Ingeometry,inversive geometry is the study ofinversion, a transformation of theEuclidean plane that mapscircles orlines 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. Inversion seems to have been discovered by a number of people contemporaneously, includingSteiner (1824),Quetelet (1825),Bellavitis (1836),Stubbs andIngram (1842–3) andKelvin (1845).^[1]The concept of inversion can begeneralized to higher-dimensional spaces.

To invert a number in arithmetic usually means to take itsreciprocal. A closely related idea in geometry is that of "inverting" a point. In theplane, theinverse of a pointP with respect to areference circle (Ø) with centerO and radiusr is a pointP', lying on the ray fromO throughP such that

${\displaystyle OP\cdot O{P}^{\mathrm{\prime }}=r^2.}$

This is calledcircle inversion orplane inversion. The inversion taking any pointP (other thanO) to its imageP' also takesP' back toP, so the result of applying the same inversion twice is the identity transformation which makes it aself-inversion (i.e. an involution).^[2][3] To make the inversion atotal function that is also defined forO, it is necessary to introduce apoint at infinity, a single point placed on all the lines, and extend the inversion, by definition, to interchange the centerO 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 (isinvariant under inversion). In summary, for a point inside the circle, the nearer the point to the center, the further away its transformation. While for any point (inside or outside the circle), the nearer the point to the circle, the closer its transformation.

Toconstruct the inverseP' of a pointP outside a circleØ:

• Draw the segment fromO (center of circleØ) toP.
• LetM be the midpoint ofOP. (Not shown)
• Draw the circlec with centerM going throughP. (Not labeled. It's the blue circle)
• LetN andN' be the points whereØ andc intersect.
• Draw segmentNN'.
• P' is whereOP andNN' intersect.

To construct the inverseP of a pointP' inside a circleØ:

• Draw rayr fromO (center of circleØ) throughP'. (Not labeled, it's the horizontal line)
• Draw lines throughP' perpendicular tor. (Not labeled. It's the vertical line)
• LetN be one of the points whereØ ands intersect.
• Draw the segmentON.
• Draw linet throughN perpendicular toON.
• P is where rayr and linet intersect.

There is a construction of the inverse point toA with respect to a circleØ that isindependent of whetherA is inside or outsideØ.^[4]

Consider a circleØ with centerO and a pointA which may lie inside or outside the circleØ.

• Take the intersection pointC of the rayOA with the circleØ.
• Connect the pointC with an arbitrary pointB on the circleØ (different fromC and from the point onØ antipodal toC)
• Leth be the reflection of rayBA in lineBC. Thenh cuts rayOC in a pointA'.A' is the inverse point ofA with respect to circleØ.^{[4]: § 3.2}

• 
  The inverse, with respect to the red circle, of a circle going throughO (blue) is a line not going throughO (green), and vice versa.
• 
  The inverse, with respect to the red circle, of a circlenot going throughO (blue) is a circle not going throughO (green), and vice versa.
• 
  Inversion with respect to a circle does not map the center of the circle to the center of its image

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.

• A circle that passes through the centerO of the reference circle inverts to a line not passing throughO, but parallel to the tangent to the original circle atO, and vice versa; whereas a line passing throughO is inverted into itself (but not pointwise invariant).^[5]
• A circle not passing throughO inverts to a circle not passing throughO. If the circle meets the reference circle, these invariant points of intersection are also on the inverse circle. A circle (or line) is unchanged by inversionif and only if it isorthogonal to the reference circle at the points of intersection.^[5]

Additional properties include:

• If a circleq passes through two distinct points A and A' which are inverses with respect to a circlek, then the circlesk andq are orthogonal.
• If the circlesk andq are orthogonal, then a straight line passing through the center O ofk and intersectingq, does so at inverse points with respect tok.
• Given a triangle OAB in which O is the center of a circlek, and points A' and B' inverses of A and B with respect tok, then

${\displaystyle \mathrm{\angle }OAB=\mathrm{\angle }O{B}^{\prime }{A}^{\prime }\text{ and }\mathrm{\angle }OBA=\mathrm{\angle }O{A}^{\prime }{B}^{\prime }.}$

• The points of intersection of two circlesp andq orthogonal to a circlek, are inverses with respect tok.
• If M and M' are inverse points with respect to a circlek on two curves m and m', also inverses with respect tok, then the tangents to m and m' at the points M and M' are either perpendicular to the straight line MM' or form with this line an isosceles triangle with base MM'.
• Inversion leaves the measure of angles unaltered, but reverses the orientation of oriented angles.^[6]

• Inversion of a line is a circle containing the center of inversion; or it is the line itself if it contains the center
• Inversion of a circle is another circle; or it is a line if the original circle contains the center
• Inversion of a parabola is acardioid
• Inversion of hyperbola is alemniscate of Bernoulli

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 arecollinear with the center of the reference circle. This fact can be used to prove that theEuler line of theintouch triangle of a triangle coincides with its OI line. The proof roughly goes as below:

Invert with respect to theincircle of triangleABC. Themedial triangle of the intouch triangle is inverted into triangleABC, meaning the circumcenter of the medial triangle, that is, the nine-point center of the intouch triangle, the incenter and circumcenter of triangleABC arecollinear.

Any two non-intersecting circles may be inverted intoconcentric circles. Then theinversive distance (usually denoted δ) is defined as thenatural logarithm of the ratio of the radii of the two concentric circles.

In addition, any two non-intersecting circles may be inverted intocongruent circles, using circle of inversion centered at a point on thecircle of antisimilitude.

ThePeaucellier–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 pointR is the inverse of pointP then the linesperpendicular to the linePR through one of the points is thepolar of the other point (thepole).

Poles and polars have several useful properties:

• If a pointP lies on a linel, then the poleL of the linel lies on the polarp of pointP.
• If a pointP moves along a linel, its polarp rotates about the poleL of the linel.
• If two tangent lines can be drawn from a pole to the circle, then its polar passes through both tangent points.
• If a point lies on the circle, its polar is the tangent through this point.
• If a pointP lies on its own polar line, thenP is on the circle.
• Each line has exactly one pole.

Circle inversion is generalizable to sphere inversion in three dimensions. The inversion of a pointP in 3D with respect to a reference sphere centered at a pointO with radiusR is a pointP ' on the ray with directionOP such that${\displaystyle OP\cdot O{P}^{\mathrm{\prime }}=||OP||\cdot ||O{P}^{\mathrm{\prime }}||=R^2}$. As with the 2D version, a sphere inverts to a sphere, except that if a sphere passes through the centerO of the reference sphere, then it inverts to a plane. Any plane passing throughO, inverts to a sphere touching atO. A circle, that is, the intersection of a sphere with a secant plane, inverts into a circle, except that if the circle passes throughO it inverts into a line. This reduces to the 2D case when the secant plane passes throughO, but is a true 3D phenomenon if the secant plane does not pass throughO.

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 aDupin cyclide.

A spheroid is asurface 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.

Astereographic projection usually projects a sphere from a point${\displaystyle N}$ (north pole) of the sphere onto the tangent plane at the opposite point${\displaystyle S}$ (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${\displaystyle x^2+y^2+z^2=-z}$ (alternately written${\displaystyle x^2+y^2+(z+\frac{1}{2}{)}^2=\frac{1}{4}}$; center${\displaystyle (0,0,-0.5)}$, radius${\displaystyle 0.5}$, green in the picture), then it will be mapped by the inversion at the unit sphere (red) onto the tangent plane at point${\displaystyle S=(0,0,-1)}$. The lines through the center of inversion (point${\displaystyle N}$) are mapped onto themselves. They are the projection lines of the stereographic projection.

The6-sphere coordinates are a coordinate system for three-dimensional space obtained by inverting theCartesian coordinates.

One of the first to consider foundations of inversive geometry wasMario Pieri in 1911 and 1912.^[7]Edward Kasner wrote his thesis on "Invariant theory of the inversion group".^[8]

More recently themathematical structure of inversive geometry has been interpreted as anincidence structure where the generalized circles are called "blocks": Inincidence geometry, anyaffine plane together with a singlepoint at infinity forms aMöbius plane, also known as aninversive 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.

Amodel for the Möbius plane that comes from the Euclidean plane is theRiemann sphere.

Thecross-ratio between 4 points${\displaystyle x,y,z,w}$ is invariant under an inversion. In particular if O is the centre of the inversion and${\displaystyle r_1}$ and${\displaystyle r_2}$ are distances to the ends of a line L, then length of the line${\displaystyle d}$ will become${\displaystyle d/(r_1{r}_2)}$ under an inversion with radius 1. The invariant is:

${\displaystyle I=\frac{|x-y||w-z|}{|x-w||y-z|}}$

According to Coxeter,^[9] the transformation by inversion in circle was invented byL. 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 oftransformation geometry soon appreciates the significance ofFelix Klein'sErlangen program, an outgrowth of certain models ofhyperbolic geometry.

The combination of two inversions in concentric circles results in asimilarity,homothetic transformation, or dilation characterized by the ratio of the circle radii.

${\displaystyle x\mapsto R^2\frac{x}{|x{|}^2}=y\mapsto T^2\frac{y}{|y{|}^2}={\left(\frac{T}{R}\right)}^{2}x.}$

When a point in the plane is interpreted as acomplex number${\displaystyle z=x+iy,}$ withcomplex conjugate${\displaystyle \overline{z}=x-iy,}$ then thereciprocal ofz is

${\displaystyle \frac{1}{z}=\frac{\overline{z}}{|z{|}^2}.}$

Consequently, the algebraic form of the inversion in a unit circle is given by${\displaystyle z\mapsto w}$ where:

${\displaystyle w=\frac{1}{\overline{z}}=\overline{\left(\frac{1}{z}\right)}}$.

Reciprocation is key in transformation theory as agenerator of theMöbius 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 theconjugation mapping. Neither conjugation nor inversion-in-a-circle are in the Möbius group since they are non-conformal (see below). Möbius group elements areanalytic functions of the whole plane and so are necessarilyconformal.

Consider, in the complex plane, the circle of radius${\displaystyle r}$ around the point${\displaystyle a}$

${\displaystyle (z-a)(z-a{)}^{\ast }=r^2}$

where without loss of generality,${\displaystyle a\in \mathbb{R}.}$ Using the definition of inversion

${\displaystyle w=\frac{1}{z^{\ast }}}$

it is straightforward to show that${\displaystyle w}$ obeys the equation

${\displaystyle w{w}^{\ast }-\frac{a}{(a^2-r^2)}(w+w^{\ast })+\frac{a^2}{(a^2-r^2{)}^2}=\frac{r^2}{(a^2-r^2{)}^2}}$

and hence that${\displaystyle w}$ describes the circle of center$\frac{a}{a^2-r^2}$ and radius$\frac{r}{|a^2-r^2|}.$

When${\displaystyle a\to r,}$ the circle transforms into the line parallel to the imaginary axis${\displaystyle w+w^{\ast }=\frac{1}{a}.}$

For${\displaystyle a\notin \mathbb{R}}$ and${\displaystyle a{a}^{\ast }\ne r^2}$ the result for${\displaystyle w}$ is

showing that the${\displaystyle w}$ describes the circle of center$\frac{a}{(a{a}^{\ast }-r^2)}$ and radius$\frac{r}{\left|a^{\ast }a-r^2\right|}$.

When${\displaystyle a^{\ast }a\to r^2,}$ the equation for${\displaystyle w}$ 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 thecomplex projective line, often called theRiemann sphere. It was subspaces and subgroups of this space and group of mappings that were applied to produce early models of hyperbolic geometry byBeltrami,Cayley, andKlein. Thus inversive geometry includes the ideas originated byLobachevsky andBolyai in their plane geometry. Furthermore,Felix Klein was so overcome by this facility of mappings to identify geometrical phenomena that he delivered a manifesto, theErlangen program, in 1872. Since then many mathematicians reserve the termgeometry for aspace together with agroup of mappings of that space. The significant properties of figures in the geometry are those that are invariant under this group.

For example, Smogorzhevsky^[10] develops several theorems of inversive geometry before beginning Lobachevskian geometry.

In a realn-dimensional Euclidean space, aninversion in the sphere of radiusr centered at the point${\displaystyle O=(o_1,...,o_n)}$ is a map of an arbitrary point${\displaystyle P=(p_1,...,p_n)}$ found by inverting the length of thedisplacement vector${\displaystyle P-O}$ and multiplying by${\displaystyle r^2}$:

The transformation by inversion inhyperplanes orhyperspheres in Eⁿ can be used to generate dilations, translations, or rotations. Indeed, two concentric hyperspheres, used to produce successive inversions, result in adilation orhomothety about the hyperspheres' center.

When two parallel hyperplanes are used to produce successive reflections, the result is atranslation. When two hyperplanes intersect in an (n−2)-flat, successive reflections produce arotation where every point of the (n−2)-flat is afixed point of each reflection and thus of the composition.

Any combination of reflections, translations, and rotations is called anisometry. Any combination of reflections, dilations, translations, and rotations is asimilarity.

All of these areconformal 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 ofconformal geometry.

The addition of apoint 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 ann-sphere as the base space. The transformations of inversive geometry are often referred to asMöbius transformations. Inversive geometry has been applied to the study of colorings, or partitionings, of ann-sphere.^[11]

The circle inversion map is anticonformal, which means that at every point it preserves angles and reverses orientation (a map is calledconformal if it preservesoriented angles). Algebraically, a map is anticonformal if at every point theJacobian is a scalar times anorthogonal matrix with negative determinant: in two dimensions the Jacobian must be a scalar times a reflection at every point. This means that ifJ is the Jacobian, then${\displaystyle J\cdot J^T=kI}$ and${\displaystyle det(J)=-\sqrt{k}.}$ Computing the Jacobian in the casez_i =x_i/‖x‖², where‖x‖² =x₁² + ... +x_n² givesJJ^T =kI, withk = 1/‖x‖⁴ⁿ, 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 takingz to 1/z. The complex analytic inverse map is conformal and its conjugate, circle inversion, is anticonformal.In this case ahomography is conformal while ananti-homography is anticonformal.

The(n − 1)-sphere with equation

${\displaystyle x_1^2+\cdots +x_n^2+2a_1{x}_1+\cdots +2a_n{x}_n+c=0}$

will have a positive radius ifa₁² + ... +a_n² is greater thanc, and on inversion gives the sphere

${\displaystyle x_1^2+\cdots +x_n^2+2\frac{a_1}{c}{x}_1+\cdots +2\frac{a_n}{c}{x}_n+\frac{1}{c}=0.}$

Hence, it will be invariant under inversion if and only ifc = 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

${\displaystyle x_1^2+\cdots +x_n^2+2a_1{x}_1+\cdots +2a_n{x}_n+1=0,}$

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 thePoincaré disk 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.

• Circle of antisimilitude
• Duality (projective geometry)
• Inverse curve
• Limiting point (geometry)
• Möbius transformation
• Projective geometry
• Soddy's hexlet
• Mohr–Mascheroni theorem
• Inversion of curves and surfaces (German)

• Altshiller-Court, Nathan (1952),College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle (2nd ed.), New York:Barnes & Noble,LCCN 52-13504
• Blair, David E. (2000),Inversion Theory and Conformal Mapping, American Mathematical Society,ISBN 0-8218-2636-0
• Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1998), "Chapter 5: Inversive Geometry",Geometry, Cambridge: Cambridge University Press, pp. 199–260,ISBN 0-521-59787-0
• Coxeter, H.S.M. (1969) [1961],Introduction to Geometry (2nd ed.), John Wiley & Sons,ISBN 0-471-18283-4
• Hartshorne, Robin (2000),"Chapter 7: Non-Euclidean Geometry, Section 37: Circular Inversion",Geometry: Euclid and Beyond, Springer,ISBN 0-387-98650-2
• Kay, David C. (1969),College Geometry, New York:Holt, Rinehart and Winston,LCCN 69-12075
• Patterson, Boyd (1941) "The Inversive Plane",American Mathematical Monthly 48: 589–99,doi:10.2307/2303867MR 0006034

• Inversion: Reflection in a Circle atcut-the-knot
• Wilson Stother's inversive geometry page
• IMO Compendium Training Materials practice problems on how to use inversion for math olympiad problems
• Weisstein, Eric W."Inversion".MathWorld.
• Visual Dictionary of Special Plane Curves Xah Lee

• Inversive geometry

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