Equations of isometric imbeddings in three-dimensional Euclidean space of two-dimensional manifolds of negative curvature

1982 ◽  
Vol 31 (4) ◽  
pp. 305-312
Author(s):  
E. V. Shikin
Keyword(s):  
Euclidean Space ◽  
Negative Curvature ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Two Dimensional ◽  
Isometric Imbeddings

scholarly journals Trading spaces: building three-dimensional nets from two-dimensional tilings

2012 ◽  
Vol 2 (5) ◽  
pp. 555-566 ◽  
Author(s):  
Toen Castle ◽  
Myfanwy E. Evans ◽  
Stephen T. Hyde ◽  
Stuart Ramsden ◽  
Vanessa Robins
Keyword(s):  
Euclidean Space ◽  
Ab Initio ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Dimensional Euclidean Space ◽  
Two Dimensional ◽  
Complex Patterns ◽  
Geometrically Complex

We construct some examples of finite and infinite crystalline three-dimensional nets derived from symmetric reticulations of homogeneous two-dimensional spaces: elliptic ( S 2 ), Euclidean ( E 2 ) and hyperbolic ( H 2 ) space. Those reticulations are edges and vertices of simple spherical, planar and hyperbolic tilings. We show that various projections of the simplest symmetric tilings of those spaces into three-dimensional Euclidean space lead to topologically and geometrically complex patterns, including multiple interwoven nets and tangled nets that are otherwise difficult to generate ab initio in three dimensions.

scholarly journals Unification and classification of two-dimensional crystalline patterns using orbifolds

2014 ◽  
Vol 70 (4) ◽  
pp. 319-337 ◽  
Author(s):  
S. T. Hyde ◽  
S. J. Ramsden ◽  
V. Robins
Keyword(s):  
Euclidean Space ◽  
Minimal Surfaces ◽  
Three Dimensional ◽  
Hyperbolic Groups ◽  
Dimensional Euclidean Space ◽  
Two Dimensional ◽  
Triply Periodic Minimal Surfaces ◽  
Triply Periodic ◽  
Simple Rules

The concept of an orbifold is particularly suited to classification and enumeration of crystalline groups in the euclidean (flat) plane and its elliptic and hyperbolic counterparts. Using Conway's orbifold naming scheme, this article explicates conventional point, frieze and plane groups, and describes the advantages of the orbifold approach, which relies on simple rules for calculating the orbifold topology. The article proposes a simple taxonomy of orbifolds into seven classes, distinguished by their underlying topological connectedness, boundedness and orientability. Simpler `crystallographic hyperbolic groups' are listed, namely groups that result from hyperbolic sponge-like sections through three-dimensional euclidean space related to all known genus-three triply periodic minimal surfaces (i.e.theP,D,Gyroid,CLPandHsurfaces) as well as the genus-fourI-WPsurface.

THE FISHER–RAO METRIC FOR LINES IN A CONVEX IMAGE

2007 ◽  
Vol 21 (06) ◽  
pp. 977-994 ◽  
Author(s):  
STEPHEN J. MAYBANK
Keyword(s):  
Euclidean Space ◽  
Parameter Space ◽  
Three Dimensional ◽  
Experimental Results ◽  
Dimensional Euclidean Space ◽  
Two Dimensional ◽  
Euclidean Metric ◽  
Sample Points

The Fisher–Rao metric on the parameter space for the set of lines in a two-dimensional convex image is approximated under the assumption that the errors in the measurements are small. The volume of the parameter space under the approximating metric is proportional to the area of the image under the Euclidean metric. In the case of a rectangular image, expressions for the approximating metric are obtained and an algorithm is given for sampling the parameter space. The sample points are used in an algorithm for detecting lines in a rectangular image. Experimental results are reported. In the case of a disc shaped image the parameter space for lines embeds isometrically, under the approximating metric, into three-dimensional Euclidean space.

scholarly journals A simple and fast hole detection algorithm for triangulated surfaces

2020 ◽  
pp. 1-9
Author(s):  
Vijai Kumar Suriyababu ◽  
Cornelis Vuik
Keyword(s):  
Data Structure ◽  
Euclidean Space ◽  
Three Dimensional ◽  
Curvature Flow ◽  
Detection Algorithm ◽  
Dimensional Euclidean Space ◽  
Two Dimensional ◽  
Local Curvature ◽  
Triangulated Surfaces ◽  
Hole Detection

Abstract We present a simple and fast algorithm for computing the exact holes in discrete two-dimensional manifolds embedded in three-dimensional euclidean space. The algorithm detects the holes in the geometry directly without any approximation. Discrete Gaussian curvature is used for approximating the local curvature flow in the geometry and for removing outliers from the collection of feature edges. We present an algorithm with varying degrees of flexibility. The algorithm is demonstrated separately for sheets and solid geometries. The article demonstrates the algorithm on triangulated surfaces. However, the algorithm and the underlying data structure are also applicable for surfaces with mixed polygons.

scholarly journals On a Question of Bourgain about Geometric Incidences

2008 ◽  
Vol 17 (4) ◽  
pp. 619-625 ◽  
Author(s):  
JÓZSEF SOLYMOSI ◽  
CSABA D. TÓTH
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Geometric Incidences

Given a set of s points and a set of n2 lines in three-dimensional Euclidean space such that each line is incident to n points but no n lines are coplanar, we show that s = Ω(n11/4). This is the first non-trivial answer to a question recently posed by Jean Bourgain.

scholarly journals Orthogonal Isomorphic Representations Of Free Groups

1956 ◽  
Vol 8 ◽  
pp. 256-262 ◽  
Author(s):  
J. De Groot
Keyword(s):  
Euclidean Space ◽  
Free Groups ◽  
Three Dimensional ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Orthogonal Matrices ◽  
Abelian Subgroups ◽  
Proper Orthogonal

1. Introduction. We consider the group of proper orthogonal transformations (rotations) in three-dimensional Euclidean space, represented by real orthogonal matrices (aik) (i, k = 1,2,3) with determinant + 1 . It is known that this rotation group contains free (non-abelian) subgroups; in fact Hausdorff (5) showed how to find two rotations P and Q generating a group with only two non-trivial relationsP2 = Q3 = I.

Gröbner basis and resultant method for the forward displacement of 3-DoF planar parallel manipulators in seven-dimensional kinematic space

Robotica ◽  
2015 ◽  
Vol 34 (11) ◽  
pp. 2610-2628 ◽  
Author(s):  
Davood Naderi ◽  
Mehdi Tale-Masouleh ◽  
Payam Varshovi-Jaghargh
Keyword(s):  
Euclidean Space ◽  
Gröbner Basis ◽  
Three Dimensional ◽  
Parallel Manipulators ◽  
Parallel Robots ◽  
Dimensional Euclidean Space ◽  
Grobner Basis ◽  
Kinematic Space ◽  
Forward Kinematic Problem ◽  
Forward Kinematic

SUMMARYIn this paper, the forward kinematic analysis of 3-degree-of-freedom planar parallel robots with identical limb structures is presented. The proposed algorithm is based on Study's kinematic mapping (E. Study, “von den Bewegungen und Umlegungen,” Math. Ann.39, 441–565 (1891)), resultant method, and the Gröbner basis in seven-dimensional kinematic space. The obtained solution in seven-dimensional kinematic space of the forward kinematic problem is mapped into three-dimensional Euclidean space. An alternative solution of the forward kinematic problem is obtained using resultant method in three-dimensional Euclidean space, and the result is compared with the obtained mapping result from seven-dimensional kinematic space. Both approaches lead to the same maximum number of solutions: 2, 6, 6, 6, 2, 2, 2, 6, 2, and 2 for the forward kinematic problem of planar parallel robots; 3-RPR, 3-RPR, 3-RRR, 3-RRR, 3-RRP, 3-RPP, 3-RPP, 3-PRR, 3-PRR, and 3-PRP, respectively.

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