scholarly journals Some results for the Hodge decomposition theorem in Euclidean three-space

2016 ◽  
Vol 11 ◽  
pp. 155-164
Author(s):  
Paul Bracken
Keyword(s):  
Decomposition Theorem ◽  
Hodge Decomposition ◽  
Euclidean Three Space ◽  
Three Space

Inflectional Convex Space Curves

1984 ◽  
Vol 36 (3) ◽  
pp. 537-549 ◽  
Author(s):  
Tibor Bisztriczky
Keyword(s):  
Convex Hull ◽  
Convex Space ◽  
Intersection Point ◽  
List Type ◽  
Multiple Points ◽  
Space Curves ◽  
Euclidean Three Space ◽  
Set Of Points ◽  
Three Space ◽  
Regular Closed

Let Φ be a regular closed C2 curve on a sphere S in Euclidean three-space. Let H(S)[H(Φ) ] denote the convex hull of S[Φ]. For any point p ∈ H(S), let O(p) be the set of points of Φ whose osculating plane at each of these points passes through p.1. THEOREM ([8]). If Φ has no multiple points and p ∈ H(Φ), then |0(p) | ≧ 3[4] when p is [is not] a vertex of Φ.2. THEOREM ( [9]). a) If the only self intersection point of Φ is a doublepoint and p ∈ H(Φ) is not a vertex of Φ, then |O(p)| ≧ 2.b) Let Φ possess exactly n vertices. Then(1)|O(p)| ≦ nforp ∈ H(S) and(2)if the osculating plane at each vertex of Φ meets Φ at exactly one point, |O(p)| = n if and only if p ∈ H(Φ) is not vertex.

Regular Skew Polyhedra in Hyperbolic Three-Space

1967 ◽  
Vol 19 ◽  
pp. 1179-1186 ◽  
Author(s):  
Cyril W. L. Garner
Keyword(s):  
Space Filling ◽  
Euclidean Three Space ◽  
Three Space

The study of regular skew polyhedra was initiated in 1926 by Petrie's discovery of two infinite polyhedra in Euclidean three-space E3 which were free of false vertices; the only other regular skew polyhedron in E3 was found by Coxeter (1, pp. 33-34). The simplest of these is denoted {4, 6 | 4} and is derived from the space-filling of cubes by omitting half the faces.

scholarly journals On the Fixed Homogeneous Circle Problem

2007 ◽  
Vol 7 (1) ◽  
Author(s):  
C. Azevêdo ◽  
P. Ontaneda ◽  
H. E. Cabral
Keyword(s):  
Gravitational Force ◽  
Circle Problem ◽  
Euclidean Three Space ◽  
Three Space

AbstractWe give some results about the dynamics of a particle moving in Euclidean three-space under the influence of the gravitational force induced by a fixed homogeneous circle. Our main results concern (1) singularities and (2) the dynamics in the plane that contains the circle. The study presented here is purely analytic.

On the Space of Oriented Lines in Euclidean Three-Space

1957 ◽  
Vol 64 (8) ◽  
pp. 79 ◽  
Author(s):  
Tibor Rado
Keyword(s):  
Euclidean Three Space ◽  
Three Space

Schwarzite nets: a wealth of 3-valent examples sharing similar topologies and symmetries

2021 ◽  
Vol 477 (2246) ◽  
pp. 20200372
Author(s):  
Stephen T. Hyde ◽  
Martin Cramer Pedersen
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Two Dimensional ◽  
Hyperbolic Surfaces ◽  
Euclidean Three Space ◽  
Three Space ◽  
Planar Graphene

We enumerate trivalent reticulations of two- and three-periodic hyperbolic surfaces by equal-sided n -gonal faces, ( n , 3), where n  = 7, 8, 9, 10, 12. These are the simplest hyperbolic generalizations of the planar graphene net, (6, 3) and dodecahedrane, (5, 3). The enumeration proceeds by deleting isometries of regular reticulations of two-dimensional hyperbolic space until the ( n , 3) nets can be embedded in euclidean three-space via periodic hyperbolic surfaces. Those nets are then symmetrized in euclidean space retaining their net topology, leading to explicit crystallographic net embeddings whose edges are as equal as possible, affording candidtae patterns for graphitic schwarzites. The resulting schwarzites are the most symmetric examples. More than one hundred topologically distinct nets are described, most of which are novel.

Rational Bézier and B-Spline Ruled Surface Patches

1996 ◽  
Author(s):  
Q. J. Ge ◽  
Donglai Kang
Keyword(s):  
Ruled Surfaces ◽  
Dual Numbers ◽  
Line Segments ◽  
B Spline ◽  
Surface Patches ◽  
Spline Curves ◽  
Oriented Line ◽  
Euclidean Three Space ◽  
Three Space ◽  
Projective Algorithms

Abstract This paper presents a geometric method for constructing bounded rational Bézier and B-spline ruled surfaces directly from line-segments. Oriented line-segments in a Euclidean three-space are represented by vectors with four homogeneous components over the ring of dual numbers. Projective algorithms for rational Bézier and B-spline curves are dualized to yield algorithms for rational Bézier and B-spline ruled surfaces.

Curves in Euclidean Three-Space

2014 ◽  
pp. 159-164
Author(s):  
Jan J. Koenderink
Keyword(s):  
Euclidean Three Space ◽  
Three Space
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