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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.
2757:. 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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4421:{\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}}}
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3917:{\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}}}
4951:
1577:
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.
1067:
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
5732:
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
5806:
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
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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
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63:
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
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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.
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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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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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4946:{\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}}}
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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.
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845:. 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
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to the space obviates the distinction between hyperplane and hypersphere; higher dimensional inversive geometry is frequently studied then in the presumed context of an
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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.
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1051:, 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'.
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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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in E can be used to generate dilations, translations, or rotations. Indeed, two concentric hyperspheres, used to produce successive inversions, result in a
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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
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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
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916:
5595:
1980:. 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.
5282:
1132:, meaning the circumcenter of the medial triangle, that is, the nine-point center of the intouch triangle, the incenter and circumcenter of triangle
2761:
mapping. Neither conjugation nor inversion-in-a-circle are in the Möbius group since they are non-conformal (see below). Möbius group elements are
1638:(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
1101:
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
383:
1397:
103:
5055:
The circle inversion map is anticonformal, which means that at every point it preserves angles and reverses orientation (a map is called
4445:. It was subspaces and subgroups of this space and group of mappings that were applied to produce early models of hyperbolic geometry by
1703:
5847:
1523:. A circle, that is, the intersection of a sphere with a secant plane, inverts into a circle, except that if the circle passes through
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6051:
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1965:
85:
5017:, and in fact, where the space has three or more dimensions, the mappings generated by inversion are the only conformal mappings.
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with negative determinant: in two dimensions the
Jacobian must be a scalar times a reflection at every point. This means that if
2579:
4485:
of mappings of that space. The significant properties of figures in the geometry are those that are invariant under this group.
3185:{\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}}}}
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1850:, green in the picture), then it will be mapped by the inversion at the unit sphere (red) onto the tangent plane at point
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5571:{\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.}
2459:{\displaystyle x\mapsto R^{2}{\frac {x}{|x|^{2}}}=y\mapsto T^{2}{\frac {y}{|y|^{2}}}=\left({\frac {T}{R}}\right)^{2}x.}
6214:
5252:. The complex analytic inverse map is conformal and its conjugate, circle inversion, is anticonformal. In this case a
1077:
Inversion of a line is a circle containing the center of inversion; or it is the line itself if it contains the center
5807:
independently discovered by Giusto
Bellavitis in 1836, by Stubbs and Ingram in 1842–3, and by Lord Kelvin in 1845.)"
4488:
For example, Smogorzhevsky develops several theorems of inversive geometry before beginning
Lobachevskian geometry.
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was so overcome by this facility of mappings to identify geometrical phenomena that he delivered a manifesto, the
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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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292:, a single point placed on all the lines, and extend the inversion, by definition, to interchange the center
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1511:. As with the 2D version, a sphere inverts to a sphere, except that if a sphere passes through the center
139:
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5585:= 1. But this is the condition of being orthogonal to the unit sphere. Hence we are led to consider the (
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Kasner, E. (1900). "The
Invariant Theory of the Inversion Group: Geometry Upon a Quadric Surface".
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Inversion of a circle is another circle; or it is a line if the original circle contains the center
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Inversion with respect to a circle does not map the center of the circle to the center of its image
77:
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Inversion leaves the measure of angles unaltered, but reverses the orientation of oriented angles.
5937:
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1961:
1148:
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Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1998), "Chapter 5: Inversive
Geometry",
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1917:) are mapped onto themselves. They are the projection lines of the stereographic projection.
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When two parallel hyperplanes are used to produce successive reflections, the result is a
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5043:. Inversive geometry has been applied to the study of colorings, or partitionings, of an
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1973:
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it inverts into a line. This reduces to the 2D case when the secant plane passes through
861:
passes through two distinct points A and A' which are inverses with respect to a circle
120:
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6006:
College
Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle
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is invariant under an inversion. In particular if O is the centre of the inversion and
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as the base space. The transformations of inversive geometry are often referred to as
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2738:{\displaystyle w={\frac {1}{\bar {z}}}={\overline {\left({\frac {1}{z}}\right)}}}
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with the center of the reference circle. This fact can be used to prove that the
1011:{\displaystyle \angle OAB=\angle OB'A'\ {\text{ and }}\ \angle OBA=\angle OA'B'.}
5714:{\displaystyle x_{1}^{2}+\cdots +x_{n}^{2}+2a_{1}x_{1}+\cdots +2a_{n}x_{n}+1=0,}
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According to
Coxeter, the transformation by inversion in circle was invented by
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5398:{\displaystyle x_{1}^{2}+\cdots +x_{n}^{2}+2a_{1}x_{1}+\cdots +2a_{n}x_{n}+c=0}
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Consequently, the algebraic form of the inversion in a unit circle is given by
1515:
of the reference sphere, then it inverts to a plane. Any plane passing through
5253:
5006:. Any combination of reflections, dilations, translations, and rotations is a
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1929:
are a coordinate system for three-dimensional space obtained by inverting the
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142:. A closely related idea in geometry is that of "inverting" a point. In the
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If a point lies on the circle, its polar is the tangent through this point.
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of a triangle coincides with its OI line. The proof roughly goes as below:
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Any combination of reflections, translations, and rotations is called an
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1084:
38:
1618:(north pole) of the sphere onto the tangent plane at the opposite point
1531:, but is a true 3D phenomenon if the secant plane does not pass through
5941:
1504:{\displaystyle OP\cdot OP^{\prime }=||OP||\cdot ||OP^{\prime }||=R^{2}}
765:
The inverse, with respect to the red circle, of a circle going through
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angles). Algebraically, a map is anticonformal if at every point the
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The combination of two inversions in concentric circles results in a
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are orthogonal, then a straight line passing through the center O of
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changing positions, whilst any point on the circle is unaffected (is
56:
6176:
practice problems on how to use inversion for math olympiad problems
6141:
6088:"Chapter 7: Non-Euclidean Geometry, Section 37: Circular Inversion"
5894:
M. Pieri (1911,12) "Nuovi principia di geometria della inversion",
3357:
the circle transforms into the line parallel to the imaginary axis
1941:
One of the first to consider foundations of inversive geometry was
1782:{\displaystyle x^{2}+y^{2}+(z+{\tfrac {1}{2}})^{2}={\tfrac {1}{4}}}
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In addition, any two non-intersecting circles may be inverted into
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119:
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1987:
for the Möbius plane that comes from the
Euclidean plane is the
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are distances to the ends of a line L, then length of the line
1949:
wrote his thesis on "Invariant theory of the inversion group".
2319:, or dilation characterized by the ratio of the circle radii.
1162:
circles, using circle of inversion centered at a point on the
822:
of the reference circle inverts to a line not passing through
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in 3D with respect to a reference sphere centered at a point
1473:
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208:
6046:, Cambridge: Cambridge University Press, pp. 199–260,
5581:
Hence, it will be invariant under inversion if and only if
4473:, in 1872. Since then many mathematicians reserve the term
4457:. Thus inversive geometry includes the ideas originated by
2638:{\displaystyle {\frac {1}{z}}={\frac {\bar {z}}{|z|^{2}}}.}
903:, and points A' and B' inverses of A and B with respect to
138:
To invert a number in arithmetic usually means to take its
5845:
A simple property of isosceles triangles with applications
899:
Given a triangle OAB in which O is the center of a circle
5241:) is negative; hence the inversive map is anticonformal.
1557:
The inversion of a cylinder, cone, or torus results in a
1155:
of the ratio of the radii of the two concentric circles.
785:
The inverse, with respect to the red circle, of a circle
1043:
If M and M' are inverse points with respect to a circle
826:, but parallel to the tangent to the original circle at
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1960:
where the generalized circles are called "blocks": In
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Any two non-intersecting circles may be inverted into
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on two curves m and m', also inverses with respect to
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to the reference circle at the points of intersection.
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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
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under an inversion with radius 1. The invariant is:
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Stereographic projection as an inversion of a sphere
4058:{\textstyle {\frac {r}{\left|a^{*}a-r^{2}\right|}}}
2749:Reciprocation is key in transformation theory as a
1897:. The lines through the center of inversion (point
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2269:{\displaystyle I={\frac {|x-y||w-z|}{|x-w||y-z|}}}
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1128:of the intouch triangle is inverted into triangle
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5911:Transactions of the American Mathematical Society
1956:of inversive geometry has been interpreted as an
1266:Poles and polars have several useful properties:
830:, and vice versa; whereas a line passing through
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4999:of each reflection and thus of the composition.
620:There is a construction of the inverse point to
280:(i.e. an involution). To make the inversion a
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2474:When a point in the plane is interpreted as a
1374:in three dimensions. The inversion of a point
1223:that is perpendicular to the line containing
1072:click or hover over a circle to highlight it.
8:
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1358:Inversion of a spheroid (at the red sphere)
655:which may lie inside or outside the circle
232:{\displaystyle OP\cdot OP^{\prime }=r^{2}.}
5264:Inversive geometry and hyperbolic geometry
2765:of the whole plane and so are necessarily
1024:The points of intersection of two circles
6199:Visual Dictionary of Special Plane Curves
5931:
5783:Inversion of curves and surfaces (German)
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27:Study of angle-preserving transformations
6169:Wilson Stother's inversive geometry page
5983:Joel C. Gibbons & Yushen Luo (2013)
4979:. When two hyperplanes intersect in an (
3998:{\textstyle {\frac {a}{(aa^{*}-r^{2})}}}
3401:{\displaystyle w+w^{*}={\tfrac {1}{a}}.}
3319:{\textstyle {\frac {r}{|a^{2}-r^{2}|}}.}
1598:usually projects a sphere from a point
1585:
1361:
1353:
1345:
841:inverts to a circle not passing through
818:A circle that passes through the center
93:generalized to higher-dimensional spaces
6069:(2nd ed.), John Wiley & Sons,
5954:
5794:
1366:Inversion of a hyperboloid of one sheet
1350:Inversion of a sphere at the red sphere
755:
560:. (Not labeled. It's the vertical line)
6026:Inversion Theory and Conformal Mapping
5589: − 1)-spheres with equation
4465:in their plane geometry. Furthermore,
3431:{\displaystyle a\not \in \mathbb {R} }
1151:(usually denoted δ) is defined as the
5898:49:49–96 & 50:106–140
5155:{\displaystyle \det(J)=-{\sqrt {k}}.}
4636:found by inverting the length of the
3257:{\textstyle {\frac {a}{a^{2}-r^{2}}}}
2292:soon appreciates the significance of
1370:Circle inversion is generalizable to
1212:
793:(blue) is a circle not going through
443:. (Not labeled. It's the blue circle)
312:Compass and straightedge construction
7:
5896:Giornal di Matematiche di Battaglini
5839:
5837:
5428:, and on inversion gives the sphere
2956:{\displaystyle w={\frac {1}{z^{*}}}}
2879:{\displaystyle (z-a)(z-a)^{*}=r^{2}}
2300:, an outgrowth of certain models of
1693:{\displaystyle x^{2}+y^{2}+z^{2}=-z}
5882:
5870:
5828:
5162:Computing the Jacobian in the case
4987:, successive reflections produce a
4956:The transformation by inversion in
4629:{\displaystyle P=(p_{1},...,p_{n})}
4565:{\displaystyle O=(o_{1},...,o_{n})}
2966:it is straightforward to show that
1199:with respect to a circle of radius
769:(blue) is a line not going through
2920:Using the definition of inversion
2913:{\displaystyle a\in \mathbb {R} .}
2889:where without loss of generality,
1519:, inverts to a sphere touching at
983:
968:
935:
920:
250:. The inversion taking any point
25:
6174:IMO Compendium Training Materials
6160:Inversion: Reflection in a Circle
6028:, American Mathematical Society,
6004:Altshiller-Court, Nathan (1952),
5924:10.1090/S0002-9947-1900-1500550-1
4500:-dimensional Euclidean space, an
2773:Transforming circles into circles
1330:lies on its own polar line, then
1255:through one of the points is the
288:, it is necessary to introduce a
4972:about the hyperspheres' center.
4107:{\displaystyle a^{*}a\to r^{2},}
3474:{\displaystyle aa^{*}\neq r^{2}}
2558:{\displaystyle {\bar {z}}=x-iy,}
1219:; the polar is the line through
802:
778:
758:
91:The concept of inversion can be
5408:will have a positive radius if
5110:{\displaystyle J\cdot J^{T}=kI}
4572:is a map of an arbitrary point
3947:describes the circle of center
3215:describes the circle of center
1337:Each line has exactly one pole.
1036:, are inverses with respect to
853:Additional properties include:
6130:(1941) "The Inversive Plane",
5989:-sphere and inversive geometry
5133:
5127:
5051:Anticonformal mapping property
4924:
4897:
4882:
4856:
4811:
4760:
4748:
4715:
4623:
4585:
4559:
4521:
4297:
4214:
4184:
4088:
3989:
3960:
3726:
3710:
3680:
3652:
3622:
3595:
3566:
3338:
3306:
3278:
3170:
3143:
3115:
3088:
3069:
3050:
3044:
3018:
2854:
2841:
2838:
2826:
2700:
2658:
2619:
2610:
2603:
2531:
2409:
2400:
2380:
2361:
2352:
2332:
2259:
2245:
2240:
2226:
2219:
2205:
2200:
2186:
2159:{\displaystyle d/(r_{1}r_{2})}
2153:
2130:
1884:
1863:
1817:
1796:
1755:
1733:
1484:
1479:
1461:
1456:
1448:
1443:
1432:
1427:
1:
6133:American Mathematical Monthly
5748:Duality (projective geometry)
1937:Axiomatics and generalization
1083:Inversion of a parabola is a
837:A circle not passing through
2730:
2280:Relation to Erlangen program
1539:Examples in three dimensions
1390:' on the ray with direction
1089:Inversion of hyperbola is a
663:Take the intersection point
615:
296:and this point at infinity.
6092:Geometry: Euclid and Beyond
5803:Curves and Their Properties
1116:Invert with respect to the
567:be one of the points where
134:with respect to the circle.
111:with different translations
6231:
6114:Holt, Rinehart and Winston
6008:(2nd ed.), New York:
5966:A.S. Smogorzhevsky (1982)
5933:2027/miun.abv0510.0001.001
5021:is a classical theorem of
4991:where every point of the (
2667:{\displaystyle z\mapsto w}
1890:{\displaystyle S=(0,0,-1)}
1823:{\displaystyle (0,0,-0.5)}
1180:
1171:Peaucellier–Lipkin linkage
1059:Examples in two dimensions
51:, a transformation of the
29:
5758:Limiting point (geometry)
2317:homothetic transformation
709:be the reflection of ray
624:with respect to a circle
504:To construct the inverse
320:To construct the inverse
284:that is also defined for
6066:Introduction to Geometry
6024:Blair, David E. (2000),
5743:Circle of antisimilitude
5729:of hyperbolic geometry.
1596:stereographic projection
1573:Hyperboloid of one sheet
1259:of the other point (the
1243:is the inverse of point
1164:circle of antisimilitude
797:(green), and vice versa.
773:(green), and vice versa.
739:is the inverse point of
682:with an arbitrary point
173:, lying on the ray from
5778:Mohr-Mascheroni theorem
5237:, and additionally det(
4502:inversion in the sphere
4439:complex projective line
3350:{\displaystyle a\to r,}
2508:{\displaystyle z=x+iy,}
2290:transformation geometry
2034:{\displaystyle x,y,z,w}
1968:together with a single
1309:rotates about the pole
1091:lemniscate of Bernoulli
1032:orthogonal to a circle
743:with respect to circle
6108:Kay, David C. (1969),
5968:Lobachevskian Geometry
5843:Dutta, Surajit (2014)
5817:Altshiller-Court (1952
5715:
5572:
5399:
5274: − 1)-sphere
5256:is conformal while an
5156:
5111:
5075:is the Jacobian, then
5041:Möbius transformations
4947:
4687:
4659:
4630:
4566:
4508:centered at the point
4422:
4128:
4108:
4059:
3999:
3941:
3918:
3495:
3475:
3432:
3402:
3351:
3320:
3258:
3209:
3186:
2980:
2957:
2914:
2880:
2811:
2791:
2739:
2668:
2639:
2559:
2509:
2460:
2270:
2160:
2109:
2089:
2062:
2035:
1954:mathematical structure
1911:
1891:
1844:
1824:
1783:
1694:
1632:
1612:
1591:
1505:
1367:
1359:
1351:
1236:
1203:centered on the point
1073:
1012:
694:and from the point on
405:Draw the segment from
374:
233:
135:
112:
99:Inversion in a circle
5763:Möbius transformation
5716:
5573:
5400:
5157:
5112:
5067:is a scalar times an
4948:
4688:
4686:{\displaystyle r^{2}}
4660:
4631:
4567:
4423:
4129:
4109:
4060:
4000:
3942:
3919:
3496:
3476:
3433:
3403:
3352:
3321:
3259:
3210:
3187:
2981:
2958:
2915:
2881:
2812:
2792:
2740:
2669:
2640:
2560:
2510:
2461:
2271:
2161:
2110:
2090:
2088:{\displaystyle r_{2}}
2063:
2061:{\displaystyle r_{1}}
2036:
1931:Cartesian coordinates
1912:
1892:
1845:
1825:
1784:
1700:(alternately written
1695:
1633:
1613:
1589:
1553:Cylinder, cone, torus
1506:
1365:
1357:
1349:
1190:
1066:
1013:
636:is inside or outside
319:
234:
123:
106:
18:Inversion in a circle
5596:
5435:
5283:
5121:
5079:
4701:
4670:
4643:
4576:
4512:
4492:In higher dimensions
4141:
4118:
4072:
4009:
3951:
3931:
3508:
3485:
3442:
3414:
3361:
3332:
3268:
3219:
3199:
2993:
2970:
2927:
2893:
2823:
2801:
2781:
2681:
2652:
2580:
2522:
2481:
2326:
2173:
2119:
2099:
2072:
2045:
2007:
1927:6-sphere coordinates
1921:6-sphere coordinates
1901:
1854:
1834:
1793:
1704:
1642:
1622:
1602:
1398:
917:
616:Dutta's construction
457:be the points where
378:Point outside circle
188:
156:reference circle (Ø)
107:Inversion of lambda
30:For other uses, see
5856:Forum Geometricorum
5768:Projective geometry
5727:Poincaré disc model
5637:
5613:
5476:
5452:
5324:
5300:
5019:Liouville's theorem
4826:
4665:and multiplying by
4658:{\displaystyle P-O}
4638:displacement vector
4441:, often called the
2986:obeys the equation
2302:hyperbolic geometry
1976:, also known as an
1958:incidence structure
1843:{\displaystyle 0.5}
1342:In three dimensions
1301:moves along a line
865:, then the circles
500:Point inside circle
424:be the midpoint of
6215:Inversive geometry
6182:Weisstein, Eric W.
6010:Barnes & Noble
5850:2018-04-21 at the
5711:
5623:
5599:
5568:
5462:
5438:
5395:
5310:
5286:
5260:is anticonformal.
5152:
5107:
5028:The addition of a
5023:conformal geometry
4943:
4941:
4896:
4814:
4683:
4655:
4626:
4562:
4418:
4416:
4124:
4104:
4055:
3995:
3937:
3914:
3912:
3491:
3471:
3428:
3398:
3393:
3347:
3316:
3254:
3205:
3182:
2976:
2953:
2910:
2876:
2807:
2787:
2763:analytic functions
2735:
2664:
2635:
2555:
2505:
2456:
2266:
2156:
2105:
2085:
2058:
2031:
1962:incidence geometry
1952:More recently the
1945:in 1911 and 1912.
1907:
1887:
1840:
1820:
1779:
1777:
1752:
1690:
1628:
1608:
1592:
1501:
1368:
1360:
1352:
1286:lies on the polar
1237:
1149:inversive distance
1147:circles. Then the
1074:
1008:
678:Connect the point
643:Consider a circle
531:(center of circle
409:(center of circle
375:
343:. Right triangles
229:
154:with respect to a
136:
130:is the inverse of
116:Inverse of a point
113:
43:inversive geometry
6084:Hartshorne, Robin
5985:Colorings of the
5560:
5537:
5498:
5147:
5069:orthogonal matrix
5030:point at infinity
5013:All of these are
4934:
4887:
4788:
4409:
4358:
4291:
4127:{\displaystyle w}
4114:the equation for
4053:
3993:
3940:{\displaystyle w}
3927:showing that the
3898:
3842:
3784:
3720:
3662:
3599:
3494:{\displaystyle w}
3392:
3311:
3252:
3208:{\displaystyle w}
3180:
3125:
3048:
2979:{\displaystyle w}
2951:
2810:{\displaystyle a}
2797:around the point
2790:{\displaystyle r}
2733:
2724:
2705:
2703:
2630:
2606:
2591:
2534:
2517:complex conjugate
2438:
2420:
2372:
2264:
2108:{\displaystyle d}
2003:between 4 points
1970:point at infinity
1910:{\displaystyle N}
1776:
1751:
1631:{\displaystyle S}
1611:{\displaystyle N}
1334:is on the circle.
1153:natural logarithm
967:
963:
959:
888:and intersecting
593:perpendicular to
578:Draw the segment
556:perpendicular to
397:outside a circle
339:be the radius of
331:outside a circle
301:point at infinity
290:point at infinity
16:(Redirected from
6222:
6195:
6194:
6124:
6110:College Geometry
6104:
6079:
6056:
6038:
6020:
5991:
5981:
5975:
5964:
5958:
5957:, pp. 77–95
5952:
5946:
5945:
5935:
5905:
5899:
5892:
5886:
5880:
5874:
5868:
5859:
5841:
5832:
5826:
5820:
5814:
5808:
5799:
5720:
5718:
5717:
5712:
5695:
5694:
5685:
5684:
5663:
5662:
5653:
5652:
5636:
5631:
5612:
5607:
5577:
5575:
5574:
5569:
5561:
5553:
5548:
5547:
5538:
5533:
5532:
5523:
5509:
5508:
5499:
5494:
5493:
5484:
5475:
5470:
5451:
5446:
5424:is greater than
5404:
5402:
5401:
5396:
5382:
5381:
5372:
5371:
5350:
5349:
5340:
5339:
5323:
5318:
5299:
5294:
5236:
5234:
5223:
5213:
5196:
5188:
5186:
5161:
5159:
5158:
5153:
5148:
5143:
5116:
5114:
5113:
5108:
5097:
5096:
5059:if it preserves
4952:
4950:
4949:
4944:
4942:
4935:
4933:
4932:
4931:
4922:
4921:
4909:
4908:
4895:
4885:
4881:
4880:
4868:
4867:
4855:
4854:
4844:
4839:
4838:
4822:
4806:
4805:
4789:
4787:
4786:
4785:
4763:
4747:
4746:
4736:
4725:
4694:
4692:
4690:
4689:
4684:
4682:
4681:
4664:
4662:
4661:
4656:
4635:
4633:
4632:
4627:
4622:
4621:
4597:
4596:
4571:
4569:
4568:
4563:
4558:
4557:
4533:
4532:
4507:
4481:together with a
4471:Erlangen program
4427:
4425:
4424:
4419:
4417:
4410:
4408:
4382:
4359:
4357:
4340:
4323:
4301:
4292:
4284:
4177:
4176:
4167:
4166:
4147:
4133:
4131:
4130:
4125:
4113:
4111:
4110:
4105:
4100:
4099:
4084:
4083:
4064:
4062:
4061:
4056:
4054:
4052:
4048:
4047:
4046:
4031:
4030:
4013:
4004:
4002:
4001:
3996:
3994:
3992:
3988:
3987:
3975:
3974:
3955:
3946:
3944:
3943:
3938:
3923:
3921:
3920:
3915:
3913:
3909:
3908:
3903:
3899:
3897:
3893:
3892:
3891:
3879:
3878:
3858:
3848:
3844:
3843:
3841:
3840:
3839:
3824:
3823:
3810:
3805:
3804:
3790:
3786:
3785:
3783:
3782:
3781:
3769:
3768:
3755:
3754:
3745:
3730:
3721:
3719:
3718:
3717:
3708:
3707:
3695:
3694:
3678:
3677:
3668:
3663:
3661:
3660:
3659:
3650:
3649:
3637:
3636:
3620:
3619:
3618:
3605:
3600:
3598:
3594:
3593:
3578:
3577:
3564:
3563:
3562:
3553:
3552:
3533:
3528:
3527:
3514:
3500:
3498:
3497:
3492:
3480:
3478:
3477:
3472:
3470:
3469:
3457:
3456:
3437:
3435:
3434:
3429:
3427:
3407:
3405:
3404:
3399:
3394:
3385:
3379:
3378:
3356:
3354:
3353:
3348:
3325:
3323:
3322:
3317:
3312:
3310:
3309:
3304:
3303:
3291:
3290:
3281:
3272:
3263:
3261:
3260:
3255:
3253:
3251:
3250:
3249:
3237:
3236:
3223:
3214:
3212:
3211:
3206:
3191:
3189:
3188:
3183:
3181:
3179:
3178:
3177:
3168:
3167:
3155:
3154:
3141:
3140:
3131:
3126:
3124:
3123:
3122:
3113:
3112:
3100:
3099:
3086:
3085:
3076:
3068:
3067:
3049:
3047:
3043:
3042:
3030:
3029:
3013:
3008:
3007:
2985:
2983:
2982:
2977:
2962:
2960:
2959:
2954:
2952:
2950:
2949:
2937:
2919:
2917:
2916:
2911:
2906:
2885:
2883:
2882:
2877:
2875:
2874:
2862:
2861:
2816:
2814:
2813:
2808:
2796:
2794:
2793:
2788:
2744:
2742:
2741:
2736:
2734:
2729:
2725:
2717:
2711:
2706:
2704:
2696:
2691:
2673:
2671:
2670:
2665:
2644:
2642:
2641:
2636:
2631:
2629:
2628:
2627:
2622:
2613:
2607:
2599:
2597:
2592:
2584:
2564:
2562:
2561:
2556:
2536:
2535:
2527:
2514:
2512:
2511:
2506:
2465:
2463:
2462:
2457:
2449:
2448:
2443:
2439:
2431:
2421:
2419:
2418:
2417:
2412:
2403:
2394:
2392:
2391:
2373:
2371:
2370:
2369:
2364:
2355:
2346:
2344:
2343:
2298:Erlangen program
2275:
2273:
2272:
2267:
2265:
2263:
2262:
2248:
2243:
2229:
2223:
2222:
2208:
2203:
2189:
2183:
2165:
2163:
2162:
2157:
2152:
2151:
2142:
2141:
2129:
2114:
2112:
2111:
2106:
2094:
2092:
2091:
2086:
2084:
2083:
2067:
2065:
2064:
2059:
2057:
2056:
2040:
2038:
2037:
2032:
1916:
1914:
1913:
1908:
1896:
1894:
1893:
1888:
1849:
1847:
1846:
1841:
1829:
1827:
1826:
1821:
1788:
1786:
1785:
1780:
1778:
1769:
1763:
1762:
1753:
1744:
1729:
1728:
1716:
1715:
1699:
1697:
1696:
1691:
1680:
1679:
1667:
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1629:
1617:
1615:
1614:
1609:
1510:
1508:
1507:
1502:
1500:
1499:
1487:
1482:
1477:
1476:
1464:
1459:
1451:
1446:
1435:
1430:
1422:
1421:
1372:sphere inversion
1278:, then the pole
1111:intouch triangle
1017:
1015:
1014:
1009:
1004:
996:
965:
964:
961:
957:
956:
948:
806:
782:
762:
737:
730:
690:(different from
671:with the circle
554:
540:
515:inside a circle
513:
493:
482:
473:
455:
431:Draw the circle
391:
371:
352:
325:
271:
264:
244:circle inversion
238:
236:
235:
230:
225:
224:
212:
211:
172:
129:
45:is the study of
32:Point reflection
21:
6230:
6229:
6225:
6224:
6223:
6221:
6220:
6219:
6205:
6204:
6180:
6179:
6156:
6142:10.2307/2303867
6128:Patterson, Boyd
6107:
6102:
6082:
6077:
6061:Coxeter, H.S.M.
6059:
6054:
6041:
6036:
6023:
6003:
6000:
5995:
5994:
5982:
5978:
5965:
5961:
5953:
5949:
5907:
5906:
5902:
5893:
5889:
5881:
5877:
5869:
5862:
5852:Wayback Machine
5842:
5835:
5827:
5823:
5815:
5811:
5800:
5796:
5791:
5739:
5686:
5676:
5654:
5644:
5594:
5593:
5539:
5524:
5500:
5485:
5433:
5432:
5423:
5414:
5373:
5363:
5341:
5331:
5281:
5280:
5266:
5258:anti-homography
5230:
5225:
5215:
5212:
5203:
5192:
5190:
5182:
5180:
5171:
5163:
5119:
5118:
5088:
5077:
5076:
5053:
4940:
4939:
4923:
4913:
4900:
4886:
4872:
4859:
4846:
4845:
4830:
4807:
4797:
4794:
4793:
4777:
4764:
4738:
4737:
4718:
4711:
4699:
4698:
4673:
4668:
4667:
4666:
4641:
4640:
4613:
4588:
4574:
4573:
4549:
4524:
4510:
4509:
4505:
4494:
4434:
4432:Higher geometry
4415:
4414:
4386:
4341:
4324:
4302:
4294:
4293:
4168:
4158:
4139:
4138:
4116:
4115:
4091:
4075:
4070:
4069:
4038:
4022:
4021:
4017:
4007:
4006:
3979:
3966:
3959:
3949:
3948:
3929:
3928:
3911:
3910:
3883:
3870:
3866:
3862:
3853:
3852:
3831:
3815:
3814:
3796:
3795:
3791:
3773:
3760:
3756:
3746:
3737:
3733:
3731:
3723:
3722:
3709:
3699:
3686:
3679:
3669:
3651:
3641:
3628:
3621:
3610:
3606:
3585:
3569:
3565:
3554:
3544:
3534:
3519:
3506:
3505:
3483:
3482:
3481:the result for
3461:
3448:
3440:
3439:
3412:
3411:
3370:
3359:
3358:
3330:
3329:
3295:
3282:
3276:
3266:
3265:
3241:
3228:
3227:
3217:
3216:
3197:
3196:
3195:and hence that
3169:
3159:
3146:
3142:
3132:
3114:
3104:
3091:
3087:
3077:
3059:
3034:
3021:
3017:
2999:
2991:
2990:
2968:
2967:
2941:
2925:
2924:
2891:
2890:
2866:
2853:
2821:
2820:
2799:
2798:
2779:
2778:
2775:
2712:
2679:
2678:
2650:
2649:
2617:
2608:
2578:
2577:
2520:
2519:
2479:
2478:
2472:
2426:
2425:
2407:
2398:
2383:
2359:
2350:
2335:
2324:
2323:
2309:
2282:
2224:
2184:
2171:
2170:
2143:
2133:
2117:
2116:
2097:
2096:
2075:
2070:
2069:
2048:
2043:
2042:
2005:
2004:
1997:
1978:inversive plane
1939:
1923:
1899:
1898:
1852:
1851:
1832:
1831:
1791:
1790:
1754:
1720:
1707:
1702:
1701:
1671:
1658:
1645:
1640:
1639:
1620:
1619:
1600:
1599:
1584:
1575:
1567:
1555:
1546:
1541:
1491:
1468:
1413:
1396:
1395:
1344:
1274:lies on a line
1247:then the lines
1213:inversion point
1191:The polar line
1185:
1179:
1126:medial triangle
1099:
1061:
997:
989:
962: and
949:
941:
915:
914:
876:If the circles
873:are orthogonal.
810:
807:
798:
783:
774:
763:
754:
735:
728:
618:
552:
538:
511:
502:
491:
480:
471:
453:
389:
380:
369:
350:
323:
314:
269:
262:
258:) to its image
248:plane inversion
242:This is called
216:
203:
186:
185:
170:
127:
118:
101:
53:Euclidean plane
35:
28:
23:
22:
15:
12:
11:
5:
6228:
6226:
6218:
6217:
6207:
6206:
6203:
6202:
6196:
6177:
6171:
6166:
6155:
6154:External links
6152:
6151:
6150:
6125:
6105:
6100:
6080:
6075:
6057:
6052:
6039:
6034:
6021:
5999:
5996:
5993:
5992:
5976:
5972:Mir Publishers
5959:
5947:
5918:(4): 430–498.
5900:
5887:
5885:, p. 269)
5875:
5873:, p. 265)
5860:
5833:
5831:, p. 264)
5821:
5819:, p. 230)
5809:
5793:
5792:
5790:
5787:
5786:
5785:
5780:
5775:
5773:Soddy's hexlet
5770:
5765:
5760:
5755:
5750:
5745:
5738:
5735:
5722:
5721:
5710:
5707:
5704:
5701:
5698:
5693:
5689:
5683:
5679:
5675:
5672:
5669:
5666:
5661:
5657:
5651:
5647:
5643:
5640:
5635:
5630:
5626:
5622:
5619:
5616:
5611:
5606:
5602:
5579:
5578:
5567:
5564:
5559:
5556:
5551:
5546:
5542:
5536:
5531:
5527:
5521:
5518:
5515:
5512:
5507:
5503:
5497:
5492:
5488:
5482:
5479:
5474:
5469:
5465:
5461:
5458:
5455:
5450:
5445:
5441:
5419:
5412:
5406:
5405:
5394:
5391:
5388:
5385:
5380:
5376:
5370:
5366:
5362:
5359:
5356:
5353:
5348:
5344:
5338:
5334:
5330:
5327:
5322:
5317:
5313:
5309:
5306:
5303:
5298:
5293:
5289:
5276:with equation
5265:
5262:
5208:
5201:
5176:
5167:
5151:
5146:
5141:
5138:
5135:
5132:
5129:
5126:
5106:
5103:
5100:
5095:
5091:
5087:
5084:
5052:
5049:
5015:conformal maps
4995:–2)-flat is a
4954:
4953:
4938:
4930:
4926:
4920:
4916:
4912:
4907:
4903:
4899:
4894:
4890:
4884:
4879:
4875:
4871:
4866:
4862:
4858:
4853:
4849:
4842:
4837:
4833:
4829:
4825:
4821:
4817:
4813:
4810:
4808:
4804:
4800:
4796:
4795:
4792:
4784:
4780:
4776:
4773:
4770:
4767:
4762:
4759:
4756:
4753:
4750:
4745:
4741:
4734:
4731:
4728:
4724:
4721:
4717:
4714:
4712:
4710:
4707:
4706:
4680:
4676:
4654:
4651:
4648:
4625:
4620:
4616:
4612:
4609:
4606:
4603:
4600:
4595:
4591:
4587:
4584:
4581:
4561:
4556:
4552:
4548:
4545:
4542:
4539:
4536:
4531:
4527:
4523:
4520:
4517:
4493:
4490:
4443:Riemann sphere
4433:
4430:
4429:
4428:
4413:
4407:
4404:
4401:
4398:
4395:
4392:
4389:
4385:
4380:
4377:
4374:
4371:
4368:
4365:
4362:
4356:
4353:
4350:
4347:
4344:
4339:
4336:
4333:
4330:
4327:
4321:
4318:
4315:
4312:
4309:
4306:
4303:
4299:
4296:
4295:
4290:
4287:
4282:
4279:
4276:
4273:
4270:
4267:
4264:
4261:
4258:
4255:
4252:
4249:
4246:
4243:
4240:
4237:
4234:
4231:
4228:
4225:
4222:
4219:
4216:
4213:
4210:
4207:
4204:
4201:
4198:
4195:
4192:
4189:
4186:
4183:
4180:
4175:
4171:
4165:
4161:
4157:
4154:
4151:
4148:
4146:
4123:
4103:
4098:
4094:
4090:
4087:
4082:
4078:
4051:
4045:
4041:
4037:
4034:
4029:
4025:
4020:
4016:
3991:
3986:
3982:
3978:
3973:
3969:
3965:
3962:
3958:
3936:
3925:
3924:
3907:
3902:
3896:
3890:
3886:
3882:
3877:
3873:
3869:
3865:
3861:
3856:
3851:
3847:
3838:
3834:
3830:
3827:
3822:
3818:
3813:
3808:
3803:
3799:
3794:
3789:
3780:
3776:
3772:
3767:
3763:
3759:
3753:
3749:
3743:
3740:
3736:
3732:
3728:
3725:
3724:
3716:
3712:
3706:
3702:
3698:
3693:
3689:
3685:
3682:
3676:
3672:
3666:
3658:
3654:
3648:
3644:
3640:
3635:
3631:
3627:
3624:
3617:
3613:
3609:
3603:
3597:
3592:
3588:
3584:
3581:
3576:
3572:
3568:
3561:
3557:
3551:
3547:
3543:
3540:
3537:
3531:
3526:
3522:
3518:
3515:
3513:
3490:
3468:
3464:
3460:
3455:
3451:
3447:
3426:
3422:
3419:
3397:
3391:
3388:
3382:
3377:
3373:
3369:
3366:
3346:
3343:
3340:
3337:
3315:
3308:
3302:
3298:
3294:
3289:
3285:
3280:
3275:
3248:
3244:
3240:
3235:
3231:
3226:
3204:
3193:
3192:
3176:
3172:
3166:
3162:
3158:
3153:
3149:
3145:
3139:
3135:
3129:
3121:
3117:
3111:
3107:
3103:
3098:
3094:
3090:
3084:
3080:
3074:
3071:
3066:
3062:
3058:
3055:
3052:
3046:
3041:
3037:
3033:
3028:
3024:
3020:
3016:
3011:
3006:
3002:
2998:
2975:
2964:
2963:
2948:
2944:
2940:
2935:
2932:
2909:
2905:
2901:
2898:
2887:
2886:
2873:
2869:
2865:
2860:
2856:
2852:
2849:
2846:
2843:
2840:
2837:
2834:
2831:
2828:
2806:
2786:
2774:
2771:
2747:
2746:
2732:
2728:
2723:
2720:
2715:
2709:
2702:
2699:
2694:
2689:
2686:
2663:
2660:
2657:
2646:
2645:
2634:
2626:
2621:
2616:
2612:
2605:
2602:
2595:
2590:
2587:
2554:
2551:
2548:
2545:
2542:
2539:
2533:
2530:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2476:complex number
2471:
2468:
2467:
2466:
2455:
2452:
2447:
2442:
2437:
2434:
2429:
2424:
2416:
2411:
2406:
2402:
2397:
2390:
2386:
2382:
2379:
2376:
2368:
2363:
2358:
2354:
2349:
2342:
2338:
2334:
2331:
2308:
2305:
2281:
2278:
2277:
2276:
2261:
2257:
2254:
2251:
2247:
2242:
2238:
2235:
2232:
2228:
2221:
2217:
2214:
2211:
2207:
2202:
2198:
2195:
2192:
2188:
2181:
2178:
2155:
2150:
2146:
2140:
2136:
2132:
2128:
2124:
2104:
2082:
2078:
2055:
2051:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
1996:
1993:
1989:Riemann sphere
1938:
1935:
1922:
1919:
1906:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1839:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1775:
1772:
1766:
1761:
1757:
1750:
1747:
1741:
1738:
1735:
1732:
1727:
1723:
1719:
1714:
1710:
1689:
1686:
1683:
1678:
1674:
1670:
1665:
1661:
1657:
1652:
1648:
1627:
1607:
1583:
1580:
1574:
1571:
1566:
1563:
1554:
1551:
1545:
1542:
1540:
1537:
1498:
1494:
1490:
1486:
1481:
1475:
1471:
1467:
1463:
1458:
1454:
1450:
1445:
1441:
1438:
1434:
1429:
1425:
1420:
1416:
1412:
1409:
1406:
1403:
1343:
1340:
1339:
1338:
1335:
1324:
1321:
1318:
1295:
1183:pole and polar
1181:Main article:
1178:
1177:Pole and polar
1175:
1098:
1095:
1094:
1093:
1087:
1081:
1078:
1060:
1057:
1056:
1055:
1052:
1041:
1021:
1020:
1019:
1018:
1007:
1003:
1000:
995:
992:
988:
985:
982:
979:
976:
973:
970:
955:
952:
947:
944:
940:
937:
934:
931:
928:
925:
922:
909:
908:
897:
874:
851:
850:
835:
812:
811:
808:
801:
799:
789:going through
784:
777:
775:
764:
757:
753:
750:
749:
748:
703:
686:on the circle
676:
617:
614:
613:
612:
598:
583:
576:
561:
543:
501:
498:
497:
496:
476:
466:
444:
439:going through
429:
418:
379:
376:
313:
310:
282:total function
278:self-inversion
240:
239:
228:
223:
219:
215:
210:
206:
202:
199:
196:
193:
117:
114:
109:Mandelbrot set
100:
97:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6227:
6216:
6213:
6212:
6210:
6200:
6197:
6192:
6191:
6186:
6183:
6178:
6175:
6172:
6170:
6167:
6165:
6161:
6158:
6157:
6153:
6149:
6146:
6143:
6139:
6135:
6134:
6129:
6126:
6123:
6119:
6115:
6111:
6106:
6103:
6101:0-387-98650-2
6097:
6093:
6089:
6085:
6081:
6078:
6076:0-471-18283-4
6072:
6068:
6067:
6062:
6058:
6055:
6053:0-521-59787-0
6049:
6045:
6040:
6037:
6035:0-8218-2636-0
6031:
6027:
6022:
6019:
6015:
6011:
6007:
6002:
6001:
5997:
5990:
5988:
5980:
5977:
5973:
5969:
5963:
5960:
5956:
5951:
5948:
5943:
5939:
5934:
5929:
5925:
5921:
5917:
5913:
5912:
5904:
5901:
5897:
5891:
5888:
5884:
5879:
5876:
5872:
5867:
5865:
5861:
5857:
5853:
5849:
5846:
5840:
5838:
5834:
5830:
5825:
5822:
5818:
5813:
5810:
5805:
5804:
5798:
5795:
5788:
5784:
5781:
5779:
5776:
5774:
5771:
5769:
5766:
5764:
5761:
5759:
5756:
5754:
5753:Inverse curve
5751:
5749:
5746:
5744:
5741:
5740:
5736:
5734:
5730:
5728:
5708:
5705:
5702:
5699:
5696:
5691:
5687:
5681:
5677:
5673:
5670:
5667:
5664:
5659:
5655:
5649:
5645:
5641:
5638:
5633:
5628:
5624:
5620:
5617:
5614:
5609:
5604:
5600:
5592:
5591:
5590:
5588:
5584:
5565:
5562:
5557:
5554:
5549:
5544:
5540:
5534:
5529:
5525:
5519:
5516:
5513:
5510:
5505:
5501:
5495:
5490:
5486:
5480:
5477:
5472:
5467:
5463:
5459:
5456:
5453:
5448:
5443:
5439:
5431:
5430:
5429:
5427:
5422:
5418:
5411:
5392:
5389:
5386:
5383:
5378:
5374:
5368:
5364:
5360:
5357:
5354:
5351:
5346:
5342:
5336:
5332:
5328:
5325:
5320:
5315:
5311:
5307:
5304:
5301:
5296:
5291:
5287:
5279:
5278:
5277:
5275:
5273:
5263:
5261:
5259:
5255:
5251:
5247:
5242:
5240:
5233:
5228:
5222:
5218:
5211:
5207:
5200:
5195:
5185:
5179:
5175:
5170:
5166:
5149:
5144:
5139:
5136:
5130:
5104:
5101:
5098:
5093:
5089:
5085:
5082:
5074:
5070:
5066:
5062:
5058:
5050:
5048:
5046:
5042:
5038:
5036:
5031:
5026:
5024:
5020:
5016:
5011:
5009:
5005:
5000:
4998:
4994:
4990:
4986:
4982:
4978:
4973:
4971:
4967:
4963:
4959:
4936:
4928:
4918:
4914:
4910:
4905:
4901:
4892:
4888:
4877:
4873:
4869:
4864:
4860:
4851:
4847:
4840:
4835:
4831:
4827:
4823:
4819:
4815:
4809:
4802:
4798:
4790:
4782:
4774:
4771:
4768:
4757:
4754:
4751:
4743:
4739:
4732:
4729:
4726:
4722:
4719:
4713:
4708:
4697:
4696:
4695:
4678:
4674:
4652:
4649:
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2528:
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2499:
2496:
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2477:
2470:Reciprocation
2469:
2453:
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2440:
2435:
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2427:
2422:
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2395:
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2148:
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2138:
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2126:
2122:
2102:
2080:
2076:
2053:
2049:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2002:
1994:
1992:
1990:
1986:
1981:
1979:
1975:
1971:
1967:
1963:
1959:
1955:
1950:
1948:
1947:Edward Kasner
1944:
1936:
1934:
1932:
1928:
1920:
1918:
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1881:
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1597:
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1579:
1572:
1570:
1564:
1562:
1560:
1559:Dupin cyclide
1552:
1550:
1543:
1538:
1536:
1534:
1530:
1526:
1522:
1518:
1514:
1496:
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1336:
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1319:
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1308:
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1296:
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1281:
1277:
1273:
1269:
1268:
1267:
1264:
1262:
1258:
1254:
1250:
1249:perpendicular
1246:
1242:
1234:
1230:
1226:
1222:
1218:
1214:
1210:
1207:. The point
1206:
1202:
1198:
1194:
1189:
1184:
1176:
1174:
1172:
1167:
1165:
1161:
1156:
1154:
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1135:
1131:
1127:
1123:
1119:
1114:
1112:
1108:
1104:
1096:
1092:
1088:
1086:
1082:
1079:
1076:
1075:
1071:
1070:the SVG file,
1065:
1058:
1053:
1050:
1046:
1042:
1039:
1035:
1031:
1027:
1023:
1022:
1005:
1001:
998:
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986:
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977:
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945:
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868:
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796:
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776:
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768:
761:
756:
751:
746:
742:
738:
731:
724:
720:
716:
712:
708:
704:
701:
698:antipodal to
697:
693:
689:
685:
681:
677:
674:
670:
666:
662:
661:
660:
658:
654:
650:
646:
641:
639:
635:
631:
627:
623:
610:
606:
603:is where ray
602:
599:
596:
592:
588:
584:
581:
577:
574:
570:
566:
562:
559:
555:
548:
544:
541:
534:
530:
526:
522:
521:
520:
518:
514:
507:
499:
494:
487:
483:
477:
474:
468:Draw segment
467:
464:
460:
456:
449:
445:
442:
438:
434:
430:
428:. (Not shown)
427:
423:
419:
416:
412:
408:
404:
403:
402:
400:
396:
392:
385:
377:
372:
365:
361:
357:
354:are similar.
353:
346:
342:
338:
334:
330:
326:
318:
311:
309:
306:
302:
297:
295:
291:
287:
283:
279:
275:
268:
261:
257:
253:
249:
245:
226:
221:
217:
213:
204:
200:
197:
194:
191:
184:
183:
182:
180:
176:
169:
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157:
153:
149:
145:
141:
133:
126:
122:
115:
110:
105:
98:
96:
94:
89:
87:
84:(1842–3) and
83:
79:
75:
71:
67:
62:
58:
54:
50:
49:
44:
40:
33:
19:
6188:
6164:cut-the-knot
6136:48: 589–99,
6131:
6112:, New York:
6109:
6094:, Springer,
6091:
6065:
6043:
6025:
6005:
5986:
5979:
5967:
5962:
5955:Coxeter 1969
5950:
5915:
5909:
5903:
5895:
5890:
5878:
5824:
5812:
5801:
5797:
5731:
5723:
5586:
5582:
5580:
5425:
5420:
5416:
5409:
5407:
5271:
5267:
5249:
5245:
5243:
5238:
5231:
5229:= 1/‖
5226:
5220:
5216:
5209:
5205:
5198:
5193:
5183:
5177:
5173:
5168:
5164:
5072:
5060:
5054:
5044:
5034:
5027:
5012:
5001:
4992:
4980:
4974:
4962:hyperspheres
4955:
4501:
4497:
4495:
4487:
4474:
4435:
4067:
3926:
3409:
3327:
3194:
2965:
2888:
2776:
2755:Möbius group
2748:
2647:
2570:
2473:
2310:
2286:L. I. Magnus
2283:
2115:will become
1998:
1982:
1977:
1974:Möbius plane
1966:affine plane
1951:
1940:
1924:
1593:
1576:
1568:
1556:
1547:
1532:
1528:
1524:
1520:
1516:
1512:
1391:
1387:
1383:
1382:with radius
1379:
1375:
1369:
1331:
1327:
1314:
1313:of the line
1310:
1306:
1305:, its polar
1302:
1298:
1291:
1287:
1283:
1282:of the line
1279:
1275:
1271:
1265:
1252:
1251:to the line
1244:
1240:
1238:
1232:
1228:
1224:
1220:
1216:
1208:
1204:
1200:
1196:
1192:
1168:
1157:
1142:
1133:
1129:
1121:
1120:of triangle
1115:
1100:
1048:
1044:
1037:
1033:
1029:
1025:
904:
900:
893:
889:
885:
881:
877:
870:
866:
862:
858:
857:If a circle
852:
842:
838:
831:
827:
823:
819:
813:
794:
790:
786:
770:
766:
744:
740:
733:
726:
722:
718:
714:
710:
706:
699:
695:
691:
687:
683:
679:
672:
668:
664:
656:
652:
651:and a point
648:
647:with center
644:
642:
637:
633:
629:
625:
621:
619:
608:
604:
600:
594:
590:
586:
579:
572:
568:
564:
557:
550:
546:
536:
532:
528:
524:
516:
509:
505:
503:
489:
485:
478:
469:
462:
458:
451:
447:
440:
436:
435:with center
432:
425:
421:
414:
410:
406:
398:
394:
387:
386:the inverse
381:
367:
363:
359:
355:
348:
344:
340:
336:
332:
328:
321:
304:
298:
293:
285:
273:
266:
259:
255:
254:(other than
251:
247:
243:
241:
178:
174:
167:
163:
159:
158:with center
155:
151:
147:
137:
131:
124:
90:
46:
42:
36:
6185:"Inversion"
5858:14: 237–240
5197:‖ =
4997:fixed point
4977:translation
4958:hyperplanes
4467:Felix Klein
4459:Lobachevsky
4005:and radius
3264:and radius
2759:conjugation
2294:Felix Klein
2001:cross-ratio
1943:Mario Pieri
1386:is a point
1326:If a point
1297:If a point
1270:If a point
1195:to a point
1097:Application
1068:circle. In
725:in a point
667:of the ray
632:of whether
630:independent
508:of a point
393:of a point
327:of a point
265:also takes
166:is a point
162:and radius
150:of a point
5998:References
5254:homography
5008:similarity
4504:of radius
4496:In a real
2567:reciprocal
2313:similarity
1394:such that
1145:concentric
1107:Euler line
847:orthogonal
752:Properties
611:intersect.
585:Draw line
575:intersect.
545:Draw line
535:) through
495:intersect.
465:intersect.
181:such that
140:reciprocal
74:Bellavitis
55:that maps
6190:MathWorld
6063:(1969) ,
5883:Kay (1969
5871:Kay (1969
5829:Kay (1969
5668:⋯
5618:⋯
5514:⋯
5457:⋯
5355:⋯
5305:⋯
5181:/‖
5140:−
5086:⋅
5057:conformal
5047:-sphere.
4970:homothety
4911:−
4889:∑
4870:−
4812:↦
4779:‖
4772:−
4766:‖
4755:−
4716:↦
4650:−
4397:
4391:⋅
4379:−
4367:
4361:⋅
4346:
4329:
4308:
4298:⟺
4269:
4254:
4248:−
4236:
4221:
4215:⟺
4194:
4185:⟺
4174:∗
4164:∗
4089:→
4081:∗
4036:−
4028:∗
3977:−
3972:∗
3881:−
3876:∗
3829:−
3821:∗
3807:−
3802:∗
3771:−
3766:∗
3752:∗
3742:−
3727:⟺
3697:−
3692:∗
3639:−
3634:∗
3616:∗
3583:−
3575:∗
3560:∗
3550:∗
3530:−
3525:∗
3459:≠
3454:∗
3376:∗
3339:→
3293:−
3239:−
3157:−
3102:−
3065:∗
3032:−
3010:−
3005:∗
2947:∗
2900:∈
2859:∗
2848:−
2833:−
2767:conformal
2751:generator
2731:¯
2701:¯
2659:↦
2604:¯
2565:then the
2544:−
2532:¯
2381:↦
2333:↦
2253:−
2234:−
2213:−
2194:−
1995:Invariant
1879:−
1830:, radius
1812:−
1789:; center
1685:−
1474:′
1453:⋅
1419:′
1408:⋅
1290:of point
1239:If point
1160:congruent
1138:collinear
1103:collinear
984:∠
969:∠
936:∠
921:∠
721:cuts ray
607:and line
523:Draw ray
484:is where
384:construct
305:invariant
209:′
198:⋅
48:inversion
6209:Category
6122:69-12075
6086:(2000),
6044:Geometry
6018:52-13504
5974:, Moscow
5848:Archived
5737:See also
5415:+ ... +
5235:‖
5204:+ ... +
5191:‖
5189:, where
5187:‖
5065:Jacobian
5061:oriented
5004:isometry
4989:rotation
4966:dilation
4824:′
4723:′
4475:geometry
4447:Beltrami
4134:becomes
3421:∉
2307:Dilation
1972:forms a
1565:Spheroid
1118:incircle
1085:cardioid
1002:′
994:′
954:′
946:′
713:in line
628:that is
589:through
549:through
272:back to
177:through
88:(1845).
76:(1836),
72:(1825),
70:Quetelet
68:(1824),
39:geometry
6201:Xah Lee
6148:0006034
5942:1986367
5224:, with
5037:-sphere
2753:of the
2674:where:
1211:is the
1109:of the
717:. Then
335:: Let
148:inverse
66:Steiner
57:circles
6120:
6098:
6073:
6050:
6032:
6016:
5940:
5214:gives
4477:for a
4463:Bolyai
4453:, and
4451:Cayley
1964:, any
1544:Sphere
1124:. The
966:
958:
907:, then
366:is to
358:is to
146:, the
86:Kelvin
82:Ingram
78:Stubbs
5938:JSTOR
5789:Notes
5248:to 1/
4483:group
4479:space
4455:Klein
4068:When
3328:When
2515:with
1985:model
1257:polar
736:'
729:'
553:'
539:'
527:from
512:'
492:'
481:'
472:'
454:'
413:) to
390:'
370:'
351:'
324:'
270:'
263:'
171:'
144:plane
128:'
61:lines
6118:LCCN
6096:ISBN
6071:ISBN
6048:ISBN
6030:ISBN
6014:LCCN
5268:The
5117:and
4985:flat
4983:–2)-
4461:and
3438:and
3410:For
2068:and
1999:The
1925:The
1261:pole
1231:and
1169:The
1136:are
1028:and
880:and
869:and
705:Let
571:and
563:Let
488:and
461:and
450:and
446:Let
420:Let
347:and
80:and
6162:at
6138:doi
5928:hdl
5920:doi
5125:det
4968:or
4960:or
3501:is
2573:is
2569:of
2296:'s
1838:0.5
1815:0.5
1263:).
1215:of
1134:ABC
1130:ABC
1122:ABC
787:not
382:To
362:as
349:ONP
345:OPN
246:or
59:or
37:In
6211::
6187:.
6145:MR
6116:,
6090:,
6012:,
5970:,
5936:.
5926:.
5914:.
5863:^
5854:,
5836:^
5566:0.
5221:kI
5219:=
5217:JJ
5172:=
5025:.
5010:.
4449:,
4394:Im
4364:Re
4343:Im
4326:Re
4305:Im
4266:Im
4251:Im
4233:Re
4218:Re
4191:Re
4065:.
2769:.
2315:,
1991:.
1983:A
1933:.
1594:A
1561:.
1535:.
1392:OP
1253:PR
1227:,
1166:.
1140:.
732:.
723:OC
715:BC
711:BA
669:OA
659:.
640:.
595:ON
580:ON
519::
490:NN
486:OP
470:NN
426:OP
401::
368:OP
356:OP
95:.
41:,
6193:.
6140::
5987:n
5944:.
5930::
5922::
5916:1
5709:,
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5703:=
5700:1
5697:+
5692:n
5688:x
5682:n
5678:a
5674:2
5671:+
5665:+
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5656:x
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5625:x
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5583:c
5563:=
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5555:1
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5541:x
5535:c
5530:n
5526:a
5520:2
5517:+
5511:+
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5491:1
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5481:2
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5444:1
5440:x
5426:c
5421:n
5417:a
5413:1
5410:a
5393:0
5390:=
5387:c
5384:+
5379:n
5375:x
5369:n
5365:a
5361:2
5358:+
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5347:1
5343:x
5337:1
5333:a
5329:2
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5321:2
5316:n
5312:x
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5270:(
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5202:1
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5150:.
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5137:=
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5131:J
5128:(
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5099:=
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5045:n
5035:n
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4937:.
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4925:)
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4915:o
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4857:(
4852:2
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4828:=
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4803:j
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4758:O
4752:P
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4727:=
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4693::
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4602:.
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4583:=
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4519:=
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4400:{
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4376:}
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4370:{
4355:}
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4349:{
4338:}
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4332:{
4320:=
4317:}
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4311:{
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4278:}
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4272:{
4263:}
4260:a
4257:{
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4206:}
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3144:(
3138:2
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3128:=
3120:2
3116:)
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3089:(
3083:2
3079:a
3073:+
3070:)
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3015:a
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2934:=
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2908:.
2904:R
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2855:)
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2842:(
2839:)
2836:a
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2827:(
2805:a
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2745:.
2727:)
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2708:=
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2688:=
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2625:2
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2611:|
2601:z
2594:=
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2538:=
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2500:y
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2488:=
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2428:(
2423:=
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2401:|
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2375:=
2367:2
2362:|
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2341:2
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2260:|
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2180:=
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2127:/
2123:d
2103:d
2081:2
2077:r
2054:1
2050:r
2029:w
2026:,
2023:z
2020:,
2017:y
2014:,
2011:x
1905:N
1885:)
1882:1
1876:,
1873:0
1870:,
1867:0
1864:(
1861:=
1858:S
1818:)
1809:,
1806:0
1803:,
1800:0
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1774:4
1771:1
1765:=
1760:2
1756:)
1749:2
1746:1
1740:+
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1734:(
1731:+
1726:2
1722:y
1718:+
1713:2
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1682:=
1677:2
1673:z
1669:+
1664:2
1660:y
1656:+
1651:2
1647:x
1626:S
1606:N
1533:O
1529:O
1525:O
1521:O
1517:O
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1497:2
1493:R
1489:=
1485:|
1480:|
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1428:|
1424:=
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1045:k
1040:.
1038:k
1034:k
1030:q
1026:p
1006:.
999:B
991:A
987:O
981:=
978:A
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933:=
930:B
927:A
924:O
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896:.
894:k
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886:k
882:q
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767:O
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745:P
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727:A
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707:h
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680:C
675:.
673:P
665:C
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626:P
622:A
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587:t
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573:s
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547:s
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222:2
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195:P
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168:P
164:r
160:O
152:P
132:P
125:P
34:.
20:)
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