93:
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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.
1053:
2746:. 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
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4410:{\displaystyle {\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}}\\\Longleftrightarrow {}&\operatorname {Im} \{w\}={\frac {\operatorname {Re} \{a\}}{\operatorname {Im} \{a\}}}\cdot \operatorname {Re} \{w\}-{\frac {1}{2\cdot \operatorname {Im} \{a\}}}.\end{aligned}}}
1177:
1344:
3496:
1336:
3906:{\displaystyle {\begin{aligned}&ww^{*}-{\frac {aw+a^{*}w^{*}}{(a^{*}a-r^{2})}}+{\frac {aa^{*}}{(aa^{*}-r^{2})^{2}}}={\frac {r^{2}}{(aa^{*}-r^{2})^{2}}}\\\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}}}
4940:
1566:
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.
1056:
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
5721:
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
5795:
by Robert C. Yates, National
Council of Teachers of Mathematics, Inc., Washington, D.C., p. 127: "Geometrical inversion seems to be due to Jakob Steiner who indicated a knowledge of the subject in 1824. He was closely followed by Adolphe Quetelet (1825) who gave some examples. Apparently
4425:
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
3179:
52:
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, including
5565:
2453:
4689:
296:
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.
1005:
1537:
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.
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5708:
768:
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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
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4935:{\displaystyle {\begin{aligned}P&\mapsto P'=O+{\frac {r^{2}(P-O)}{\|P-O\|^{2}}},\\p_{j}&\mapsto p_{j}'=o_{j}+{\frac {r^{2}(p_{j}-o_{j})}{\sum _{k}(p_{k}-o_{k})^{2}}}.\end{aligned}}}
2314:
226:
1558:
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.
3992:
3395:
3313:
3425:
834:. 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 is
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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
2153:
5021:
to the space obviates the distinction between hyperplane and hypersphere; higher dimensional inversive geometry is frequently studied then in the presumed context of an
2661:
1884:
1817:
792:
803:
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.
3344:
2502:
2028:
4680:
2082:
2055:
1040:, 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'.
4652:
1837:
1162:
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.
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2102:
1904:
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1605:
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in E can be used to generate dilations, translations, or rotations. Indeed, two concentric hyperspheres, used to produce successive inversions, result in a
2277:
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
288:
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
2669:
905:
5584:
1969:. 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.
5899:
5271:
1121:, meaning the circumcenter of the medial triangle, that is, the nine-point center of the intouch triangle, the incenter and circumcenter of triangle
2750:
mapping. Neither conjugation nor inversion-in-a-circle are in the Möbius group since they are non-conformal (see below). Möbius group elements are
1627:(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
1090:
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
1386:
5044:
The circle inversion map is anticonformal, which means that at every point it preserves angles and reverses orientation (a map is called
5007:
4434:. It was subspaces and subgroups of this space and group of mappings that were applied to produce early models of hyperbolic geometry by
1692:
5836:
1512:. A circle, that is, the intersection of a sphere with a secant plane, inverts into a circle, except that if the circle passes through
5006:, and in fact, where the space has three or more dimensions, the mappings generated by inversion are the only conformal mappings.
5060:
with negative determinant: in two dimensions the
Jacobian must be a scalar times a reflection at every point. This means that if
2568:
4474:
of mappings of that space. The significant properties of figures in the geometry are those that are invariant under this group.
3174:{\displaystyle 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}}}}
372:
1839:, green in the picture), then it will be mapped by the inversion at the unit sphere (red) onto the tangent plane at point
3997:
2161:
5560:{\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.}
2448:{\displaystyle x\mapsto R^{2}{\frac {x}{|x|^{2}}}=y\mapsto T^{2}{\frac {y}{|y|^{2}}}=\left({\frac {T}{R}}\right)^{2}x.}
5241:. The complex analytic inverse map is conformal and its conjugate, circle inversion, is anticonformal. In this case a
1066:
Inversion of a line is a circle containing the center of inversion; or it is the line itself if it contains the center
6088:
6063:
6040:
6022:
1954:
74:
5796:
independently discovered by Giusto
Bellavitis in 1836, by Stubbs and Ingram in 1842–3, and by Lord Kelvin in 1845.)"
5029:
4477:
For example, Smogorzhevsky develops several theorems of inversive geometry before beginning
Lobachevskian geometry.
4458:
was so overcome by this facility of mappings to identify geometrical phenomena that he delivered a manifesto, the
5053:
176:
92:
6049:
1309:
If two tangent lines can be drawn from a pole to the circle, then its polar passes through both tangent points.
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6121:
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281:, a single point placed on all the lines, and extend the inversion, by definition, to interchange the center
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1500:. As with the 2D version, a sphere inverts to a sphere, except that if a sphere passes through the center
6203:
6148:
5574:= 1. But this is the condition of being orthogonal to the unit sphere. Hence we are led to consider the (
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5746:
5067:
4985:
2739:
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1058:
265:, so the result of applying the same inversion twice is the identity transformation which makes it a
2107:
5897:
Kasner, E. (1900). "The
Invariant Theory of the Inversion Group: Geometry Upon a Quadric Surface".
5771:
5731:
1584:
1152:
1069:
Inversion of a circle is another circle; or it is a line if the original circle contains the center
798:
Inversion with respect to a circle does not map the center of the circle to the center of its image
5751:
1043:
Inversion leaves the measure of angles unaltered, but reverses the orientation of oriented angles.
5766:
4954:
4427:
2278:
1079:
6031:
Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1998), "Chapter 5: Inversive
Geometry",
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1942:
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128:
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4996:
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1995:
1919:
1906:) are mapped onto themselves. They are the projection lines of the stereographic projection.
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36:
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1973:
1915:
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When two parallel hyperplanes are used to produce successive reflections, the result is a
8:
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5032:. Inversive geometry has been applied to the study of colorings, or partitionings, of an
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1946:
1822:
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it inverts into a line. This reduces to the 2D case when the secant plane passes through
66:
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passes through two distinct points A and A' which are inverses with respect to a circle
5995:
College
Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle
5926:
5011:
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3187:
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is invariant under an inversion. In particular if O is the centre of the inversion and
1950:
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as the base space. The transformations of inversive geometry are often referred to as
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2727:{\displaystyle w={\frac {1}{\bar {z}}}={\overline {\left({\frac {1}{z}}\right)}}}
1114:
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with the center of the reference circle. This fact can be used to prove that the
1000:{\displaystyle \angle OAB=\angle OB'A'\ {\text{ and }}\ \angle OBA=\angle OA'B'.}
49:
41:
5703:{\displaystyle x_{1}^{2}+\cdots +x_{n}^{2}+2a_{1}x_{1}+\cdots +2a_{n}x_{n}+1=0,}
2743:
2273:
According to
Coxeter, the transformation by inversion in circle was invented by
1962:
6173:
6162:
6116:
5960:
5387:{\displaystyle x_{1}^{2}+\cdots +x_{n}^{2}+2a_{1}x_{1}+\cdots +2a_{n}x_{n}+c=0}
4431:
2637:
Consequently, the algebraic form of the inversion in a unit circle is given by
2464:
1977:
1504:
of the reference sphere, then it inverts to a plane. Any plane passing through
1249:
1245:
1171:
270:
266:
97:
4995:. Any combination of reflections, dilations, translations, and rotations is a
1918:
are a coordinate system for three-dimensional space obtained by inverting the
131:. A closely related idea in geometry is that of "inverting" a point. In the
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2755:
1935:
1547:
1237:
54:
6152:
1312:
If a point lies on the circle, its polar is the tangent through this point.
1102:
of a triangle coincides with its OI line. The proof roughly goes as below:
109:
5258:
4991:
Any combination of reflections, translations, and rotations is called an
4950:
4455:
4443:
2282:
1989:
1931:
1607:(north pole) of the sphere onto the tangent plane at the opposite point
1570:
1520:, but is a true 3D phenomenon if the secant plane does not pass through
5930:
5242:
4946:
1493:{\displaystyle OP\cdot OP^{\prime }=||OP||\cdot ||OP^{\prime }||=R^{2}}
1133:
1095:
754:
The inverse, with respect to the red circle, of a circle going through
305:
6178:
5052:
angles). Algebraically, a map is anticonformal if at every point the
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2300:
The combination of two inversions in concentric circles results in a
1126:
1091:
873:
are orthogonal, then a straight line passing through the center O of
292:
changing positions, whilst any point on the circle is unaffected (is
6165:
practice problems on how to use inversion for math olympiad problems
6130:
1575:
6077:"Chapter 7: Non-Euclidean Geometry, Section 37: Circular Inversion"
5883:
M. Pieri (1911,12) "Nuovi principia di geometria della inversion",
5022:
4992:
4977:
3346:
the circle transforms into the line parallel to the imaginary axis
1930:
One of the first to consider foundations of inversive geometry was
1771:{\displaystyle x^{2}+y^{2}+(z+{\tfrac {1}{2}})^{2}={\tfrac {1}{4}}}
1147:
In addition, any two non-intersecting circles may be inverted into
1106:
1073:
27:
5973:
1351:
1176:
6110:
6006:
1976:
for the Möbius plane that comes from the
Euclidean plane is the
1343:
4451:
2084:
are distances to the ends of a line L, then length of the line
1938:
wrote his thesis on "Invariant theory of the inversion group".
45:
2308:, or dilation characterized by the ratio of the circle radii.
1151:
circles, using circle of inversion centered at a point on the
811:
of the reference circle inverts to a line not passing through
5252:
1367:
in 3D with respect to a reference sphere centered at a point
1335:
6035:, Cambridge: Cambridge University Press, pp. 199–260,
5570:
Hence, it will be invariant under inversion if and only if
4462:, in 1872. Since then many mathematicians reserve the term
4446:. Thus inversive geometry includes the ideas originated by
2627:{\displaystyle {\frac {1}{z}}={\frac {\bar {z}}{|z|^{2}}}.}
892:, and points A' and B' inverses of A and B with respect to
127:
To invert a number in arithmetic usually means to take its
5834:
A simple property of isosceles triangles with applications
888:
Given a triangle OAB in which O is the center of a circle
5230:) is negative; hence the inversive map is anticonformal.
1546:
The inversion of a cylinder, cone, or torus results in a
1144:
of the ratio of the radii of the two concentric circles.
774:
The inverse, with respect to the red circle, of a circle
1032:
If M and M' are inverse points with respect to a circle
815:, but parallel to the tangent to the original circle at
300:
4000:
3942:
3373:
3259:
3210:
1949:
where the generalized circles are called "blocks": In
1757:
1732:
1132:
Any two non-intersecting circles may be inverted into
1036:
on two curves m and m', also inverses with respect to
838:
to the reference circle at the points of intersection.
823:
is inverted into itself (but not pointwise invariant).
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Consider, in the complex plane, the circle of radius
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Stereographic projection as the inversion of a sphere
1389:
908:
179:
2761:
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under an inversion with radius 1. The invariant is:
1579:
Stereographic projection as an inversion of a sphere
4047:{\textstyle {\frac {r}{\left|a^{*}a-r^{2}\right|}}}
2738:Reciprocation is key in transformation theory as a
1886:. The lines through the center of inversion (point
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2258:{\displaystyle I={\frac {|x-y||w-z|}{|x-w||y-z|}}}
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2022:
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1117:of the intouch triangle is inverted into triangle
999:
220:
6030:
5900:Transactions of the American Mathematical Society
5039:
1945:of inversive geometry has been interpreted as an
1255:Poles and polars have several useful properties:
819:, and vice versa; whereas a line passing through
6195:
5992:
5805:
5113:
4988:of each reflection and thus of the composition.
1925:
609:There is a construction of the inverse point to
2268:
1527:
269:(i.e. an involution). To make the inversion a
5855:
5853:
2463:When a point in the plane is interpreted as a
1363:in three dimensions. The inversion of a point
1212:that is perpendicular to the line containing
1061:click or hover over a circle to highlight it.
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881:, does so at inverse points with respect to
5850:
1561:
1347:Inversion of a spheroid (at the red sphere)
644:which may lie inside or outside the circle
221:{\displaystyle OP\cdot OP^{\prime }=r^{2}.}
6071:
5253:Inversive geometry and hyperbolic geometry
2754:of the whole plane and so are necessarily
1013:The points of intersection of two circles
6188:Visual Dictionary of Special Plane Curves
5920:
5772:Inversion of curves and surfaces (German)
3413:
2892:
531:. (Not labeled, it's the horizontal line)
81:
16:Study of angle-preserving transformations
6158:Wilson Stother's inversive geometry page
5972:Joel C. Gibbons & Yushen Luo (2013)
4968:. When two hyperplanes intersect in an (
3987:{\textstyle {\frac {a}{(aa^{*}-r^{2})}}}
3390:{\displaystyle w+w^{*}={\tfrac {1}{a}}.}
3308:{\textstyle {\frac {r}{|a^{2}-r^{2}|}}.}
1587:usually projects a sphere from a point
1574:
1541:
1350:
1342:
1334:
1175:
1051:
830:inverts to a circle not passing through
807:A circle that passes through the center
304:
108:
91:
87:
82:generalized to higher-dimensional spaces
6058:(2nd ed.), John Wiley & Sons,
6048:
5943:
4480:
1909:
1355:Inversion of a hyperboloid of one sheet
1339:Inversion of a sphere at the red sphere
604:
549:. (Not labeled. It's the vertical line)
366:
6196:
6015:Inversion Theory and Conformal Mapping
5896:
5578: − 1)-spheres with equation
4454:in their plane geometry. Furthermore,
3420:{\displaystyle a\not \in \mathbb {R} }
1330:
1140:(usually denoted δ) is defined as the
488:
6169:
6012:
5887:49:49–96 & 50:106–140
5144:{\displaystyle \det(J)=-{\sqrt {k}}.}
4625:found by inverting the length of the
3246:{\textstyle {\frac {a}{a^{2}-r^{2}}}}
2281:soon appreciates the significance of
1359:Circle inversion is generalizable to
1201:
782:(blue) is a circle not going through
432:. (Not labeled. It's the blue circle)
301:Compass and straightedge construction
104:
5885:Giornal di Matematiche di Battaglini
5828:
5826:
5417:, and on inversion gives the sphere
2945:{\displaystyle w={\frac {1}{z^{*}}}}
2868:{\displaystyle (z-a)(z-a)^{*}=r^{2}}
2289:, an outgrowth of certain models of
1682:{\displaystyle x^{2}+y^{2}+z^{2}=-z}
6096:
5871:
5859:
5817:
5151:Computing the Jacobian in the case
4976:, successive reflections produce a
4945:The transformation by inversion in
4618:{\displaystyle P=(p_{1},...,p_{n})}
4554:{\displaystyle O=(o_{1},...,o_{n})}
2955:it is straightforward to show that
1188:with respect to a circle of radius
758:(blue) is a line not going through
13:
4420:
2909:Using the definition of inversion
2902:{\displaystyle a\in \mathbb {R} .}
2878:where without loss of generality,
1508:, inverts to a sphere touching at
1462:
1407:
972:
957:
924:
909:
239:. The inversion taking any point
197:
14:
6215:
6163:IMO Compendium Training Materials
6149:Inversion: Reflection in a Circle
6142:
6017:, American Mathematical Society,
5993:Altshiller-Court, Nathan (1952),
5913:10.1090/S0002-9947-1900-1500550-1
5823:
4489:-dimensional Euclidean space, an
2762:Transforming circles into circles
1319:lies on its own polar line, then
1244:through one of the points is the
1165:
277:, it is necessary to introduce a
4961:about the hyperspheres' center.
4096:{\displaystyle a^{*}a\to r^{2},}
3463:{\displaystyle aa^{*}\neq r^{2}}
2547:{\displaystyle {\bar {z}}=x-iy,}
2458:
1208:; the polar is the line through
791:
767:
747:
80:The concept of inversion can be
5966:
5949:
5397:will have a positive radius if
5099:{\displaystyle J\cdot J^{T}=kI}
4561:is a map of an arbitrary point
3936:describes the circle of center
3204:describes the circle of center
1326:Each line has exactly one pole.
1025:, are inverses with respect to
842:Additional properties include:
6119:(1941) "The Inversive Plane",
5978:-sphere and inversive geometry
5937:
5890:
5877:
5865:
5811:
5799:
5784:
5122:
5116:
5040:Anticonformal mapping property
4913:
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4871:
4845:
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4749:
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4548:
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3039:
3033:
3007:
2843:
2830:
2827:
2815:
2689:
2647:
2608:
2599:
2592:
2520:
2398:
2389:
2369:
2350:
2341:
2321:
2248:
2234:
2229:
2215:
2208:
2194:
2189:
2175:
2148:{\displaystyle d/(r_{1}r_{2})}
2142:
2119:
1873:
1852:
1806:
1785:
1744:
1722:
1473:
1468:
1450:
1445:
1437:
1432:
1421:
1416:
1085:
1:
6122:American Mathematical Monthly
5986:
5737:Duality (projective geometry)
1926:Axiomatics and generalization
1072:Inversion of a parabola is a
826:A circle not passing through
740:
2719:
2269:Relation to Erlangen program
1983:
1528:Examples in three dimensions
1379:' on the ray with direction
1078:Inversion of hyperbola is a
652:Take the intersection point
285:and this point at infinity.
7:
6081:Geometry: Euclid and Beyond
5792:Curves and Their Properties
5725:
2295:
1553:
1105:Invert with respect to the
556:be one of the points where
123:with respect to the circle.
100:with different translations
10:
6220:
6103:Holt, Rinehart and Winston
5997:(2nd ed.), New York:
5955:A.S. Smogorzhevsky (1982)
5922:2027/miun.abv0510.0001.001
5010:is a classical theorem of
4980:where every point of the (
2656:{\displaystyle z\mapsto w}
1879:{\displaystyle S=(0,0,-1)}
1812:{\displaystyle (0,0,-0.5)}
1169:
1160:Peaucellier–Lipkin linkage
1048:Examples in two dimensions
40:, a transformation of the
18:
5747:Limiting point (geometry)
2306:homothetic transformation
1532:
698:be the reflection of ray
613:with respect to a circle
493:To construct the inverse
309:To construct the inverse
273:that is also defined for
6055:Introduction to Geometry
6013:Blair, David E. (2000),
5777:
5732:Circle of antisimilitude
5718:of hyperbolic geometry.
1585:stereographic projection
1562:Hyperboloid of one sheet
1248:of the other point (the
1232:is the inverse of point
1153:circle of antisimilitude
786:(green), and vice versa.
762:(green), and vice versa.
728:is the inverse point of
671:with an arbitrary point
162:, lying on the ray from
5767:Mohr-Mascheroni theorem
5226:, and additionally det(
4491:inversion in the sphere
4428:complex projective line
3339:{\displaystyle a\to r,}
2497:{\displaystyle z=x+iy,}
2279:transformation geometry
2023:{\displaystyle x,y,z,w}
1957:together with a single
1298:rotates about the pole
1080:lemniscate of Bernoulli
1021:orthogonal to a circle
732:with respect to circle
6097:Kay, David C. (1969),
5957:Lobachevskian Geometry
5832:Dutta, Surajit (2014)
5806:Altshiller-Court (1952
5704:
5561:
5388:
5263: − 1)-sphere
5245:is conformal while an
5145:
5100:
5064:is the Jacobian, then
5030:Möbius transformations
4936:
4676:
4648:
4619:
4555:
4497:centered at the point
4411:
4117:
4097:
4048:
3988:
3930:
3907:
3484:
3464:
3421:
3391:
3340:
3309:
3247:
3198:
3175:
2969:
2946:
2903:
2869:
2800:
2780:
2728:
2657:
2628:
2548:
2498:
2449:
2259:
2149:
2098:
2078:
2051:
2024:
1943:mathematical structure
1900:
1880:
1833:
1813:
1772:
1683:
1621:
1601:
1580:
1494:
1356:
1348:
1340:
1225:
1192:centered on the point
1062:
1001:
683:and from the point on
394:Draw the segment from
363:
222:
124:
101:
88:Inversion in a circle
5752:Möbius transformation
5705:
5562:
5389:
5146:
5101:
5056:is a scalar times an
4937:
4677:
4675:{\displaystyle r^{2}}
4649:
4620:
4556:
4412:
4118:
4098:
4049:
3989:
3931:
3908:
3485:
3465:
3422:
3392:
3341:
3310:
3248:
3199:
3176:
2970:
2947:
2904:
2870:
2801:
2781:
2729:
2658:
2629:
2549:
2499:
2450:
2260:
2150:
2099:
2079:
2077:{\displaystyle r_{2}}
2052:
2050:{\displaystyle r_{1}}
2025:
1920:Cartesian coordinates
1901:
1881:
1834:
1814:
1773:
1689:(alternately written
1684:
1622:
1602:
1578:
1542:Cylinder, cone, torus
1495:
1354:
1346:
1338:
1179:
1055:
1002:
625:is inside or outside
308:
223:
112:
95:
5585:
5424:
5272:
5110:
5068:
4690:
4659:
4632:
4565:
4501:
4481:In higher dimensions
4130:
4107:
4061:
3998:
3940:
3920:
3497:
3474:
3431:
3403:
3350:
3321:
3257:
3208:
3188:
2982:
2959:
2916:
2882:
2812:
2790:
2770:
2670:
2641:
2569:
2511:
2470:
2315:
2162:
2108:
2088:
2061:
2034:
1996:
1916:6-sphere coordinates
1910:6-sphere coordinates
1890:
1843:
1823:
1782:
1693:
1631:
1611:
1591:
1387:
906:
605:Dutta's construction
446:be the points where
367:Point outside circle
177:
145:reference circle (Ø)
96:Inversion of lambda
19:For other uses, see
5845:Forum Geometricorum
5757:Projective geometry
5716:Poincaré disc model
5626:
5602:
5465:
5441:
5313:
5289:
5008:Liouville's theorem
4815:
4654:and multiplying by
4647:{\displaystyle P-O}
4627:displacement vector
4430:, often called the
2975:obeys the equation
2291:hyperbolic geometry
1965:, also known as an
1947:incidence structure
1832:{\displaystyle 0.5}
1331:In three dimensions
1290:moves along a line
854:, then the circles
489:Point inside circle
413:be the midpoint of
6204:Inversive geometry
6171:Weisstein, Eric W.
5999:Barnes & Noble
5839:2018-04-21 at the
5700:
5612:
5588:
5557:
5451:
5427:
5384:
5299:
5275:
5249:is anticonformal.
5141:
5096:
5017:The addition of a
5012:conformal geometry
4932:
4930:
4885:
4803:
4672:
4644:
4615:
4551:
4407:
4405:
4113:
4093:
4044:
3984:
3926:
3903:
3901:
3480:
3460:
3417:
3387:
3382:
3336:
3305:
3243:
3194:
3171:
2965:
2942:
2899:
2865:
2796:
2776:
2752:analytic functions
2724:
2653:
2624:
2544:
2494:
2445:
2255:
2145:
2094:
2074:
2047:
2020:
1951:incidence geometry
1941:More recently the
1934:in 1911 and 1912.
1896:
1876:
1829:
1809:
1768:
1766:
1741:
1679:
1617:
1597:
1581:
1490:
1357:
1349:
1341:
1275:lies on the polar
1226:
1138:inversive distance
1136:circles. Then the
1063:
997:
667:Connect the point
632:Consider a circle
520:(center of circle
398:(center of circle
364:
332:. Right triangles
218:
143:with respect to a
125:
119:is the inverse of
105:Inverse of a point
102:
32:inversive geometry
6073:Hartshorne, Robin
5974:Colorings of the
5549:
5526:
5487:
5136:
5058:orthogonal matrix
5019:point at infinity
5002:All of these are
4923:
4876:
4777:
4398:
4347:
4280:
4116:{\displaystyle w}
4103:the equation for
4042:
3982:
3929:{\displaystyle w}
3916:showing that the
3887:
3831:
3773:
3709:
3651:
3588:
3483:{\displaystyle w}
3381:
3300:
3241:
3197:{\displaystyle w}
3169:
3114:
3037:
2968:{\displaystyle w}
2940:
2799:{\displaystyle a}
2786:around the point
2779:{\displaystyle r}
2722:
2713:
2694:
2692:
2619:
2595:
2580:
2523:
2506:complex conjugate
2427:
2409:
2361:
2253:
2097:{\displaystyle d}
1992:between 4 points
1959:point at infinity
1899:{\displaystyle N}
1765:
1740:
1620:{\displaystyle S}
1600:{\displaystyle N}
1323:is on the circle.
1142:natural logarithm
956:
952:
948:
877:and intersecting
582:perpendicular to
567:Draw the segment
545:perpendicular to
386:outside a circle
328:be the radius of
320:outside a circle
290:point at infinity
279:point at infinity
6211:
6184:
6183:
6113:
6099:College Geometry
6093:
6068:
6045:
6027:
6009:
5980:
5970:
5964:
5953:
5947:
5946:, pp. 77–95
5941:
5935:
5934:
5924:
5894:
5888:
5881:
5875:
5869:
5863:
5857:
5848:
5830:
5821:
5815:
5809:
5803:
5797:
5788:
5709:
5707:
5706:
5701:
5684:
5683:
5674:
5673:
5652:
5651:
5642:
5641:
5625:
5620:
5601:
5596:
5566:
5564:
5563:
5558:
5550:
5542:
5537:
5536:
5527:
5522:
5521:
5512:
5498:
5497:
5488:
5483:
5482:
5473:
5464:
5459:
5440:
5435:
5413:is greater than
5393:
5391:
5390:
5385:
5371:
5370:
5361:
5360:
5339:
5338:
5329:
5328:
5312:
5307:
5288:
5283:
5225:
5223:
5212:
5202:
5185:
5177:
5175:
5150:
5148:
5147:
5142:
5137:
5132:
5105:
5103:
5102:
5097:
5086:
5085:
5048:if it preserves
4941:
4939:
4938:
4933:
4931:
4924:
4922:
4921:
4920:
4911:
4910:
4898:
4897:
4884:
4874:
4870:
4869:
4857:
4856:
4844:
4843:
4833:
4828:
4827:
4811:
4795:
4794:
4778:
4776:
4775:
4774:
4752:
4736:
4735:
4725:
4714:
4683:
4681:
4679:
4678:
4673:
4671:
4670:
4653:
4651:
4650:
4645:
4624:
4622:
4621:
4616:
4611:
4610:
4586:
4585:
4560:
4558:
4557:
4552:
4547:
4546:
4522:
4521:
4496:
4470:together with a
4460:Erlangen program
4416:
4414:
4413:
4408:
4406:
4399:
4397:
4371:
4348:
4346:
4329:
4312:
4290:
4281:
4273:
4166:
4165:
4156:
4155:
4136:
4122:
4120:
4119:
4114:
4102:
4100:
4099:
4094:
4089:
4088:
4073:
4072:
4053:
4051:
4050:
4045:
4043:
4041:
4037:
4036:
4035:
4020:
4019:
4002:
3993:
3991:
3990:
3985:
3983:
3981:
3977:
3976:
3964:
3963:
3944:
3935:
3933:
3932:
3927:
3912:
3910:
3909:
3904:
3902:
3898:
3897:
3892:
3888:
3886:
3882:
3881:
3880:
3868:
3867:
3847:
3837:
3833:
3832:
3830:
3829:
3828:
3813:
3812:
3799:
3794:
3793:
3779:
3775:
3774:
3772:
3771:
3770:
3758:
3757:
3744:
3743:
3734:
3719:
3710:
3708:
3707:
3706:
3697:
3696:
3684:
3683:
3667:
3666:
3657:
3652:
3650:
3649:
3648:
3639:
3638:
3626:
3625:
3609:
3608:
3607:
3594:
3589:
3587:
3583:
3582:
3567:
3566:
3553:
3552:
3551:
3542:
3541:
3522:
3517:
3516:
3503:
3489:
3487:
3486:
3481:
3469:
3467:
3466:
3461:
3459:
3458:
3446:
3445:
3426:
3424:
3423:
3418:
3416:
3396:
3394:
3393:
3388:
3383:
3374:
3368:
3367:
3345:
3343:
3342:
3337:
3314:
3312:
3311:
3306:
3301:
3299:
3298:
3293:
3292:
3280:
3279:
3270:
3261:
3252:
3250:
3249:
3244:
3242:
3240:
3239:
3238:
3226:
3225:
3212:
3203:
3201:
3200:
3195:
3180:
3178:
3177:
3172:
3170:
3168:
3167:
3166:
3157:
3156:
3144:
3143:
3130:
3129:
3120:
3115:
3113:
3112:
3111:
3102:
3101:
3089:
3088:
3075:
3074:
3065:
3057:
3056:
3038:
3036:
3032:
3031:
3019:
3018:
3002:
2997:
2996:
2974:
2972:
2971:
2966:
2951:
2949:
2948:
2943:
2941:
2939:
2938:
2926:
2908:
2906:
2905:
2900:
2895:
2874:
2872:
2871:
2866:
2864:
2863:
2851:
2850:
2805:
2803:
2802:
2797:
2785:
2783:
2782:
2777:
2733:
2731:
2730:
2725:
2723:
2718:
2714:
2706:
2700:
2695:
2693:
2685:
2680:
2662:
2660:
2659:
2654:
2633:
2631:
2630:
2625:
2620:
2618:
2617:
2616:
2611:
2602:
2596:
2588:
2586:
2581:
2573:
2553:
2551:
2550:
2545:
2525:
2524:
2516:
2503:
2501:
2500:
2495:
2454:
2452:
2451:
2446:
2438:
2437:
2432:
2428:
2420:
2410:
2408:
2407:
2406:
2401:
2392:
2383:
2381:
2380:
2362:
2360:
2359:
2358:
2353:
2344:
2335:
2333:
2332:
2287:Erlangen program
2264:
2262:
2261:
2256:
2254:
2252:
2251:
2237:
2232:
2218:
2212:
2211:
2197:
2192:
2178:
2172:
2154:
2152:
2151:
2146:
2141:
2140:
2131:
2130:
2118:
2103:
2101:
2100:
2095:
2083:
2081:
2080:
2075:
2073:
2072:
2056:
2054:
2053:
2048:
2046:
2045:
2029:
2027:
2026:
2021:
1905:
1903:
1902:
1897:
1885:
1883:
1882:
1877:
1838:
1836:
1835:
1830:
1818:
1816:
1815:
1810:
1777:
1775:
1774:
1769:
1767:
1758:
1752:
1751:
1742:
1733:
1718:
1717:
1705:
1704:
1688:
1686:
1685:
1680:
1669:
1668:
1656:
1655:
1643:
1642:
1626:
1624:
1623:
1618:
1606:
1604:
1603:
1598:
1499:
1497:
1496:
1491:
1489:
1488:
1476:
1471:
1466:
1465:
1453:
1448:
1440:
1435:
1424:
1419:
1411:
1410:
1361:sphere inversion
1267:, then the pole
1100:intouch triangle
1006:
1004:
1003:
998:
993:
985:
954:
953:
950:
946:
945:
937:
795:
771:
751:
726:
719:
679:(different from
660:with the circle
543:
529:
504:inside a circle
502:
482:
471:
462:
444:
420:Draw the circle
380:
360:
341:
314:
260:
253:
233:circle inversion
227:
225:
224:
219:
214:
213:
201:
200:
161:
118:
34:is the study of
21:Point reflection
6219:
6218:
6214:
6213:
6212:
6210:
6209:
6208:
6194:
6193:
6145:
6131:10.2307/2303867
6117:Patterson, Boyd
6091:
6066:
6050:Coxeter, H.S.M.
6043:
6025:
5989:
5984:
5983:
5971:
5967:
5954:
5950:
5942:
5938:
5895:
5891:
5882:
5878:
5870:
5866:
5858:
5851:
5841:Wayback Machine
5831:
5824:
5816:
5812:
5804:
5800:
5789:
5785:
5780:
5728:
5679:
5675:
5669:
5665:
5647:
5643:
5637:
5633:
5621:
5616:
5597:
5592:
5586:
5583:
5582:
5541:
5532:
5528:
5517:
5513:
5511:
5493:
5489:
5478:
5474:
5472:
5460:
5455:
5436:
5431:
5425:
5422:
5421:
5412:
5403:
5366:
5362:
5356:
5352:
5334:
5330:
5324:
5320:
5308:
5303:
5284:
5279:
5273:
5270:
5269:
5255:
5247:anti-homography
5219:
5214:
5204:
5201:
5192:
5181:
5179:
5171:
5169:
5160:
5152:
5131:
5111:
5108:
5107:
5081:
5077:
5069:
5066:
5065:
5042:
4929:
4928:
4916:
4912:
4906:
4902:
4893:
4889:
4880:
4875:
4865:
4861:
4852:
4848:
4839:
4835:
4834:
4832:
4823:
4819:
4807:
4796:
4790:
4786:
4783:
4782:
4770:
4766:
4753:
4731:
4727:
4726:
4724:
4707:
4700:
4693:
4691:
4688:
4687:
4666:
4662:
4660:
4657:
4656:
4655:
4633:
4630:
4629:
4606:
4602:
4581:
4577:
4566:
4563:
4562:
4542:
4538:
4517:
4513:
4502:
4499:
4498:
4494:
4483:
4423:
4421:Higher geometry
4404:
4403:
4375:
4370:
4330:
4313:
4311:
4291:
4289:
4283:
4282:
4272:
4161:
4157:
4151:
4147:
4133:
4131:
4128:
4127:
4108:
4105:
4104:
4084:
4080:
4068:
4064:
4062:
4059:
4058:
4031:
4027:
4015:
4011:
4010:
4006:
4001:
3999:
3996:
3995:
3972:
3968:
3959:
3955:
3948:
3943:
3941:
3938:
3937:
3921:
3918:
3917:
3900:
3899:
3893:
3876:
3872:
3863:
3859:
3855:
3851:
3846:
3842:
3841:
3824:
3820:
3808:
3804:
3803:
3798:
3789:
3785:
3784:
3780:
3766:
3762:
3753:
3749:
3745:
3739:
3735:
3733:
3726:
3722:
3720:
3718:
3712:
3711:
3702:
3698:
3692:
3688:
3679:
3675:
3668:
3662:
3658:
3656:
3644:
3640:
3634:
3630:
3621:
3617:
3610:
3603:
3599:
3595:
3593:
3578:
3574:
3562:
3558:
3554:
3547:
3543:
3537:
3533:
3523:
3521:
3512:
3508:
3500:
3498:
3495:
3494:
3475:
3472:
3471:
3470:the result for
3454:
3450:
3441:
3437:
3432:
3429:
3428:
3412:
3404:
3401:
3400:
3372:
3363:
3359:
3351:
3348:
3347:
3322:
3319:
3318:
3294:
3288:
3284:
3275:
3271:
3266:
3265:
3260:
3258:
3255:
3254:
3234:
3230:
3221:
3217:
3216:
3211:
3209:
3206:
3205:
3189:
3186:
3185:
3184:and hence that
3162:
3158:
3152:
3148:
3139:
3135:
3131:
3125:
3121:
3119:
3107:
3103:
3097:
3093:
3084:
3080:
3076:
3070:
3066:
3064:
3052:
3048:
3027:
3023:
3014:
3010:
3006:
3001:
2992:
2988:
2983:
2980:
2979:
2960:
2957:
2956:
2934:
2930:
2925:
2917:
2914:
2913:
2891:
2883:
2880:
2879:
2859:
2855:
2846:
2842:
2813:
2810:
2809:
2791:
2788:
2787:
2771:
2768:
2767:
2764:
2705:
2701:
2699:
2684:
2679:
2671:
2668:
2667:
2642:
2639:
2638:
2612:
2607:
2606:
2598:
2597:
2587:
2585:
2572:
2570:
2567:
2566:
2515:
2514:
2512:
2509:
2508:
2471:
2468:
2467:
2461:
2433:
2419:
2415:
2414:
2402:
2397:
2396:
2388:
2387:
2382:
2376:
2372:
2354:
2349:
2348:
2340:
2339:
2334:
2328:
2324:
2316:
2313:
2312:
2298:
2271:
2247:
2233:
2228:
2214:
2213:
2207:
2193:
2188:
2174:
2173:
2171:
2163:
2160:
2159:
2136:
2132:
2126:
2122:
2114:
2109:
2106:
2105:
2089:
2086:
2085:
2068:
2064:
2062:
2059:
2058:
2041:
2037:
2035:
2032:
2031:
1997:
1994:
1993:
1986:
1967:inversive plane
1928:
1912:
1891:
1888:
1887:
1844:
1841:
1840:
1824:
1821:
1820:
1783:
1780:
1779:
1756:
1747:
1743:
1731:
1713:
1709:
1700:
1696:
1694:
1691:
1690:
1664:
1660:
1651:
1647:
1638:
1634:
1632:
1629:
1628:
1612:
1609:
1608:
1592:
1589:
1588:
1573:
1564:
1556:
1544:
1535:
1530:
1484:
1480:
1472:
1467:
1461:
1457:
1449:
1444:
1436:
1431:
1420:
1415:
1406:
1402:
1388:
1385:
1384:
1333:
1263:lies on a line
1236:then the lines
1202:inversion point
1180:The polar line
1174:
1168:
1115:medial triangle
1088:
1050:
986:
978:
951: and
949:
938:
930:
907:
904:
903:
865:If the circles
862:are orthogonal.
799:
796:
787:
772:
763:
752:
743:
724:
717:
607:
541:
527:
500:
491:
480:
469:
460:
442:
378:
369:
358:
339:
312:
303:
258:
251:
247:) to its image
237:plane inversion
231:This is called
209:
205:
196:
192:
178:
175:
174:
159:
116:
107:
90:
42:Euclidean plane
24:
17:
12:
11:
5:
6217:
6207:
6206:
6192:
6191:
6185:
6166:
6160:
6155:
6144:
6143:External links
6141:
6140:
6139:
6114:
6094:
6089:
6069:
6064:
6046:
6041:
6028:
6023:
6010:
5988:
5985:
5982:
5981:
5965:
5961:Mir Publishers
5948:
5936:
5907:(4): 430–498.
5889:
5876:
5874:, p. 269)
5864:
5862:, p. 265)
5849:
5822:
5820:, p. 264)
5810:
5808:, p. 230)
5798:
5782:
5781:
5779:
5776:
5775:
5774:
5769:
5764:
5762:Soddy's hexlet
5759:
5754:
5749:
5744:
5739:
5734:
5727:
5724:
5711:
5710:
5699:
5696:
5693:
5690:
5687:
5682:
5678:
5672:
5668:
5664:
5661:
5658:
5655:
5650:
5646:
5640:
5636:
5632:
5629:
5624:
5619:
5615:
5611:
5608:
5605:
5600:
5595:
5591:
5568:
5567:
5556:
5553:
5548:
5545:
5540:
5535:
5531:
5525:
5520:
5516:
5510:
5507:
5504:
5501:
5496:
5492:
5486:
5481:
5477:
5471:
5468:
5463:
5458:
5454:
5450:
5447:
5444:
5439:
5434:
5430:
5408:
5401:
5395:
5394:
5383:
5380:
5377:
5374:
5369:
5365:
5359:
5355:
5351:
5348:
5345:
5342:
5337:
5333:
5327:
5323:
5319:
5316:
5311:
5306:
5302:
5298:
5295:
5292:
5287:
5282:
5278:
5265:with equation
5254:
5251:
5197:
5190:
5165:
5156:
5140:
5135:
5130:
5127:
5124:
5121:
5118:
5115:
5095:
5092:
5089:
5084:
5080:
5076:
5073:
5041:
5038:
5004:conformal maps
4984:–2)-flat is a
4943:
4942:
4927:
4919:
4915:
4909:
4905:
4901:
4896:
4892:
4888:
4883:
4879:
4873:
4868:
4864:
4860:
4855:
4851:
4847:
4842:
4838:
4831:
4826:
4822:
4818:
4814:
4810:
4806:
4802:
4799:
4797:
4793:
4789:
4785:
4784:
4781:
4773:
4769:
4765:
4762:
4759:
4756:
4751:
4748:
4745:
4742:
4739:
4734:
4730:
4723:
4720:
4717:
4713:
4710:
4706:
4703:
4701:
4699:
4696:
4695:
4669:
4665:
4643:
4640:
4637:
4614:
4609:
4605:
4601:
4598:
4595:
4592:
4589:
4584:
4580:
4576:
4573:
4570:
4550:
4545:
4541:
4537:
4534:
4531:
4528:
4525:
4520:
4516:
4512:
4509:
4506:
4482:
4479:
4432:Riemann sphere
4422:
4419:
4418:
4417:
4402:
4396:
4393:
4390:
4387:
4384:
4381:
4378:
4374:
4369:
4366:
4363:
4360:
4357:
4354:
4351:
4345:
4342:
4339:
4336:
4333:
4328:
4325:
4322:
4319:
4316:
4310:
4307:
4304:
4301:
4298:
4295:
4292:
4288:
4285:
4284:
4279:
4276:
4271:
4268:
4265:
4262:
4259:
4256:
4253:
4250:
4247:
4244:
4241:
4238:
4235:
4232:
4229:
4226:
4223:
4220:
4217:
4214:
4211:
4208:
4205:
4202:
4199:
4196:
4193:
4190:
4187:
4184:
4181:
4178:
4175:
4172:
4169:
4164:
4160:
4154:
4150:
4146:
4143:
4140:
4137:
4135:
4112:
4092:
4087:
4083:
4079:
4076:
4071:
4067:
4040:
4034:
4030:
4026:
4023:
4018:
4014:
4009:
4005:
3980:
3975:
3971:
3967:
3962:
3958:
3954:
3951:
3947:
3925:
3914:
3913:
3896:
3891:
3885:
3879:
3875:
3871:
3866:
3862:
3858:
3854:
3850:
3845:
3840:
3836:
3827:
3823:
3819:
3816:
3811:
3807:
3802:
3797:
3792:
3788:
3783:
3778:
3769:
3765:
3761:
3756:
3752:
3748:
3742:
3738:
3732:
3729:
3725:
3721:
3717:
3714:
3713:
3705:
3701:
3695:
3691:
3687:
3682:
3678:
3674:
3671:
3665:
3661:
3655:
3647:
3643:
3637:
3633:
3629:
3624:
3620:
3616:
3613:
3606:
3602:
3598:
3592:
3586:
3581:
3577:
3573:
3570:
3565:
3561:
3557:
3550:
3546:
3540:
3536:
3532:
3529:
3526:
3520:
3515:
3511:
3507:
3504:
3502:
3479:
3457:
3453:
3449:
3444:
3440:
3436:
3415:
3411:
3408:
3386:
3380:
3377:
3371:
3366:
3362:
3358:
3355:
3335:
3332:
3329:
3326:
3304:
3297:
3291:
3287:
3283:
3278:
3274:
3269:
3264:
3237:
3233:
3229:
3224:
3220:
3215:
3193:
3182:
3181:
3165:
3161:
3155:
3151:
3147:
3142:
3138:
3134:
3128:
3124:
3118:
3110:
3106:
3100:
3096:
3092:
3087:
3083:
3079:
3073:
3069:
3063:
3060:
3055:
3051:
3047:
3044:
3041:
3035:
3030:
3026:
3022:
3017:
3013:
3009:
3005:
3000:
2995:
2991:
2987:
2964:
2953:
2952:
2937:
2933:
2929:
2924:
2921:
2898:
2894:
2890:
2887:
2876:
2875:
2862:
2858:
2854:
2849:
2845:
2841:
2838:
2835:
2832:
2829:
2826:
2823:
2820:
2817:
2795:
2775:
2763:
2760:
2736:
2735:
2721:
2717:
2712:
2709:
2704:
2698:
2691:
2688:
2683:
2678:
2675:
2652:
2649:
2646:
2635:
2634:
2623:
2615:
2610:
2605:
2601:
2594:
2591:
2584:
2579:
2576:
2543:
2540:
2537:
2534:
2531:
2528:
2522:
2519:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2465:complex number
2460:
2457:
2456:
2455:
2444:
2441:
2436:
2431:
2426:
2423:
2418:
2413:
2405:
2400:
2395:
2391:
2386:
2379:
2375:
2371:
2368:
2365:
2357:
2352:
2347:
2343:
2338:
2331:
2327:
2323:
2320:
2297:
2294:
2270:
2267:
2266:
2265:
2250:
2246:
2243:
2240:
2236:
2231:
2227:
2224:
2221:
2217:
2210:
2206:
2203:
2200:
2196:
2191:
2187:
2184:
2181:
2177:
2170:
2167:
2144:
2139:
2135:
2129:
2125:
2121:
2117:
2113:
2093:
2071:
2067:
2044:
2040:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1985:
1982:
1978:Riemann sphere
1927:
1924:
1911:
1908:
1895:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1828:
1808:
1805:
1802:
1799:
1796:
1793:
1790:
1787:
1764:
1761:
1755:
1750:
1746:
1739:
1736:
1730:
1727:
1724:
1721:
1716:
1712:
1708:
1703:
1699:
1678:
1675:
1672:
1667:
1663:
1659:
1654:
1650:
1646:
1641:
1637:
1616:
1596:
1572:
1569:
1563:
1560:
1555:
1552:
1543:
1540:
1534:
1531:
1529:
1526:
1487:
1483:
1479:
1475:
1470:
1464:
1460:
1456:
1452:
1447:
1443:
1439:
1434:
1430:
1427:
1423:
1418:
1414:
1409:
1405:
1401:
1398:
1395:
1392:
1332:
1329:
1328:
1327:
1324:
1313:
1310:
1307:
1284:
1172:pole and polar
1170:Main article:
1167:
1166:Pole and polar
1164:
1087:
1084:
1083:
1082:
1076:
1070:
1067:
1049:
1046:
1045:
1044:
1041:
1030:
1010:
1009:
1008:
1007:
996:
992:
989:
984:
981:
977:
974:
971:
968:
965:
962:
959:
944:
941:
936:
933:
929:
926:
923:
920:
917:
914:
911:
898:
897:
886:
863:
840:
839:
824:
801:
800:
797:
790:
788:
778:going through
773:
766:
764:
753:
746:
742:
739:
738:
737:
692:
675:on the circle
665:
606:
603:
602:
601:
587:
572:
565:
550:
532:
490:
487:
486:
485:
465:
455:
433:
428:going through
418:
407:
368:
365:
302:
299:
271:total function
267:self-inversion
229:
228:
217:
212:
208:
204:
199:
195:
191:
188:
185:
182:
106:
103:
98:Mandelbrot set
89:
86:
15:
9:
6:
4:
3:
2:
6216:
6205:
6202:
6201:
6199:
6189:
6186:
6181:
6180:
6175:
6172:
6167:
6164:
6161:
6159:
6156:
6154:
6150:
6147:
6146:
6138:
6135:
6132:
6128:
6124:
6123:
6118:
6115:
6112:
6108:
6104:
6100:
6095:
6092:
6090:0-387-98650-2
6086:
6082:
6078:
6074:
6070:
6067:
6065:0-471-18283-4
6061:
6057:
6056:
6051:
6047:
6044:
6042:0-521-59787-0
6038:
6034:
6029:
6026:
6024:0-8218-2636-0
6020:
6016:
6011:
6008:
6004:
6000:
5996:
5991:
5990:
5979:
5977:
5969:
5962:
5958:
5952:
5945:
5940:
5932:
5928:
5923:
5918:
5914:
5910:
5906:
5902:
5901:
5893:
5886:
5880:
5873:
5868:
5861:
5856:
5854:
5846:
5842:
5838:
5835:
5829:
5827:
5819:
5814:
5807:
5802:
5794:
5793:
5787:
5783:
5773:
5770:
5768:
5765:
5763:
5760:
5758:
5755:
5753:
5750:
5748:
5745:
5743:
5742:Inverse curve
5740:
5738:
5735:
5733:
5730:
5729:
5723:
5719:
5717:
5697:
5694:
5691:
5688:
5685:
5680:
5676:
5670:
5666:
5662:
5659:
5656:
5653:
5648:
5644:
5638:
5634:
5630:
5627:
5622:
5617:
5613:
5609:
5606:
5603:
5598:
5593:
5589:
5581:
5580:
5579:
5577:
5573:
5554:
5551:
5546:
5543:
5538:
5533:
5529:
5523:
5518:
5514:
5508:
5505:
5502:
5499:
5494:
5490:
5484:
5479:
5475:
5469:
5466:
5461:
5456:
5452:
5448:
5445:
5442:
5437:
5432:
5428:
5420:
5419:
5418:
5416:
5411:
5407:
5400:
5381:
5378:
5375:
5372:
5367:
5363:
5357:
5353:
5349:
5346:
5343:
5340:
5335:
5331:
5325:
5321:
5317:
5314:
5309:
5304:
5300:
5296:
5293:
5290:
5285:
5280:
5276:
5268:
5267:
5266:
5264:
5262:
5250:
5248:
5244:
5240:
5236:
5231:
5229:
5222:
5217:
5211:
5207:
5200:
5196:
5189:
5184:
5174:
5168:
5164:
5159:
5155:
5138:
5133:
5128:
5125:
5119:
5093:
5090:
5087:
5082:
5078:
5074:
5071:
5063:
5059:
5055:
5051:
5047:
5037:
5035:
5031:
5027:
5025:
5020:
5015:
5013:
5009:
5005:
5000:
4998:
4994:
4989:
4987:
4983:
4979:
4975:
4971:
4967:
4962:
4960:
4956:
4952:
4948:
4925:
4917:
4907:
4903:
4899:
4894:
4890:
4881:
4877:
4866:
4862:
4858:
4853:
4849:
4840:
4836:
4829:
4824:
4820:
4816:
4812:
4808:
4804:
4798:
4791:
4787:
4779:
4771:
4763:
4760:
4757:
4746:
4743:
4740:
4732:
4728:
4721:
4718:
4715:
4711:
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2466:
2459:Reciprocation
2442:
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2137:
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2127:
2123:
2115:
2111:
2091:
2069:
2065:
2042:
2038:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1991:
1981:
1979:
1975:
1970:
1968:
1964:
1960:
1956:
1952:
1948:
1944:
1939:
1937:
1936:Edward Kasner
1933:
1923:
1921:
1917:
1907:
1893:
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1568:
1559:
1551:
1549:
1548:Dupin cyclide
1539:
1525:
1523:
1519:
1515:
1511:
1507:
1503:
1485:
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1270:
1266:
1262:
1258:
1257:
1256:
1253:
1251:
1247:
1243:
1239:
1238:perpendicular
1235:
1231:
1223:
1219:
1215:
1211:
1207:
1203:
1199:
1196:. The point
1195:
1191:
1187:
1183:
1178:
1173:
1163:
1161:
1156:
1154:
1150:
1145:
1143:
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1128:
1124:
1120:
1116:
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1108:
1103:
1101:
1097:
1093:
1081:
1077:
1075:
1071:
1068:
1065:
1064:
1060:
1059:the SVG file,
1054:
1042:
1039:
1035:
1031:
1028:
1024:
1020:
1016:
1012:
1011:
994:
990:
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745:
744:
735:
731:
727:
720:
713:
709:
705:
701:
697:
693:
690:
687:antipodal to
686:
682:
678:
674:
670:
666:
663:
659:
655:
651:
650:
649:
647:
643:
639:
635:
630:
628:
624:
620:
616:
612:
599:
595:
592:is where ray
591:
588:
585:
581:
577:
573:
570:
566:
563:
559:
555:
551:
548:
544:
537:
533:
530:
523:
519:
515:
511:
510:
509:
507:
503:
496:
483:
476:
472:
466:
463:
457:Draw segment
456:
453:
449:
445:
438:
434:
431:
427:
423:
419:
417:. (Not shown)
416:
412:
408:
405:
401:
397:
393:
392:
391:
389:
385:
381:
374:
361:
354:
350:
346:
343:are similar.
342:
335:
331:
327:
323:
319:
315:
307:
298:
295:
291:
286:
284:
280:
276:
272:
268:
264:
257:
250:
246:
242:
238:
234:
215:
210:
206:
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193:
189:
186:
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173:
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158:
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146:
142:
138:
134:
130:
122:
115:
111:
99:
94:
85:
83:
78:
76:
73:(1842–3) and
72:
68:
64:
60:
56:
51:
47:
43:
39:
38:
33:
29:
22:
6177:
6153:cut-the-knot
6125:48: 589–99,
6120:
6101:, New York:
6098:
6083:, Springer,
6080:
6054:
6032:
6014:
5994:
5975:
5968:
5956:
5951:
5944:Coxeter 1969
5939:
5904:
5898:
5892:
5884:
5879:
5867:
5813:
5801:
5790:
5786:
5720:
5712:
5575:
5571:
5569:
5414:
5409:
5405:
5398:
5396:
5260:
5256:
5238:
5234:
5232:
5227:
5220:
5218:= 1/‖
5215:
5209:
5205:
5198:
5194:
5187:
5182:
5172:
5166:
5162:
5157:
5153:
5061:
5049:
5043:
5033:
5023:
5016:
5001:
4990:
4981:
4969:
4963:
4951:hyperspheres
4944:
4490:
4486:
4484:
4476:
4463:
4424:
4056:
3915:
3398:
3316:
3183:
2954:
2877:
2765:
2744:Möbius group
2737:
2636:
2559:
2462:
2299:
2275:L. I. Magnus
2272:
2104:will become
1987:
1971:
1966:
1963:Möbius plane
1955:affine plane
1940:
1929:
1913:
1582:
1565:
1557:
1545:
1536:
1521:
1517:
1513:
1509:
1505:
1501:
1380:
1376:
1372:
1371:with radius
1368:
1364:
1358:
1320:
1316:
1303:
1302:of the line
1299:
1295:
1294:, its polar
1291:
1287:
1280:
1276:
1272:
1271:of the line
1268:
1264:
1260:
1254:
1241:
1240:to the line
1233:
1229:
1227:
1221:
1217:
1213:
1209:
1205:
1197:
1193:
1189:
1185:
1181:
1157:
1146:
1131:
1122:
1118:
1110:
1109:of triangle
1104:
1089:
1037:
1033:
1026:
1022:
1018:
1014:
893:
889:
882:
878:
874:
870:
866:
859:
855:
851:
847:
846:If a circle
841:
831:
827:
820:
816:
812:
808:
802:
783:
779:
775:
759:
755:
733:
729:
722:
715:
711:
707:
703:
699:
695:
688:
684:
680:
676:
672:
668:
661:
657:
653:
645:
641:
640:and a point
637:
636:with center
633:
631:
626:
622:
618:
614:
610:
608:
597:
593:
589:
583:
579:
575:
568:
561:
557:
553:
546:
539:
535:
525:
521:
517:
513:
505:
498:
494:
492:
478:
474:
467:
458:
451:
447:
440:
436:
429:
425:
424:with center
421:
414:
410:
403:
399:
395:
387:
383:
376:
375:the inverse
370:
356:
352:
348:
344:
337:
333:
329:
325:
321:
317:
310:
293:
287:
282:
274:
262:
255:
248:
244:
243:(other than
240:
236:
232:
230:
167:
163:
156:
152:
148:
147:with center
144:
140:
136:
126:
120:
113:
79:
35:
31:
25:
6174:"Inversion"
5847:14: 237–240
5186:‖ =
4986:fixed point
4966:translation
4947:hyperplanes
4456:Felix Klein
4448:Lobachevsky
3994:and radius
3253:and radius
2748:conjugation
2283:Felix Klein
1990:cross-ratio
1932:Mario Pieri
1375:is a point
1315:If a point
1286:If a point
1259:If a point
1184:to a point
1086:Application
1057:circle. In
714:in a point
656:of the ray
621:of whether
619:independent
497:of a point
382:of a point
316:of a point
254:also takes
155:is a point
151:and radius
139:of a point
5987:References
5243:homography
4997:similarity
4493:of radius
4485:In a real
2556:reciprocal
2302:similarity
1383:such that
1134:concentric
1096:Euler line
836:orthogonal
741:Properties
600:intersect.
574:Draw line
564:intersect.
534:Draw line
524:) through
484:intersect.
454:intersect.
170:such that
129:reciprocal
63:Bellavitis
44:that maps
6179:MathWorld
6052:(1969) ,
5872:Kay (1969
5860:Kay (1969
5818:Kay (1969
5657:⋯
5607:⋯
5503:⋯
5446:⋯
5344:⋯
5294:⋯
5170:/‖
5129:−
5075:⋅
5046:conformal
5036:-sphere.
4959:homothety
4900:−
4878:∑
4859:−
4801:↦
4768:‖
4761:−
4755:‖
4744:−
4705:↦
4639:−
4386:
4380:⋅
4368:−
4356:
4350:⋅
4335:
4318:
4297:
4287:⟺
4258:
4243:
4237:−
4225:
4210:
4204:⟺
4183:
4174:⟺
4163:∗
4153:∗
4078:→
4070:∗
4025:−
4017:∗
3966:−
3961:∗
3870:−
3865:∗
3818:−
3810:∗
3796:−
3791:∗
3760:−
3755:∗
3741:∗
3731:−
3716:⟺
3686:−
3681:∗
3628:−
3623:∗
3605:∗
3572:−
3564:∗
3549:∗
3539:∗
3519:−
3514:∗
3448:≠
3443:∗
3365:∗
3328:→
3282:−
3228:−
3146:−
3091:−
3054:∗
3021:−
2999:−
2994:∗
2936:∗
2889:∈
2848:∗
2837:−
2822:−
2756:conformal
2740:generator
2720:¯
2690:¯
2648:↦
2593:¯
2554:then the
2533:−
2521:¯
2370:↦
2322:↦
2242:−
2223:−
2202:−
2183:−
1984:Invariant
1868:−
1819:, radius
1801:−
1778:; center
1674:−
1463:′
1442:⋅
1408:′
1397:⋅
1279:of point
1228:If point
1149:congruent
1127:collinear
1092:collinear
973:∠
958:∠
925:∠
910:∠
710:cuts ray
596:and line
512:Draw ray
473:is where
373:construct
294:invariant
198:′
187:⋅
37:inversion
6198:Category
6111:69-12075
6075:(2000),
6033:Geometry
6007:52-13504
5963:, Moscow
5837:Archived
5726:See also
5404:+ ... +
5224:‖
5193:+ ... +
5180:‖
5178:, where
5176:‖
5054:Jacobian
5050:oriented
4993:isometry
4978:rotation
4955:dilation
4813:′
4712:′
4464:geometry
4436:Beltrami
4123:becomes
3410:∉
2296:Dilation
1961:forms a
1554:Spheroid
1107:incircle
1074:cardioid
991:′
983:′
943:′
935:′
702:in line
617:that is
578:through
538:through
261:back to
166:through
77:(1845).
65:(1836),
61:(1825),
59:Quetelet
57:(1824),
28:geometry
6190:Xah Lee
6137:0006034
5931:1986367
5213:, with
5026:-sphere
2742:of the
2663:where:
1200:is the
1098:of the
706:. Then
324:: Let
137:inverse
55:Steiner
46:circles
6109:
6087:
6062:
6039:
6021:
6005:
5929:
5203:gives
4466:for a
4452:Bolyai
4442:, and
4440:Cayley
1953:, any
1533:Sphere
1113:. The
955:
947:
896:, then
355:is to
347:is to
135:, the
75:Kelvin
71:Ingram
67:Stubbs
5927:JSTOR
5778:Notes
5237:to 1/
4472:group
4468:space
4444:Klein
4057:When
3317:When
2504:with
1974:model
1246:polar
725:'
718:'
542:'
528:'
516:from
501:'
481:'
470:'
461:'
443:'
402:) to
379:'
359:'
340:'
313:'
259:'
252:'
160:'
133:plane
117:'
50:lines
6107:LCCN
6085:ISBN
6060:ISBN
6037:ISBN
6019:ISBN
6003:LCCN
5257:The
5106:and
4974:flat
4972:–2)-
4450:and
3427:and
3399:For
2057:and
1988:The
1914:The
1250:pole
1220:and
1158:The
1125:are
1017:and
869:and
858:and
694:Let
560:and
552:Let
477:and
450:and
439:and
435:Let
409:Let
336:and
69:and
6151:at
6127:doi
5917:hdl
5909:doi
5114:det
4957:or
4949:or
3490:is
2562:is
2558:of
2285:'s
1827:0.5
1804:0.5
1252:).
1204:of
1123:ABC
1119:ABC
1111:ABC
776:not
371:To
351:as
338:ONP
334:OPN
235:or
48:or
26:In
6200::
6176:.
6134:MR
6105:,
6079:,
6001:,
5959:,
5925:.
5915:.
5903:.
5852:^
5843:,
5825:^
5555:0.
5210:kI
5208:=
5206:JJ
5161:=
5014:.
4999:.
4438:,
4383:Im
4353:Re
4332:Im
4315:Re
4294:Im
4255:Im
4240:Im
4222:Re
4207:Re
4180:Re
4054:.
2758:.
2304:,
1980:.
1972:A
1922:.
1583:A
1550:.
1524:.
1381:OP
1242:PR
1216:,
1155:.
1129:.
721:.
712:OC
704:BC
700:BA
658:OA
648:.
629:.
584:ON
569:ON
508::
479:NN
475:OP
459:NN
415:OP
390::
357:OP
345:OP
84:.
30:,
6182:.
6129::
5976:n
5933:.
5919::
5911::
5905:1
5698:,
5695:0
5692:=
5689:1
5686:+
5681:n
5677:x
5671:n
5667:a
5663:2
5660:+
5654:+
5649:1
5645:x
5639:1
5635:a
5631:2
5628:+
5623:2
5618:n
5614:x
5610:+
5604:+
5599:2
5594:1
5590:x
5576:n
5572:c
5552:=
5547:c
5544:1
5539:+
5534:n
5530:x
5524:c
5519:n
5515:a
5509:2
5506:+
5500:+
5495:1
5491:x
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5480:1
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5470:2
5467:+
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5443:+
5438:2
5433:1
5429:x
5415:c
5410:n
5406:a
5402:1
5399:a
5382:0
5379:=
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5373:+
5368:n
5364:x
5358:n
5354:a
5350:2
5347:+
5341:+
5336:1
5332:x
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5191:1
5188:x
5183:x
5173:x
5167:i
5163:x
5158:i
5154:z
5139:.
5134:k
5126:=
5123:)
5120:J
5117:(
5094:I
5091:k
5088:=
5083:T
5079:J
5072:J
5062:J
5034:n
5024:n
4982:n
4970:n
4926:.
4918:2
4914:)
4908:k
4904:o
4895:k
4891:p
4887:(
4882:k
4872:)
4867:j
4863:o
4854:j
4850:p
4846:(
4841:2
4837:r
4830:+
4825:j
4821:o
4817:=
4809:j
4805:p
4792:j
4788:p
4780:,
4772:2
4764:O
4758:P
4750:)
4747:O
4741:P
4738:(
4733:2
4729:r
4722:+
4719:O
4716:=
4709:P
4698:P
4682::
4668:2
4664:r
4642:O
4636:P
4613:)
4608:n
4604:p
4600:,
4597:.
4594:.
4591:.
4588:,
4583:1
4579:p
4575:(
4572:=
4569:P
4549:)
4544:n
4540:o
4536:,
4533:.
4530:.
4527:.
4524:,
4519:1
4515:o
4511:(
4508:=
4505:O
4495:r
4487:n
4401:.
4395:}
4392:a
4389:{
4377:2
4373:1
4365:}
4362:w
4359:{
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4341:a
4338:{
4327:}
4324:a
4321:{
4309:=
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4303:w
4300:{
4278:2
4275:1
4270:=
4267:}
4264:w
4261:{
4252:}
4249:a
4246:{
4234:}
4231:w
4228:{
4219:}
4216:a
4213:{
4201:1
4198:=
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4192:w
4189:a
4186:{
4177:2
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4168:=
4159:w
4149:a
4145:+
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3839:=
3835:)
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3815:a
3806:a
3801:a
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3777:)
3768:2
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3724:(
3704:2
3700:)
3694:2
3690:r
3677:a
3673:a
3670:(
3664:2
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3654:=
3646:2
3642:)
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3632:r
3619:a
3615:a
3612:(
3601:a
3597:a
3591:+
3585:)
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3569:a
3560:a
3556:(
3545:w
3535:a
3531:+
3528:w
3525:a
3510:w
3506:w
3478:w
3456:2
3452:r
3439:a
3435:a
3414:R
3407:a
3385:.
3379:a
3376:1
3370:=
3361:w
3357:+
3354:w
3334:,
3331:r
3325:a
3303:.
3296:|
3290:2
3286:r
3277:2
3273:a
3268:|
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3236:2
3232:r
3223:2
3219:a
3214:a
3192:w
3164:2
3160:)
3154:2
3150:r
3141:2
3137:a
3133:(
3127:2
3123:r
3117:=
3109:2
3105:)
3099:2
3095:r
3086:2
3082:a
3078:(
3072:2
3068:a
3062:+
3059:)
3050:w
3046:+
3043:w
3040:(
3034:)
3029:2
3025:r
3016:2
3012:a
3008:(
3004:a
2990:w
2986:w
2963:w
2932:z
2928:1
2923:=
2920:w
2897:.
2893:R
2886:a
2861:2
2857:r
2853:=
2844:)
2840:a
2834:z
2831:(
2828:)
2825:a
2819:z
2816:(
2794:a
2774:r
2734:.
2716:)
2711:z
2708:1
2703:(
2697:=
2687:z
2682:1
2677:=
2674:w
2651:w
2645:z
2622:.
2614:2
2609:|
2604:z
2600:|
2590:z
2583:=
2578:z
2575:1
2560:z
2542:,
2539:y
2536:i
2530:x
2527:=
2518:z
2492:,
2489:y
2486:i
2483:+
2480:x
2477:=
2474:z
2443:.
2440:x
2435:2
2430:)
2425:R
2422:T
2417:(
2412:=
2404:2
2399:|
2394:y
2390:|
2385:y
2378:2
2374:T
2367:y
2364:=
2356:2
2351:|
2346:x
2342:|
2337:x
2330:2
2326:R
2319:x
2249:|
2245:z
2239:y
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2220:x
2216:|
2209:|
2205:z
2199:w
2195:|
2190:|
2186:y
2180:x
2176:|
2169:=
2166:I
2143:)
2138:2
2134:r
2128:1
2124:r
2120:(
2116:/
2112:d
2092:d
2070:2
2066:r
2043:1
2039:r
2018:w
2015:,
2012:z
2009:,
2006:y
2003:,
2000:x
1894:N
1874:)
1871:1
1865:,
1862:0
1859:,
1856:0
1853:(
1850:=
1847:S
1807:)
1798:,
1795:0
1792:,
1789:0
1786:(
1763:4
1760:1
1754:=
1749:2
1745:)
1738:2
1735:1
1729:+
1726:z
1723:(
1720:+
1715:2
1711:y
1707:+
1702:2
1698:x
1677:z
1671:=
1666:2
1662:z
1658:+
1653:2
1649:y
1645:+
1640:2
1636:x
1615:S
1595:N
1522:O
1518:O
1514:O
1510:O
1506:O
1502:O
1486:2
1482:R
1478:=
1474:|
1469:|
1459:P
1455:O
1451:|
1446:|
1438:|
1433:|
1429:P
1426:O
1422:|
1417:|
1413:=
1404:P
1400:O
1394:P
1391:O
1377:P
1373:R
1369:O
1365:P
1321:P
1317:P
1306:.
1304:l
1300:L
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1292:l
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1281:P
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1269:L
1265:l
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1234:P
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1224:.
1222:Q
1218:P
1214:O
1210:P
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1198:P
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1186:Q
1182:q
1038:k
1034:k
1029:.
1027:k
1023:k
1019:q
1015:p
995:.
988:B
980:A
976:O
970:=
967:A
964:B
961:O
940:A
932:B
928:O
922:=
919:B
916:A
913:O
894:k
890:k
885:.
883:k
879:q
875:k
871:q
867:k
860:q
856:k
852:k
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828:O
821:O
817:O
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780:O
760:O
756:O
736:.
734:P
730:A
723:A
716:A
708:h
696:h
691:)
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685:P
681:C
677:P
673:B
669:C
664:.
662:P
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634:P
627:P
623:A
615:P
611:A
598:t
594:r
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586:.
580:N
576:t
571:.
562:s
558:Ø
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547:r
540:P
536:s
526:P
522:Ø
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452:c
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430:P
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422:c
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400:Ø
396:O
388:Ø
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362:.
353:r
349:r
330:Ø
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322:Ø
318:P
311:P
283:O
275:O
263:P
256:P
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216:.
211:2
207:r
203:=
194:P
190:O
184:P
181:O
168:P
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157:P
153:r
149:O
141:P
121:P
114:P
23:.
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