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1.
Walter Benz 《Journal of Geometry》2004,79(1-2):19-26
Suppose that X is a real inner product
space of (finite or infinite) dimension at least 2. A distance preserving mapping
, where
is a (finite or infinite) subset of a
finite-dimensional subspace of X, can be extended
to an isometry
of X. This holds true for
euclidean as well as for hyperbolic geometry. To both geometries there exist examples
of non-extentable distance preserving
, where S
is not contained in a finite-dimensional subspace of
X. 相似文献
2.
Summary Abstract regular polytopes are complexes which generalize the classical regular polytopes. This paper discusses the topology of abstract regular polytopes whose vertex-figures are spherical and whose facets are topologically distinct from balls. The case of toroidal facets is particularly interesting and was studied earlier by Coxeter, Shephard and Grünbaum. Ann-dimensional manifold is associated with many abstract (n + 1)-polytopes. This is decomposed inton-dimensional manifolds-with-boundary (such as solid tori). For some polytopes with few faces the topological type or certain topological invariants of these manifolds are determined. For 4-polytopes with toroidal facets the manifolds include the 3-sphereS
3, connected sums of handlesS
1
× S
2
, euclidean and spherical space forms, and other examples with non-trivial fundamental group. 相似文献
3.
In this paper we give a combinatorial characterization of the Corrado Segre variety of type {n,m} in terms of its incidence structure of points and lines. 相似文献
4.
5.
Kurt Leichtweiß 《Journal of Geometry》2003,78(1-2):92-121
Analogue to the definition $K + L := \bigcup_{x\in K}(x + L)$ of the
Minkowski addition in the euclidean geometry it is proposed to define the
(noncommutative) addition $K \vdash L := \bigcup_{0\, \leqsl\, \rho\,\leqsl\,
a(\varphi),0\,\leqsl\,\varphi\,<\, 2\pi}T_{\rho}^{(\varphi)}(L)$ for compact,
convex and smoothly bounded sets K and
L in the hyperbolic plane $\Omega$
(Kleins model). Here $\rho = a(\varphi)$ is the representation of the boundary
$\partial$ K in geodesic polar coordinates
and $T_{\rho}^{(\varphi)}$ is the hyperbolic translation of $\Omega$ of length
$\rho$ along the line through the origin o of
direction $\varphi$. In general this addition does not preserve
convexity but nevertheless we may prove as main results: (1) $o \in$ int
$K, o \in$ int L and K,L horocyclic convex imply the strict
convexity of $K \vdash L$, and (2) in this case there exists a hyperbolic mixed
volume $V_h(K,L)$ of K and L which has a representation by a suitable
integral over the unit circle. 相似文献
6.
Maurizio Ternullo 《Journal of Geometry》2007,87(1-2):179-187
Let H be an ellipse with semiaxes a and b (a > b). Two circles concentric with H, and with radii a − b and a + b, are described, each of them being the locus of the intersections between couples of noteworthy H-related lines (Theorems 1 and 2). Tight, as well as unexpected links among such circles and Monge’s circle are shown (Theorems
4, 5, and 6). A surprising pythagorean relationship involving segments related to the ellipse is shown (Theorem 3). A set
of 10 concyclic points is associated with any general point of H (Theorem 9). New properties of Fagnano’s point are described (Theorems 10 through 13). Only elementary facts from trigonometry
and analytic geometry are used.
相似文献
7.
Given a matroid and an integer n 0, eleven conditions are shown to be equivalent to the validity of the rank formula r(E F) + r(E F = r(E) + r(F) for subspaces satisfying r(EF)n. For n=0 one finds the projective geometries. The case n=1 also includes the affine and the hyperbolic geometries, the case n=2 the Möbius geometries. The general case covers the incidence geometries of grade n of Wille. 相似文献
8.
B. Ettaoui 《Geometriae Dedicata》1994,50(3):235-241
Résumé On montre l'existence, dans un espace projectif complexe, d'un tétraèdre régulier ayant un groupe de symétrie isomorphe à celui du tétraèdre régulier euclidien. On précise la classe d'homologie de dimension 2, qui contient le complexe simplicial qui lui est associé.
We show the existence, in a complex projective space, of a regular tetrahedron which has its symmetry group isomorphic to that of the euclidean regular tetrahedron. We give precisely the two-dimensional homology class, containing the simplicial complexe associated to it.相似文献
9.
Yang Shiguo 《Geometriae Dedicata》1996,62(2):157-160
In this paper we present a geometric inequality for a finite number of points on an (n–1)-dimensional sphere S
n–1(R). As an application, we obtain a geometric inequality for finitely many points in the n-dimensional Euclidean space E
n. 相似文献
10.
Three geometric inequalities for a simplex 总被引:3,自引:0,他引:3
Yang Shiguo 《Geometriae Dedicata》1995,57(1):105-110
In this paper, we obtain three new geometric inequalities for ann-dimensional simplex in then-dimensional Euclidean spaceE
n
. As special cases we find two known inequalities from L. Fejes Tóth and M. S. Klamkin, respectively. 相似文献
11.
A point P in the plane of triangle ABC is said to be an isoperimetric point if PA + PB + AB = PB + PC + BC = PC + PA + CA, and is said to be a point of equal detour if PA + PB − AB = PB + PC − BC = PC + PA − CA. Incorrect conditions for the existence and uniqueness of such points were given by G. R. Veldkamp in Amer. Math. Monthly
92
(1985) 546-558. In this paper, we use a much simpler approach that yields correct versions of these conditions and that exhibits
the relations of these points to the centers of the Soddy circles.
Mowaffaq Hajja: This work is supported by a research grant from Yarmouk University. 相似文献
12.
G. C. Shephard 《Geometriae Dedicata》1995,56(1):75-84
LetQ
1,Q
2,Q
3 be plane quadrilaterals such that the sides ofQ
i
are the perpendicular bisectors of the sides ofQ
i–1 (i=2, 3). This note gives a simple trigonometrical solution of the long outstanding problem of showing thatQ
1 andQ
3 are similar. 相似文献
13.
Yang Shiguo 《Geometriae Dedicata》1996,62(2):161-165
In this paper we prove some inequalities on areas of bisection planes of dihedral angles of a simplex in E
n. 相似文献
14.
R. J. Gregorac 《Geometriae Dedicata》1996,60(1):65-88
Ptolemy's equality for four points on a circle is related to a Feuerbach-type area relation. This suggested an extension of Ptolemy's inequality to a Feuerbach type volume relation between simplexes formed from n+2 points in R
n
(n2). Extensions of the Möbius-Neuberg and Pompeiu Theorems in R
2 are given for R
n
. Ptolemy's inequality is also extended to convex n-gons in R
2 yielding an extension of Fuhrmann's hexagon theorem to 2n-gons in R
2 (n3). 相似文献
15.
I. Bárány 《Combinatorica》1987,7(2):161-169
The existence of a functionn(ε) (ε>0) is established such that given a finite setV in the plane there exists a subsetW⊆V, |W|<n(ε) with the property that for anyv εV\
W there are two pointsw
1,w
2 εW such that the angle ∢(w
1
vw
2)>π-ε. 相似文献
16.
B. Et-Taoui 《Geometriae Dedicata》1996,63(3):297-308
Résumé On étudie dans P
n
les m-uples de points, appelés F-réguliers, dont les sous-triplets ordonnés sont deux à deux isométriques. On montre qu'il existe au plus deux classes d'isométrie de quintuplets F-réguliers contenant un triangle équilatère T donné. On étudie aussi les m-uples F-réguliers, dont les sous k-uples (k<m) non ordonnés sont deux à deux isométriques. Ces m-uples sont appelés k-réguliers. On montre que la 4-régularité implique la k-régularité pour tous les k5.
We investigate in P n m-tuples of points in which all ordered triples are pairwise isometric. Such m-tuples are called F-regular. We show that for a given triangle T there exist at most two isometry classes of F-regular quintuples containing T. We also investigate F-regular m-tuples in which all (unordered) k-tuples (k<m) are pairwise isometric. Such m-tuples are called k-regular. We show that 4-regularity implies k-regularity for all k5.相似文献
17.
Dimitri Leemans 《Journal of Geometry》2004,79(1-2):146-155
We construct nine rank five incidence geometries that are firm and residually connected
and on which the Mathieu group M22 acts flag-transitively. The constructions use
mainly objects arising from the Steiner systemS(3, 6, 22).
One of these geometries was constructed by Meixner and Pasini in [10]. Three of them
are obtained from the geometry of Meixner and Pasini using doubling (see [8] or [12]) or similar
constructions. The remaining five are new and four of them have a star diagram. These
latter four geometries are constructed using special partitions of the 22 points of
the Steiner system S(3, 6, 22). 相似文献
18.
Peter McMullen 《Aequationes Mathematicae》1994,47(2-3):223-239
Summary In an earlier paper, a theory of realizations of (finite) regular polytopes in euclidean spaces was developed. Here, the analogous problem of realizing regular apeirotopes (infinite polytopes) is investigated. While no complete theory is expounded, several basic results are established. Among these is the curious fact that, if a regular apeirotope has a discrete realization, then it has one with no translations in its symmetry group. 相似文献
19.
Sheng Jiang 《Geometriae Dedicata》1996,63(2):113-121
The angle between two subspaces of dimensions p and q in a Euclidean space is considered by using exterior algebra. Some properties of angles are obtained. The relation between such a higher dimensional angle and the usual principal angles is also given. And finally, an application to Grassmann manifolds is briefly discussed.Supported by the National Natural Science Fund of P.R. China. 相似文献
20.
Małgorzata Buba-Brzozowa 《Journal of Geometry》2004,81(1-2):21-29
This article presents a generalization of some theorems connected with the nine-point circle for n-dimensional Euclidean space. 相似文献