977:
545:
20:
2453:
1230:(i.e., directed) line through that point. The point and the line are incident cycles. The key observation is that the set of all cycles incident with both the point and the line is a Lie invariant object: in addition to the point and the line, it consists of all the circles which make oriented contact with the line at the given point. It is called a
989:. This problem concerns a configuration of three distinct circles (which may be points or lines): the aim is to find every other circle (including points or lines) which is tangent to all three of the original circles. For a generic configuration of circles, there are at most eight such tangent circles.
1021:, then the corresponding points , , lie on a line in projective space. Since a nontrivial quadratic equation has at most two solutions, this line actually lies in the Lie quadric, and any point on this line defines a cycle incident with , and . Thus there are infinitely many solutions in this case.
104:
Lie sphere geometry can be defined in any dimension, but the case of the plane and 3-dimensional space are the most important. In the latter case, Lie noticed a remarkable similarity between the Lie quadric of spheres in 3-dimensions, and the space of lines in 3-dimensional projective space, which is
1043:
orthogonal to all three is 2-dimensional. It can have signature (2,0), (1,0), or (1,1), in which case there are zero, one or two solutions for respectively. (The signature cannot be (0,1) or (0,2) because it is orthogonal to a space containing more than one null line.) In the case that the subspace
980:
The eight solutions of the generic
Apollonian problem. The three given circles are labeled C1, C2 and C3 and colored red, green and blue, respectively. The solutions are arranged in four pairs, with one pink and one black solution circle each, labeled as 1A/1B, 2A/2B, 3A/3B, and 4A/4B. Each pair
1190:
and so all cycles are Lie equivalent. In particular, points are not preserved by general Lie transformations. The subgroup of Lie transformations preserving the point cycles is essentially the subgroup of orthogonal transformations which preserve the chosen timelike direction. This subgroup is
539:
1218:
The fact that Lie transformations do not preserve points in general can also be a hindrance to understanding Lie sphere geometry. In particular, the notion of a curve is not Lie invariant. This difficulty can be mitigated by the observation that there is a Lie invariant notion of
896:∩ (1,0,0,0,0) are signature (2,1) subspaces of (1,0,0,0,0). They therefore either coincide or intersect in a 2-dimensional subspace. In the latter case, the 2-dimensional subspace can either have signature (2,0), (1,0), (1,1), in which case the corresponding two circles in
992:
The solution, using Lie sphere geometry, proceeds as follows. Choose an orientation for each of the three circles (there are eight ways to do this, but there are only four up to reversing the orientation of all three). This defines three points , , on the Lie quadric
74:). Lie sphere geometry is the geometry of the Lie quadric and the Lie transformations which preserve it. This geometry can be difficult to visualize because Lie transformations do not preserve points in general: points can be transformed into circles (or spheres).
1244:
Note that the cycles are all incident with each other as well. In terms of the Lie quadric, this means that a pencil of cycles is a (projective) line lying entirely on the Lie quadric, i.e., it is the projectivization of a totally null two dimensional subspace of
54:
in the nineteenth century. The main idea which leads to Lie sphere geometry is that lines (or planes) should be regarded as circles (or spheres) of infinite radius and that points in the plane (or space) should be regarded as circles (or spheres) of zero radius.
1723:
2605:
This provides one way to see that Dupin cyclides are cyclides, in the sense that they are zero-sets of quartics of a particular form. For this, note that as in the planar case, 3-dimensional
Euclidean space embeds into the Lie quadric
1892:
2564:. Conversely such a decomposition uniquely determines a contact lift of a surface which envelops two one parameter families of spheres; the image of this contact lift is given by the null 2-dimensional subspaces which intersect
152:
with automorphisms known as "Mobius transformations". Second, points are regarded as circles (or spheres) of zero radius. Finally, for technical reasons, the circles (or spheres), including the lines (or planes) are given
1154:), but with an overall reversal of orientation. Thus there are at most 8 solution circles to the Apollonian problem unless all three circles meet tangentially at a single point, when there are infinitely many solutions.
160:
These objects, i.e., the points, oriented circles and oriented lines in the plane, or the points, oriented spheres and oriented planes in space, are sometimes called cycles or Lie cycles. It turns out that they form a
812:
in the plane, where a cycle is either an oriented circle (or straight line) or a point in the plane (or the point at infinity); the points can be thought of as circles of radius zero, but they are not oriented.
267:
900:
intersect in zero, one or two points respectively. Hence they have first order contact if and only if the 2-dimensional subspace is degenerate (signature (1,0)), which holds if and only if the span of
2902:
2718:
181:
of the plane/sphere plus point at infinity. The point-preserving transformations are precisely the Möbius transformations. The Lie transformations which fix the ideal point at infinity are the
189:. These two subgroups generate the group of Lie transformations, and their intersection are the Möbius transforms that fix the ideal point at infinity, namely the affine conformal maps.
1342:, and the contact structure is Lie invariant. It follows that oriented curves can be studied in a Lie invariant way via their contact lifts, which may be characterized, generically as
1542:
3017:
An elementary video introducing concepts in
Laguerre geometry (whose transformation group is a subgroup of the group of Lie transformations). The video is presented from the
845:∪ {∞}, then this just means that lies on the circle corresponding to ; this case is immediate from the definition of this circle (if corresponds to a point circle then
1729:
743:
nonzero correspond to oriented circles (or oriented lines, which are circles through infinity) in the
Euclidean plane. This is easier to see in terms of the
144:
is added to
Euclidean space so that lines (or planes) can be regarded as circles (or spheres) passing through the point at infinity (i.e., having infinite
1063:
The general solution to the
Apollonian problem is obtained by reversing orientations of some of the circles, or equivalently, by considering the triples (
1186:
on the Lie quadric. These transformations of cycles are called "Lie transformations". They preserve the incidence relation between cycles. The action is
186:
225:
90:
981:
makes oriented contact with C1, C2, and C3, for a suitable choice of orientations; there are four such choices up to an overall orientation reversal.
109:. This similarity led Lie to his famous "line-sphere correspondence" between the space of lines and the space of spheres in 3-dimensional space.
2613:
as the set of point spheres apart from the ideal point at infinity. Explicitly, the point (x,y,z) in
Euclidean space corresponds to the point
534:{\displaystyle (x_{0},x_{1},x_{2},x_{3},x_{4})\cdot (y_{0},y_{1},y_{2},y_{3},y_{4})=-x_{0}y_{0}-x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{4}+x_{4}y_{3}.}
3014:
177:
known as the Lie transformations. These transformations do not preserve points in general: they are transforms of the Lie quadric,
2736:). The class of cyclides is a natural family of surfaces in Lie sphere geometry, and the Dupin cyclides form a natural subfamily.
997:. By the incidence of cycles, a solution to the Apollonian problem compatible with the chosen orientations is given by a point ∈
3008:
805:
of the Lie quadric which reverses the orientation of circles and lines, and fixes the points of the plane (including infinity).
3009:"On complexes - in particular, line and sphere complexes - with applications to the theory of partial differential equations"
2971:
2891:
2862:
2257:
defining the Plücker relation comes from a symmetric bilinear form of signature (3,3). In other words, the space of lines in
2265:). Although this is not the same as the Lie quadric, a "correspondence" can be defined between lines and spheres using the
2640:
1969:= 0. The group of Lie transformations is now O(n + 1, 2) and the Lie transformations preserve incidence of Lie cycles.
1210:
Lie transformations can often be used to simplify a geometrical problem, by transforming circles into lines or points.
149:
2958:
2602:
such that the associated symmetric endomorphism has square equal to the identity and eigenspaces of signature (2,1).
1980:: in terms of the given choice of point spheres, these contact elements correspond to pairs consisting of a point in
58:
The space of circles in the plane (or spheres in space), including points and lines (or planes) turns out to be a
1503:. In other words, after taking contact lifts, much of the basic theory of curves in the plane is Lie invariant.
3034:
2951:
2059:) describes the (Lie) geometry of spheres in Euclidean 3-space. Lie noticed a remarkable similarity with the
197:
833:= 0 if and only if the corresponding cycles "kiss", that is they meet each other with oriented first order
735:= 0 correspond to points in the Euclidean plane with an ideal point at infinity. On the other hand, points
1718:{\displaystyle (x_{0},x_{1},\ldots x_{n},x_{n+1},x_{n+2})\cdot (y_{0},y_{1},\ldots y_{n},y_{n+1},y_{n+2})}
3044:
3039:
1167:
692:. This is the Euclidean plane with an ideal point at infinity, which we take to be : the finite points (
1183:
229:
1227:
154:
121:
in the plane (resp. in space) which only depend on the concepts of circles (resp. spheres) and their
1456:
has a preferred Lie cycle associated to each point on the curve: the derivative of the immersion at
799:
182:
1484:); the kernel of any nonzero element of this subspace is a well defined 1-dimensional subspace of
1196:
2075:
2000:-sphere. This identification is not invariant under Lie transformations: in Lie invariant terms,
1445:
258:
254:
2575:
Such a decomposition is equivalently given, up to a sign choice, by a symmetric endomorphism of
228:). Eventually, there is also an isomorphism between the Möbius group and the Lorentz group (see
1343:
909:
201:
170:
2190:
681:
The orthogonal space to (1,0,0,0,0), intersected with the Lie quadric, is the two dimensional
3018:
2745:
1952:
The incidence relation carries over without change: the spheres corresponding to points , ∈
1220:
986:
834:
140:
and points) are treated on an equal footing. This is achieved in three steps. First an ideal
1249:: the representative vectors for the cycles in the pencil are all orthogonal to each other.
2060:
1949:
is an (n + 1)-dimensional manifold (spheres are parameterized by their center and radius).
1232:
689:
561:
174:
162:
67:
2464:. These are characterized as the common envelope of two one parameter families of spheres
1276:
whose fibres are circles. This map is not Lie invariant, as points are not Lie invariant.
8:
1887:{\displaystyle =-x_{0}y_{0}+x_{1}y_{1}+\cdots +x_{n}y_{n}+x_{n+1}y_{n+2}+x_{n+2}y_{n+1}.}
1163:
808:
To summarize: there is a one-to-one correspondence between points on the Lie quadric and
782:
has a definite sign; represents the same circle with the opposite orientation. Thus the
137:
2966:
2019:. There is no longer a preferred Lie cycle associated to each point: instead, there are
1264:. For a given choice of point cycles (the points orthogonal to a chosen timelike vector
2488:
are maps from intervals into the Lie quadric. In order for a common envelope to exist,
221:
125:
118:
105:
also a quadric hypersurface in a 5-dimensional projective space, called the Plücker or
98:
1311:) and the oriented line tangent to the curve at that point (the line in the direction
2990:
2954:
2935:
2918:
Knight, Robert D. (2005), "The
Apollonius contact problem and Lie contact geometry",
2887:
2858:
1500:
1187:
1018:
141:
86:
23:
Sophus Lie, the originator of Lie sphere geometry and the line-sphere correspondence.
209:
2980:
2927:
2869:
1993:
1414:
1339:
744:
682:
213:
166:
133:
71:
39:
1984:-dimensional space (which may be the point at infinity) together with an oriented
77:
To handle this, curves in the plane and surfaces in space are studied using their
2178:
1260:. The Lie transformations preserve the contact elements, and act transitively on
1037:
985:
The incidence of cycles in Lie sphere geometry provides a simple solution to the
129:
2946:
Milson, R. (2000) "An overview of Lie’s line-sphere correspondence", pp 1–10 of
2512:, i.e., their representative vectors must span a null 2-dimensional subspace of
2189:. There are six independent coordinates and they satisfy a single relation, the
1226:
An oriented contact element in the plane is a pair consisting of a point and an
2266:
2254:
2023:– 1 such cycles, corresponding to the curvature spheres in Euclidean geometry.
35:
2985:
2931:
2851:
Classical
Geometries in Modern Contexts: Geometry of Real Inner Product Spaces
205:
3028:
2994:
2939:
2461:
2243:
2016:
217:
193:
106:
94:
82:
2854:
128:
have a more natural formulation in a more general context in which circles,
2751:
976:
192:
These groups also have a direct physical interpretation: As pointed out by
1268:), every contact element contains a unique point. This defines a map from
117:
The key observation that leads to Lie sphere geometry is that theorems of
2846:
1925:
1200:
1175:
549:
63:
2748:, also can involve considering a line as a circle with infinite radius.
1985:
1937:
1363:
1192:
659:
51:
2876:, Grundlehren der mathematischen Wissenschaften, vol. 3, Springer
2238:
It follows that there is a one-to-one correspondence between lines in
2037:
2910:
Proceedings of the Sophus Lie
Memorial Conference, Oslo, August, 1992
1506:
885:
634:
To relate this to planar geometry it is necessary to fix an oriented
856:
It therefore remains to consider the case that neither nor are in
544:
169:
of dimension 4 or 5, which is known as the Lie quadric. The natural
1253:
783:
635:
619:
consists of the points in projective space represented by vectors
59:
31:
2598:
To summarize, Dupin cyclides are determined by quadratic forms on
2444:
defines a point on the complexified Klein quadric (where i = –1).
2803:; for a classification of solutions using Laguerre geometry, see
2026:
The problem of Apollonius has a natural generalization involving
700:) in the plane are then represented by the points = ; note that
122:
2853:, chapter 3: Sphere geometries of Möbius and Lie, pages 93–174,
2536:), i.e., if and only if there is an orthogonal decomposition of
216:
pointed out that the Laguerre group is simply isomorphic to the
145:
47:
43:
2774:. Almost all the material in this article can be found there.
2516:. Hence they define a map into the space of contact elements
19:
1499:
in the plane, then the preferred cycle at each point is the
1354:
at the point corresponding to a null 2-dimensional subspace
2520:. This map is Legendrian if and only if the derivatives of
1252:
The set of all lines on the Lie quadric is a 3-dimensional
2579:
whose square is the identity and whose ±1 eigenspaces are
2452:
2315:) is a point on the (complexified) Lie quadric (i.e., the
2770:
The definitive modern textbook on Lie sphere geometry is
2786:
Lie was particularly pleased with this achievement: see
638:
line. The chosen coordinates suggest using the point ∈
245:
The Lie quadric of the plane is defined as follows. Let
196:, the Lie sphere transformations are identical with the
93:
of a surface. It also allows for a natural treatment of
2872:(1929), "Differentialgeometrie der Kreise und Kugeln",
2004:
is the space of (projective) lines on the Lie quadric.
2460:
Lie sphere geometry provides a natural description of
3011:
English translation of Lie's key paper on the subject
2967:"Configurations of cycles and the Apollonius problem"
2643:
1732:
1545:
1213:
270:
2912:, Oslo: Scandinavian University Press, pp. 3–21
2063:
for lines in 3-dimensional space (more precisely in
2713:{\displaystyle \sum _{i,j=1}^{5}a_{ij}x_{i}x_{j}=0}
2038:
Three dimensions and the line-sphere correspondence
1199:of the sphere. It can also be characterized as the
235:
2964:
2817:
2800:
2712:
2246:, which is the quadric hypersurface of points in
1886:
1717:
1507:Lie sphere geometry in space and higher dimensions
1303:to the contact element corresponding to the point
821:Suppose two cycles are represented by points , ∈
533:
2965:Zlobec, Borut Jurčič; Mramor Kosta, Neža (2001),
1961:have oriented first order contact if and only if
1182:to another such subspace. Hence the group O(3,2)
3026:
952:= 1 ± 1. The contact is oriented if and only if
3015:"Oriented circles and 3D relativistic geometry"
2634:which satisfy an additional quadratic relation
2177:. These are the homogeneous coordinates of the
1940:and point spheres as limiting cases. Note that
860:. Without loss of generality, we can then take
552:is a 2-dimensional analogue of the Lie quadric.
253:of 5-tuples of real numbers, equipped with the
2816:This problem and its solution is discussed by
2948:The Geometric Study of Differential Equations
2782:
2780:
2540:into a direct sum of 3-dimensional subspaces
1992:is therefore isomorphic to the projectivized
85:. This provides a natural realisation of the
2591:, this is determined by a quadratic form on
971:
2886:, Universitext, Springer-Verlag, New York,
2777:
1536:equipped with the symmetric bilinear form
226:Laguerre group isomorphic to Lorentz group
2984:
2950:, J.A. Leslie & T.P. Robart editors,
2011:-dimensional space has a contact lift to
1429:) of this tangent space in the space Hom(
1044:has signature (1,0), the unique solution
2900:
2868:
2830:
2799:The Lie sphere approach is discussed in
2787:
2451:
1924:= 0. The quadric parameterizes oriented
1207:, which is itself a Lie transformation.
975:
543:
42:in which the fundamental concept is the
18:
2324:are taken to be complex numbers), then
1350:. More precisely, the tangent space to
3027:
2917:
2874:Vorlesungen über Differentialgeometrie
2804:
2627:. A cyclide consists of the points ∈
2007:Any immersed oriented hypersurface in
1988:passing through that point. The space
1157:
816:
798:· (1,0,0,0,0)) (1,0,0,0,0) induces an
2972:Rocky Mountain Journal of Mathematics
2881:
2829:The following discussion is based on
2771:
1972:The space of contact elements is a (2
1524:(corresponding to the Lie quadric in
1520:-dimensions is obtained by replacing
1256:called the space of contact elements
1174:maps any one-dimensional subspace of
1488:, i.e., a point on the Lie quadric.
774:= 0. The circle is oriented because
646:can then be represented by a vector
568:and is the space of nonzero vectors
560:P is the space of lines through the
2926:(1–2), Basel: Birkhäuser: 137–152,
2250:P satisfying the Plücker relation.
1460:is a 1-dimensional subspace of Hom(
888:unit vectors in (1,0,0,0,0). Thus
230:Möbius group#Lorentz transformation
13:
1976:– 1)-dimensional contact manifold
1908:is again defined as the set of ∈
1214:Contact elements and contact lifts
1036:are linearly independent then the
240:
14:
3056:
3002:
2723:for some symmetric 5 ×; 5 matrix
2447:
1511:
1142:)) yields the same solutions as (
750:: the circle corresponding to ∈
112:
97:and a conceptual solution of the
642:P. Any point in the Lie quadric
236:Lie sphere geometry in the plane
81:, which are determined by their
2903:"Sophus Lie, the Mathematician"
1495:is the contact lift of a curve
2823:
2818:Zlobec & Mramor Kosta 2001
2810:
2801:Zlobec & Mramor Kosta 2001
2793:
2764:
2548:of signature (2,1), such that
1936:-dimensional space, including
1712:
1632:
1626:
1546:
407:
342:
336:
271:
198:spherical wave transformations
148:). This extension is known as
1:
2952:American Mathematical Society
2840:
2587:. Using the inner product on
1017:. If these three vectors are
912:, this holds if and only if (
936:) = 1, i.e., if and only if
7:
2901:Helgason, Sigurdur (1994),
2739:
2504:) must be incident for all
2015:determined by its oriented
1491:In more familiar terms, if
1287:be an oriented curve. Then
16:Geometry founded on spheres
10:
3061:
1168:orthogonal transformations
758:≠ 0) is the set of points
716:· (0,0,0,0,1) = −1.
615:). The planar Lie quadric
2932:10.1007/s00022-005-0009-x
2882:Cecil, Thomas E. (1992),
2572:in a pair of null lines.
1362:is the subspace of those
972:The problem of Apollonius
853:= 0 if and only if = ).
662:to (1,0,0,0,0). Since ∈
2757:
731:on the Lie quadric with
204:invariant. In addition,
183:Laguerre transformations
175:group of transformations
2986:10.1216/rmjm/1020171586
2076:homogeneous coordinates
1516:Lie sphere geometry in
1195:to the group O(3,1) of
259:symmetric bilinear form
200:that leave the form of
173:of this quadric form a
50:. It was introduced by
2714:
2670:
2457:
2261:P is the quadric in P(
1888:
1719:
1295:from the interval to
1197:Möbius transformations
1118:Note that the triple (
982:
712:· (1,0,0,0,0) = 0 and
553:
535:
24:
3035:Differential geometry
3019:rational trigonometry
2715:
2644:
2455:
1889:
1720:
987:problem of Apollonius
979:
556:The projective space
547:
536:
99:problem of Apollonius
22:
2641:
2528:) are orthogonal to
2242:P and points on the
2061:Klein correspondence
2030:+ 1 hyperspheres in
1730:
1543:
1417:is the subspace Hom(
1415:contact distribution
1048:lies in the span of
690:Minkowski space-time
268:
163:quadric hypersurface
89:to a curve, and the
68:quadric hypersurface
2920:Journal of Geometry
2884:Lie sphere geometry
1528:= 2 dimensions) by
1444:It follows that an
1162:Any element of the
1158:Lie transformations
910:Lagrange's identity
817:Incidence of cycles
576:up to scale, where
202:Maxwell's equations
28:Lie sphere geometry
3045:Conformal geometry
3040:Incidence geometry
2746:Descartes' theorem
2710:
2458:
1884:
1715:
1441:) of linear maps.
1203:of the involution
1019:linearly dependent
983:
908:is degenerate. By
892:∩ (1,0,0,0,0) and
554:
531:
222:special relativity
150:inversive geometry
119:Euclidean geometry
25:
2893:978-0-387-97747-8
2870:Blaschke, Wilhelm
2863:978-3-7643-8541-5
1501:osculating circle
1448:Legendrian curve
1344:Legendrian curves
1291:determines a map
1005:is orthogonal to
650:= λ(1,0,0,0,0) +
249:denote the space
187:Laguerre geometry
142:point at infinity
91:curvature spheres
87:osculating circle
3052:
2997:
2988:
2942:
2913:
2907:
2896:
2877:
2834:
2827:
2821:
2814:
2808:
2797:
2791:
2784:
2775:
2768:
2719:
2717:
2716:
2711:
2703:
2702:
2693:
2692:
2683:
2682:
2669:
2664:
2560:takes values in
2552:takes values in
2456:A Dupin cyclide.
2191:Plücker relation
2046:=3, the quadric
1994:cotangent bundle
1899:The Lie quadric
1893:
1891:
1890:
1885:
1880:
1879:
1864:
1863:
1845:
1844:
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1643:
1625:
1624:
1606:
1605:
1587:
1586:
1571:
1570:
1558:
1557:
1340:contact manifold
1272:to the 2-sphere
872:= (1,0,0,0,0) +
864:= (1,0,0,0,0) +
745:celestial sphere
683:celestial sphere
540:
538:
537:
532:
527:
526:
517:
516:
504:
503:
494:
493:
481:
480:
471:
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380:
379:
367:
366:
354:
353:
335:
334:
322:
321:
309:
308:
296:
295:
283:
282:
214:Wilhelm Blaschke
167:projective space
136:(resp. spheres,
72:projective space
40:spatial geometry
3060:
3059:
3055:
3054:
3053:
3051:
3050:
3049:
3025:
3024:
3005:
2905:
2894:
2843:
2838:
2837:
2833:, pp. 4–5.
2828:
2824:
2815:
2811:
2798:
2794:
2785:
2778:
2769:
2765:
2760:
2742:
2735:
2698:
2694:
2688:
2684:
2675:
2671:
2665:
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2642:
2639:
2638:
2633:
2626:
2612:
2450:
2440:
2433:
2426:
2419:
2411:
2404:
2397:
2390:
2383:
2376:
2368:
2361:
2354:
2347:
2340:
2333:
2323:
2314:
2307:
2300:
2293:
2286:
2279:
2267:complex numbers
2233:
2227:
2220:
2214:
2207:
2201:
2179:projective line
2176:
2168:
2159:
2151:
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1239:contact element
1221:contact element
1216:
1160:
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786:reflection map
614:
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241:The Lie quadric
238:
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17:
12:
11:
5:
3058:
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3012:
3004:
3003:External links
3001:
3000:
2999:
2979:(2): 725–744,
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2462:Dupin cyclides
2449:
2448:Dupin cyclides
2446:
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2255:quadratic form
2236:
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2218:
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2199:
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2155:
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2110:
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2096:
2089:
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2050:
2039:
2036:
2017:tangent spaces
1956:
1944:
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1561:
1556:
1552:
1548:
1513:
1512:General theory
1510:
1508:
1505:
1411:
1410:
1323:is called the
1237:, or simply a
1215:
1212:
1159:
1156:
973:
970:
818:
815:
727:(1,0,0,0,0) +
719:Hence points
612:
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591:
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239:
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234:
210:Henri Poincaré
114:
113:Basic concepts
111:
95:Dupin cyclides
83:tangent spaces
15:
9:
6:
4:
3:
2:
3057:
3046:
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3016:
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3010:
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3006:
2996:
2992:
2987:
2982:
2978:
2974:
2973:
2968:
2963:
2960:
2959:0-8218-2964-5
2956:
2953:
2949:
2945:
2941:
2937:
2933:
2929:
2925:
2921:
2916:
2911:
2904:
2899:
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2889:
2885:
2880:
2875:
2871:
2867:
2864:
2860:
2856:
2852:
2848:
2845:
2844:
2832:
2831:Helgason 1994
2826:
2819:
2813:
2806:
2802:
2796:
2789:
2788:Helgason 1994
2783:
2781:
2773:
2767:
2763:
2753:
2750:
2747:
2744:
2743:
2737:
2734:
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2726:
2707:
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2689:
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2649:
2645:
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2609:
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2523:
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2515:
2511:
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2475:
2471:
2467:
2463:
2454:
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2430:
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2268:
2264:
2260:
2256:
2251:
2249:
2245:
2244:Klein quadric
2241:
2230:
2224:
2217:
2211:
2204:
2198:
2195:
2194:
2193:
2192:
2188:
2184:
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2130:
2123:
2116:
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2095:
2088:
2081:
2077:
2073:
2070:Suppose , ∈
2068:
2066:
2062:
2058:
2053:
2049:
2045:
2035:
2033:
2029:
2024:
2022:
2018:
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1995:
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1983:
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1968:
1964:
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1939:
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1931:
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1923:
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1736:
1733:
1726:
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1707:
1704:
1701:
1697:
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1602:
1599:
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1349:
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1332:
1330:
1326:
1322:
1319:)). This map
1318:
1314:
1310:
1306:
1302:
1298:
1294:
1290:
1286:
1282:
1277:
1275:
1271:
1267:
1263:
1259:
1255:
1250:
1248:
1242:
1240:
1236:
1235:of Lie cycles
1234:
1229:
1224:
1222:
1211:
1208:
1206:
1202:
1198:
1194:
1189:
1185:
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1022:
1020:
1016:
1012:
1008:
1004:
1000:
996:
990:
988:
978:
969:
967:
963:
960:= – 1, i.e.,
959:
955:
951:
947:
944:= ± 1, i.e.,
943:
939:
935:
931:
927:
923:
919:
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911:
907:
903:
899:
895:
891:
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867:
863:
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854:
852:
848:
844:
840:
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832:
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797:
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368:
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346:
339:
331:
327:
323:
318:
314:
310:
305:
301:
297:
292:
288:
284:
279:
275:
264:
263:
262:
260:
256:
252:
248:
233:
231:
227:
223:
219:
218:Lorentz group
215:
211:
207:
203:
199:
195:
194:Harry Bateman
190:
188:
184:
180:
176:
172:
168:
164:
158:
156:
151:
147:
143:
139:
135:
131:
127:
124:
120:
110:
108:
107:Klein quadric
102:
100:
96:
92:
88:
84:
80:
79:contact lifts
75:
73:
69:
65:
62:known as the
61:
56:
53:
49:
45:
41:
37:
33:
29:
21:
2976:
2970:
2947:
2923:
2919:
2909:
2883:
2873:
2850:
2825:
2812:
2795:
2790:, p. 7.
2766:
2752:Quasi-sphere
2732:
2728:
2724:
2722:
2628:
2621:
2619:
2607:
2604:
2599:
2597:
2592:
2588:
2584:
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2549:
2545:
2541:
2537:
2533:
2529:
2525:
2521:
2517:
2513:
2509:
2505:
2501:
2497:
2493:
2489:
2485:
2481:
2477:
2473:
2469:
2465:
2459:
2443:
2435:
2428:
2421:
2414:
2406:
2399:
2392:
2385:
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2363:
2356:
2349:
2342:
2335:
2328:
2320:
2316:
2309:
2302:
2295:
2288:
2281:
2274:
2270:
2262:
2258:
2252:
2247:
2239:
2237:
2228:
2222:
2215:
2209:
2202:
2196:
2186:
2182:
2173:
2169:
2165:
2161:
2156:
2152:
2148:
2144:
2139:
2135:
2128:
2121:
2114:
2107:
2100:
2093:
2086:
2079:
2071:
2069:
2064:
2056:
2051:
2047:
2043:
2042:In the case
2041:
2034:dimensions.
2031:
2027:
2025:
2020:
2012:
2008:
2006:
2001:
1997:
1989:
1981:
1977:
1973:
1971:
1966:
1962:
1957:
1953:
1951:
1945:
1941:
1933:
1930:– 1)-spheres
1927:
1921:
1917:
1913:
1909:
1904:
1900:
1898:
1533:
1529:
1525:
1521:
1517:
1515:
1496:
1492:
1490:
1485:
1481:
1477:
1473:
1469:
1465:
1461:
1457:
1453:
1449:
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1438:
1434:
1430:
1426:
1422:
1418:
1412:
1406:
1402:
1398:
1394:
1390:
1386:
1379:
1375:
1371:
1367:
1359:
1355:
1351:
1347:
1335:
1333:
1328:
1325:contact lift
1324:
1320:
1316:
1312:
1308:
1304:
1300:
1296:
1292:
1288:
1284:
1280:
1278:
1273:
1269:
1265:
1261:
1257:
1251:
1246:
1243:
1238:
1231:
1225:
1217:
1209:
1204:
1179:
1176:null vectors
1171:
1161:
1151:
1147:
1143:
1139:
1135:
1131:
1127:
1123:
1119:
1117:
1112:
1108:
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1100:
1096:
1092:
1088:
1084:
1080:
1076:
1072:
1068:
1064:
1062:
1057:
1053:
1049:
1045:
1040:
1033:
1029:
1025:
1023:
1014:
1010:
1006:
1002:
998:
994:
991:
984:
965:
961:
957:
953:
949:
945:
941:
937:
933:
929:
925:
921:
917:
913:
905:
901:
897:
893:
889:
881:
877:
873:
869:
865:
861:
857:
855:
850:
846:
842:
838:
830:
826:
822:
820:
809:
807:
802:
795:
791:
787:
779:
775:
771:
767:
763:
759:
755:
751:
747:
740:
736:
732:
728:
724:
720:
718:
713:
709:
705:
701:
697:
693:
685:
680:
675:
671:
667:
663:
655:
651:
647:
643:
639:
633:
628:
624:
620:
616:
609:
602:
595:
588:
581:
577:
573:
569:
565:
557:
555:
250:
246:
244:
191:
178:
159:
155:orientations
116:
103:
78:
76:
57:
27:
26:
3021:perspective
2847:Walter Benz
2805:Knight 2005
1938:hyperplanes
1364:linear maps
1299:by sending
1201:centralizer
1024:If instead
550:hyperboloid
261:defined by
206:Élie Cartan
64:Lie quadric
32:geometrical
3029:Categories
2855:Birkhäuser
2841:References
2772:Cecil 1992
1986:hyperplane
1532:. This is
1193:isomorphic
1188:transitive
1166:O(3,2) of
1001:such that
800:involution
660:orthogonal
171:symmetries
123:tangential
52:Sophus Lie
34:theory of
2995:0035-7596
2940:0047-2468
2646:∑
2480:), where
1786:⋯
1737:−
1662:…
1630:⋅
1576:…
886:spacelike
784:isometric
437:−
414:−
340:⋅
255:signature
2740:See also
2181:joining
2074:P, with
1472:) where
1446:immersed
1413:and the
1334:In fact
1254:manifold
1228:oriented
1095:)) and (
1038:subspace
876:, where
837:. If ∈
654:, where
636:timelike
548:A ruled
60:manifold
2849:(2007)
2134:). Put
2106:) and (
1996:of the
1916:) with
1366:(A mod
835:contact
825:. Then
126:contact
2993:
2957:
2938:
2890:
2861:
2496:) and
2472:) and
1912:P = P(
1233:pencil
810:cycles
754:(with
562:origin
257:(3,2)
146:radius
138:planes
134:points
48:sphere
44:circle
36:planar
2906:(PDF)
2758:Notes
2269:: if
2055:in P(
1409:) = 0
1382:with
1338:is a
1164:group
1079:), (
968:= 0.
920:) = (
794:+ 2 (
766:with
739:with
708:= 0,
678:≥ 0.
631:= 0.
623:with
224:(see
165:in a
130:lines
30:is a
2991:ISSN
2955:ISBN
2936:ISSN
2888:ISBN
2859:ISBN
2583:and
2568:and
2556:and
2544:and
2532:(or
2524:(or
2508:and
2484:and
2253:The
2234:= 0.
2185:and
2067:P).
1393:) ·
1283:: →
1279:Let
1184:acts
1115:)).
1056:and
1032:and
1013:and
904:and
884:are
880:and
868:and
212:and
132:and
2981:doi
2928:doi
2727:= (
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