the sphere (to be projected) has the equation (alternately written ; center , radius , green in the picture), then it will be mapped by the inversion at the unit sphere (red) onto the tangent plane at point . The lines through the center of inversion (point ) are mapped onto themselves. They are the projection lines of the stereographic projection.
The 6-sphere coordinates are a coordinate system for three-dimensional space obtained by inverting the Cartesian coordinates.
One of the first to consider foundations of inversive geometry was Mario Pieri in 1911 and 1912.Edward Kasner wrote his thesis on "Invariant theory of the inversion group".
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 Möbius plane, also known as an inversive plane. The point at infinity is added to all the lines. These Möbius planes can be described axiomatically and exist in both finite and infinite versions.
A model for the Möbius plane that comes from the Euclidean plane is the Riemann sphere.
The cross-ratio between 4 points is invariant under an inversion. In particular if O is the centre of the inversion and and are distances to the ends of a line L, then length of the line will become under an inversion with centre O. The invariant is:
According to Coxeter,the transformation by inversion in circle was invented by L. I. Magnus in 1831. Since then this mapping has become an avenue to higher mathematics. Through some steps of application of the circle inversion map, a student of transformation geometry soon appreciates the significance of Felix Klein’s Erlangen program, an outgrowth of certain models of hyperbolic geometry
The combination of two inversions in concentric circles results in a similarity, homothetic transformation, or dilation characterized by the ratio of the circle radii.
When a point in the plane is interpreted as a complex number with complex conjugate then the reciprocal of z is
Consequently, the algebraic form of the inversion in a unit circle is given by where:
Reciprocation is key in transformation theory as a generator of the Möbius group. The other generators are translation and rotation, both familiar through physical manipulations in the ambient 3-space. Introduction of reciprocation (dependent upon circle inversion) is what produces the peculiar nature of Möbius geometry, which is sometimes identified with inversive geometry (of the Euclidean plane). However, inversive geometry is the larger study since it includes the raw inversion in a circle (not yet made, with conjugation, into reciprocation). Inversive geometry also includes the conjugation mapping. Neither conjugation nor inversion-in-a-circle are in the Möbius group since they are non-conformal (see below). Möbius group elements are analytic functions of the whole plane and so are necessarily conformal.
Consider, in the complex plane, the circle of radius around the point
where without loss of generality, Using the definition of inversion
it is straightforward to show that obeys the equation
and hence that describes the circle of center and radius
When the circle transforms into the line parallel to the imaginary axis
For and the result for is
showing that the describes the circle of center and radius .
When the equation for becomes
As mentioned above, zero, the origin, requires special consideration in the circle inversion mapping. The approach is to adjoin a point at infinity designated ∞ or 1/0 . In the complex number approach, where reciprocation is the apparent operation, this procedure leads to the complex projective line, often called the Riemann sphere. It was subspaces and subgroups of this space and group of mappings that were applied to produce early models of hyperbolic geometry by Beltrami, Cayley, and Klein. Thus inversive geometry includes the ideas originated by Lobachevsky and Bolyai in their plane geometry. Furthermore, Felix Klein was so overcome by this facility of mappings to identify geometrical phenomena that he delivered a manifesto, the Erlangen program, in 1872. Since then many mathematicians reserve the term geometry for a space together with a group of mappings of that space. The significant properties of figures in the geometry are those that are invariant under this group.
For example, Smogorzhevskydevelops several theorems of inversive geometry before beginning Lobachevskian geometry.
In n-dimensional space where there is a sphere of radius r, inversion in the sphere is given by
The transformation by inversion in hyperplanes or hyperspheres in En can be used to generate dilations, translations, or rotations. Indeed, two concentric hyperspheres, used to produce successive inversions, result in a dilation or contraction on the hyperspheres' center. Such a mapping is called a similarity.
When two parallel hyperplanes are used to produce successive reflections, the result is a translation. When two hyperplanes intersect in an (n–2)-flat, successive reflections produce a rotation where every point of the (n–2)-flat is a fixed point of each reflection and thus of the composition.
All of these are conformal maps, and in fact, where the space has three or more dimensions, the mappings generated by inversion are the only conformal mappings. Liouville's theorem is a classical theorem of conformal geometry.
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 n-sphere as the base space. The transformations of inversive geometry are often referred to as Möbius transformations. Inversive geometry has been applied to the study of colorings, or partitionings, of an n-sphere.
The circle inversion map is anticonformal, which means that at every point it preserves angles and reverses orientation (a map is called conformal if it preserves oriented angles). Algebraically, a map is anticonformal if at every point the Jacobian is a scalar times an orthogonal matrix with negative determinant: in two dimensions the Jacobian must be a scalar times a reflection at every point. This means that if J is the Jacobian, then and Computing the Jacobian in the case zi = xi/||x||2, where ||x||2 = x12 + ... + xn2 gives JJT = kI, with k = 1/||x||4, and additionally det(J) is negative; hence the inversive map is anticonformal.
In the complex plane, the most obvious circle inversion map (i.e., using the unit circle centered at the origin) is the complex conjugate of the complex inverse map taking z to 1/z. The complex analytic inverse map is conformal and its conjugate, circle inversion, is anticonformal. In this case a homography is conformal while an anti-homography is anticonformal.
The (n − 1)-sphere with equation
will have a positive radius if 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.
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.
A sphere is a geometrical object in three-dimensional space that is the surface of a ball.
In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles at which curves meet. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.
In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points F1 and F2, known as foci, at distance 2c from each other as the locus of points P so that PF1·PF2 = c2. The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from lemniscatus, which is Latin for "decorated with hanging ribbons". It is a special case of the Cassini oval and is a rational algebraic curve of degree 4.
In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection. For example the mirror image of the small Latin letter p for a reflection with respect to a vertical axis would look like q. Its image by reflection in a horizontal axis would look like b. A reflection is an involution: when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state.
In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space.
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.
In mathematics, a Dupin cyclide or cyclide of Dupin is any geometric inversion of a standard torus, cylinder or double cone. In particular, these latter are themselves examples of Dupin cyclides. They were discovered by Charles Dupin in his 1803 dissertation under Gaspard Monge. The key property of a Dupin cyclide is that it is a channel surface in two different ways. This property means that Dupin cyclides are natural objects in Lie sphere geometry.
In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a group structure, known as a rotation group.
In geometry, hyperbolic motions are isometric automorphisms of a hyperbolic space. Under composition of mappings, the hyperbolic motions form a continuous group. This group is said to characterize the hyperbolic space. Such an approach to geometry was cultivated by Felix Klein in his Erlangen program. The idea of reducing geometry to its characteristic group was developed particularly by Mario Pieri in his reduction of the primitive notions of geometry to merely point and motion.
A generalized circle, also referred to as a "cline" or "circline", is a straight line or a circle. The concept is mainly used in inversive geometry, because straight lines and circles have very similar properties in that geometry and are best treated together.
In geometry, a point reflection or inversion in a point is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invariant under point reflection through its center, it is said to possess central symmetry or to be centrally symmetric.
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 = k2. The inverse of the curve C is then the locus of P as Q runs over C. The point O in this construction is called the center of inversion, the circle the circle of inversion, and k the radius of inversion.
Lie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century. The main idea which leads to Lie sphere geometry is that lines should be regarded as circles of infinite radius and that points in the plane should be regarded as circles of zero radius.
In geometry, a Steiner chain is a set of n circles, all of which are tangent to two given non-intersecting circles, where n is finite and each circle in the chain is tangent to the previous and next circles in the chain. In the usual closed Steiner chains, the first and last (nth) circles are also tangent to each other; by contrast, in open Steiner chains, they need not be. The given circles α and β do not intersect, but otherwise are unconstrained; the smaller circle may lie completely inside or outside of the larger circle. In these cases, the centers of Steiner-chain circles lie on an ellipse or a hyperbola, respectively.
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which the points of the geometry are inside the unit disk, and the straight lines consist of all circular arcs contained within that disk that are orthogonal to the boundary of the disk, plus all diameters of the disk.
In geometry, inversion in a sphere is a transformation of Euclidean space that fixes the points of a sphere while sending the points inside of the sphere to the outside of the sphere, and vice versa. Intuitively, it "swaps the inside and outside" of the sphere while leaving the points on the sphere unchanged. Inversion is a conformal transformation, and is the basic operation of inversive geometry; it is a generalization of inversion in a circle.
In mathematics, a Möbius plane is one of the Benz planes: Möbius plane, Laguerre plane and Minkowski plane. The classical example is based on the geometry of lines and circles in the real affine plane.
Spherical wave transformations leave the form of spherical waves as well as the laws of optics and electrodynamics invariant in all inertial frames. They were defined between 1908 and 1909 by Harry Bateman and Ebenezer Cunningham, with Bateman giving the transformation its name. They correspond to the conformal group of "transformations by reciprocal radii" in relation to the framework of Lie sphere geometry, which were already known in the 19th century. Time is used as fourth dimension as in Minkowski space, so spherical wave transformations are connected to the Lorentz transformation of special relativity, and it turns out that the conformal group of spacetime includes the Lorentz group and the Poincaré group as subgroups. However, only the Lorentz/Poincaré groups represent symmetries of all laws of nature including mechanics, whereas the conformal group is related to certain areas such as electrodynamics. In addition, it can be shown that the conformal group of the plane is isomorphic to the Lorentz group.