Inversive geometry

For other uses, seePoint reflection.

Ingeometry,inversive geometryis the study ofinversion,a transformation of theEuclidean planethat mapscirclesorlinesto 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),StubbsandIngram(1842–3) andKelvin(1845).

The concept of inversion can begeneralized to higher-dimensional spaces.

Inversion in a circle

Inversion of lambdaMandelbrot setwith different translations

Inverse of a point

P'is the inverse ofPwith respect to the circle.

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,theinverseof a pointPwith respect to areference circle (Ø)with centerOand radiusris a pointP',lying on the ray fromOthroughPsuch that

$OP\cdot OP^{\prime }=r^{2}.$

This is calledcircle inversionorplane 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).To make the inversion atotal functionthat 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 centerOand 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 thepoint at infinitychanging positions, whilst any point on the circle is unaffected (isinvariantunder 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.

Compass and straightedge construction

To construct the inverseP'of a pointPoutside a circleØ:Letrbe the radius ofØ.Right trianglesOPNandONP'are similar.OPis torasris toOP'.

Point outside circle

Toconstructthe inverseP'of a pointPoutside a circleØ:

• Draw the segment fromO(center of circleØ) toP.
• LetMbe the midpoint ofOP.(Not shown)
• Draw the circlecwith centerMgoing throughP.(Not labeled. It's the blue circle)
• LetNandN'be the points whereØandcintersect.
• Draw segmentNN'.
• P'is whereOPandNN'intersect.

Point inside circle

To construct the inversePof a pointP'inside a circleØ:

• Draw rayrfromO(center of circleØ) throughP'.(Not labeled, it's the horizontal line)
• Draw linesthroughP'perpendicular tor.(Not labeled. It's the vertical line)
• LetNbe one of the points whereØandsintersect.
• Draw the segmentON.
• Draw linetthroughNperpendicular toON.
• Pis where rayrand linetintersect.

Dutta's construction

There is a construction of the inverse point toAwith respect to a circlePthat isindependentof whetherAis inside or outsideP.

Consider a circlePwith centerOand a pointAwhich may lie inside or outside the circleP.

• Take the intersection pointCof the rayOAwith the circleP.
• Connect the pointCwith an arbitrary pointBon the circleP(different fromCand from the point onPantipodal toC)
• Lethbe the reflection of rayBAin lineBC.Thenhcuts rayOCin a pointA'.A'is the inverse point ofAwith respect to circleP.

Properties

• 
  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 circlenotgoing 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 centerOof 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 throughOis inverted into itself (but not pointwise invariant).
• A circle not passing throughOinverts 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 inversion if and only if it isorthogonalto the reference circle at the points of intersection.

Additional properties include:

• If a circleqpasses through two distinct points A and A' which are inverses with respect to a circlek,then the circleskandqare orthogonal.
• If the circleskandqare orthogonal, then a straight line passing through the center O ofkand 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

$\angle OAB=\angle OB'A'\ {\text{ and }}\ \angle OBA=\angle OA'B'.$

• The points of intersection of two circlespandqorthogonal to a circlek,are inverses with respect tok.
• If M and M' are inverse points with respect to a circlekon 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.

Examples in two dimensions

Examples of inversion of circles A to J with respect to the red circle at O. Circles A to F, which pass through O, map to straight lines. Circles G to J, which do not, map to other circles. The reference circle and line L map to themselves. Circles intersect their inverses, if any, on the reference circle. Inthe SVG file,click or hover over a circle to highlight it.

• 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

Application

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

Invert with respect to theincircleof triangleABC.Themedial triangleof 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 triangleABCarecollinear.

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

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

ThePeaucellier–Lipkin linkageis a mechanical implementation of inversion in a circle. It provides an exact solution to the important problem of converting between linear and circular motion.

Pole and polar

Main article:pole and polar

The polar lineqto a pointQwith respect to a circle of radiusrcentered on the pointO.The pointPis theinversion pointofQ;the polar is the line throughPthat is perpendicular to the line containingO,PandQ.

If pointRis the inverse of pointPthen the linesperpendicularto the linePRthrough one of the points is thepolarof the other point (thepole).

Poles and polars have several useful properties:

• If a pointPlies on a linel,then the poleLof the linellies on the polarpof pointP.
• If a pointPmoves along a linel,its polarprotates about the poleLof 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 pointPlies on its own polar line, thenPis on the circle.
• Each line has exactly one pole.

In three dimensions

Inversion of a sphere at the red sphere

Inversion of a spheroid (at the red sphere)

Inversion of a hyperboloid of one sheet

Circle inversion is generalizable tosphere inversionin three dimensions. The inversion of a pointPin 3D with respect to a reference sphere centered at a pointOwith radiusRis a pointP' on the ray with directionOPsuch that$OP\cdot OP^{\prime }=||OP||\cdot ||OP^{\prime }||=R^{2}$.As with the 2D version, a sphere inverts to a sphere, except that if a sphere passes through the centerOof 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 throughOit 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.

Examples in three dimensions

Sphere

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.

Cylinder, cone, torus

The inversion of a cylinder, cone, or torus results in aDupin cyclide.

Spheroid

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.

Hyperboloid of one sheet

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.

Stereographic projection as the inversion of a sphere

Stereographic projection as an inversion of a sphere

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

6-sphere coordinates

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

Axiomatics and generalization

One of the first to consider foundations of inversive geometry wasMario Pieriin 1911 and 1912.Edward Kasnerwrote his thesis on "Invariant theory of the inversion group".

More recently themathematical structureof inversive geometry has been interpreted as anincidence structurewhere the generalized circles are called "blocks": Inincidence geometry,anyaffine planetogether with a singlepoint at infinityforms 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.

Amodelfor the Möbius plane that comes from the Euclidean plane is theRiemann sphere.

Invariant

Thecross-ratiobetween 4 points$x,y,z,w$is invariant under an inversion. In particular if O is the centre of the inversion and$r_{1}$and$r_{2}$are distances to the ends of a line L, then length of the line$d$will become$d/(r_{1}r_{2})$under an inversion with radius 1. The invariant is:

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

Relation to Erlangen program

According to Coxeter,the transformation by inversion in circle was invented byL. I. Magnusin 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 geometrysoon appreciates the significance ofFelix Klein'sErlangen program,an outgrowth of certain models ofhyperbolic geometry

Dilation

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

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

Reciprocation

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

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

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

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

Reciprocation is key in transformation theory as ageneratorof 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 theconjugationmapping. Neither conjugation nor inversion-in-a-circle are in the Möbius group since they are non-conformal (see below). Möbius group elements areanalytic functionsof the whole plane and so are necessarilyconformal.

Transforming circles into circles

Consider, in the complex plane, the circle of radius$r$around the point$a$

$(z-a)(z-a)^{*}=r^{2}$

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

$w={\frac {1}{z^{*}}}$

it is straightforward to show that$w$obeys the equation

$ww^{*}-{\frac {a}{(a^{2}-r^{2})}}(w+w^{*})+{\frac {a^{2}}{(a^{2}-r^{2})^{2}}}={\frac {r^{2}}{(a^{2}-r^{2})^{2}}}$

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

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

For$a\not \in \mathbb {R}$and$aa^{*}\neq r^{2}$the result for$w$is

${\begin{aligned}&ww^{*}-{\frac {aw+a^{*}w^{*}}{(a^{*}a-r^{2})}}+{\frac {aa^{*}}{(aa^{*}-r^{2})^{2}}}={\frac {r^{2}}{(aa^{*}-r^{2})^{2}}}\\[4pt]\Longleftrightarrow {}&\left(w-{\frac {a^{*}}{aa^{*}-r^{2}}}\right)\left(w^{*}-{\frac {a}{a^{*}a-r^{2}}}\right)=\left({\frac {r}{\left|aa^{*}-r^{2}\right|}}\right)^{2}\end{aligned}}$

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

When$a^{*}a\to r^{2},$the equation for$w$becomes

${\begin{aligned}&aw+a^{*}w^{*}=1\Longleftrightarrow 2\operatorname {Re} \{aw\}=1\Longleftrightarrow \operatorname {Re} \{a\}\operatorname {Re} \{w\}-\operatorname {Im} \{a\}\operatorname {Im} \{w\}={\frac {1}{2}}\\[4pt]\Longleftrightarrow {}&\operatorname {Im} \{w\}={\frac {\operatorname {Re} \{a\}}{\operatorname {Im} \{a\}}}\cdot \operatorname {Re} \{w\}-{\frac {1}{2\cdot \operatorname {Im} \{a\}}}.\end{aligned}}$

Higher geometry

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 byLobachevskyandBolyaiin their plane geometry. Furthermore,Felix Kleinwas 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 termgeometryfor aspacetogether with agroupof 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 higher dimensions

In a realn-dimensional Euclidean space, aninversion in the sphereof radiusrcentered at the point$O=(o_{1},...,o_{n})$is a map of an arbitrary point$P=(p_{1},...,p_{n})$found by inverting the length of thedisplacement vector$P-O$and multiplying by$r^{2}$:

${\begin{aligned}P&\mapsto P'=O+{\frac {r^{2}(P-O)}{\|P-O\|^{2}}},\\[5mu]p_{j}&\mapsto p_{j}'=o_{j}+{\frac {r^{2}(p_{j}-o_{j})}{\sum _{k}(p_{k}-o_{k})^{2}}}.\end{aligned}}$

The transformation by inversion inhyperplanesorhyperspheresin Eⁿcan be used to generate dilations, translations, or rotations. Indeed, two concentric hyperspheres, used to produce successive inversions, result in adilationorhomothetyabout 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 arotationwhere every point of the (n–2)-flat is afixed pointof 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 theoremis a classical theorem ofconformal geometry.

The addition of apoint at infinityto the space obviates the distinction between hyperplane and hypersphere; higher dimensional inversive geometry is frequently studied then in the presumed context of ann-sphereas 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.

Anticonformal mapping property

The circle inversion map is anticonformal, which means that at every point it preserves angles and reverses orientation (a map is calledconformalif it preservesorientedangles). Algebraically, a map is anticonformal if at every point theJacobianis a scalar times anorthogonal matrixwith negative determinant: in two dimensions the Jacobian must be a scalar times a reflection at every point. This means that ifJis the Jacobian, then$J\cdot J^{T}=kI$and$\det(J)=-{\sqrt {k}}.$Computing the Jacobian in the casez_i=x_i/‖x‖²,where‖x‖²=x₁²+... +x_n²givesJJ^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 takingzto 1/z.The complex analytic inverse map is conformal and its conjugate, circle inversion, is anticonformal. In this case ahomographyis conformal while ananti-homographyis anticonformal.

Inversive geometry and hyperbolic geometry

The(n− 1)-spherewith equation

$x_{1}^{2}+\cdots +x_{n}^{2}+2a_{1}x_{1}+\cdots +2a_{n}x_{n}+c=0$

will have a positive radius ifa₁²+... +a_n²is greater thanc,and on inversion gives the sphere

$x_{1}^{2}+\cdots +x_{n}^{2}+2{\frac {a_{1}}{c}}x_{1}+\cdots +2{\frac {a_{n}}{c}}x_{n}+{\frac {1}{c}}=0.$

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

$x_{1}^{2}+\cdots +x_{n}^{2}+2a_{1}x_{1}+\cdots +2a_{n}x_{n}+1=0,$

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 disc modelof 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.

See also

• 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)

References

• Altshiller-Court, Nathan (1952),College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle(2nd ed.), New York:Barnes & Noble,LCCN52-13504
• Blair, David E. (2000),Inversion Theory and Conformal Mapping,American Mathematical Society,ISBN0-8218-2636-0
• Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1998), "Chapter 5: Inversive Geometry",Geometry,Cambridge: Cambridge University Press, pp. 199–260,ISBN0-521-59787-0
• Coxeter, H.S.M.(1969) [1961],Introduction to Geometry(2nd ed.), John Wiley & Sons,ISBN0-471-18283-4
• Hartshorne, Robin(2000),"Chapter 7: Non-Euclidean Geometry, Section 37: Circular Inversion",Geometry: Euclid and Beyond,Springer,ISBN0-387-98650-2
• Kay, David C. (1969),College Geometry,New York:Holt, Rinehart and Winston,LCCN69-12075
• Patterson, Boyd(1941) "The Inversive Plane",American Mathematical Monthly48: 589–99,doi:10.2307/2303867MR0006034

External links

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