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.

Finite rotation groups and crystal classes in four dimensions

1951 ◽  
Vol 47 (4) ◽  
pp. 650-661 ◽  
Author(s):  
A. C. Hurley
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Explicit Calculation ◽  
Dimensional Euclidean Space ◽  
Finite Rotation ◽  
Four Dimensions ◽  
Higher Dimensional ◽  
Motion Groups ◽  
Rotation Groups

The groups of symmetries of three-dimensional lattices have been known for some time. They consist of finite rotation groups, the crystal classes, and infinite discrete motion groups, which include both rotations and translations. The general theory of the corresponding groups in higher dimensional Euclidean spaces has also been developed. This theory includes a demonstration that in Euclidean space of n dimensions the number of motion groups is finite, and leads to a method† for calculating the motion groups, the first step being to determine the crystal classes. The explicit calculation of the various groups by the general method is not simple, and has so far been confined to the case of two and three dimensions. In the special case of the crystal classes in four dimensions, however, we may make use of the results of a paper by Goursat‡. In this paper Goursat sets up a correspondence between the finite rotation groups in four-dimensional Euclidean space, and a set of groups each of which is formed by associating two of Klein's groups of linear non-homogeneous substitutions in one variable. Using this result he is able to evaluate explicitly all the proper and improper finite four-dimensional rotation groups which include the element −I, where I is the four-rowed unit matrix.

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.

Mine Impact Burial Prediction From One to Three Dimensions

2008 ◽  
Vol 62 (1) ◽  
Author(s):  
Peter C. Chu
Keyword(s):  
Three Dimensional ◽  
Time History ◽  
Three Dimensions ◽  
Vertical Position ◽  
Burial Depth ◽  
Prediction Errors ◽  
Two Dimensional ◽  
Dimensional Model ◽  
One Dimensional ◽  
Dimensional Models

The Navy’s mine impact burial prediction model creates a time history of a cylindrical or a noncylindrical mine as it falls through air, water, and sediment. The output of the model is the predicted mine trajectory in air and water columns, burial depth/orientation in sediment, as well as height, area, and volume protruding. Model inputs consist of parameters of environment, mine characteristics, and initial release. This paper reviews near three decades’ effort on model development from one to three dimensions: (1) one-dimensional models predict the vertical position of the mine’s center of mass (COM) with the assumption of constant falling angle, (2) two-dimensional models predict the COM position in the (x,z) plane and the rotation around the y-axis, and (3) three-dimensional models predict the COM position in the (x,y,z) space and the rotation around the x-, y-, and z-axes. These models are verified using the data collected from mine impact burial experiments. The one-dimensional model only solves one momentum equation (in the z-direction). It cannot predict the mine trajectory and burial depth well. The two-dimensional model restricts the mine motion in the (x,z) plane (which requires motionless for the environmental fluids) and uses incorrect drag coefficients and inaccurate sediment dynamics. The prediction errors are large in the mine trajectory and burial depth prediction (six to ten times larger than the observed depth in sand bottom of the Monterey Bay). The three-dimensional model predicts the trajectory and burial depth relatively well for cylindrical, near-cylindrical mines, and operational mines such as Manta and Rockan mines.

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.

Monkey superior colliculus represents rapid eye movements in a two-dimensional motor map

1993 ◽  
Vol 69 (3) ◽  
pp. 965-979 ◽  
Author(s):  
K. Hepp ◽  
A. J. Van Opstal ◽  
D. Straumann ◽  
B. J. Hess ◽  
V. Henn
Keyword(s):  
Eye Movements ◽  
Superior Colliculus ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Eye Position ◽  
Two Dimensional ◽  
Vestibular Nystagmus ◽  
Rapid Eye Movements ◽  
Q Model

1. Although the eye has three rotational degrees of freedom, eye positions, during fixations, saccades, and smooth pursuit, with the head stationary and upright, are constrained to a plane by ListingR's law. We investigated whether Listing's law for rapid eye movements is implemented at the level of the deeper layers of the superior colliculus (SC). 2. In three alert rhesus monkeys we tested whether the saccadic motor map of the SC is two dimensional, representing oculocentric target vectors (the vector or V-model), or three dimensional, representing the coordinates of the rotation of the eye from initial to final position (the quaternion or Q-model). 3. Monkeys made spontaneous saccadic eye movements both in the light and in the dark. They were also rotated about various axes to evoke quick phases of vestibular nystagmus, which have three degrees of freedom. Eye positions were measured in three dimensions with the magnetic search coil technique. 4. While the monkey made spontaneous eye movements, we electrically stimulated the deeper layers of the SC and elicited saccades from a wide range of initial positions. According to the Q-model, the torsional component of eye position after stimulation should be uniquely related to saccade onset position. However, stimulation at 110 sites induced no eye torsion, in line with the prediction of the V-model. 5. Activity of saccade-related burst neurons in the deeper layers of the SC was analyzed during rapid eye movements in three dimensions. No systematic eye-position dependence of the movement fields, as predicted by the Q-model, could be detected for these cells. Instead, the data fitted closely the predictions made by the V-model. 6. In two monkeys, both SC were reversibly inactivated by symmetrical bilateral injections of muscimol. The frequency of spontaneous saccades in the light decreased dramatically. Although the remaining spontaneous saccades were slow, Listing's law was still obeyed, both during fixations and saccadic gaze shifts. In the dark, vestibularly elicited fast phases of nystagmus could still be generated in three dimensions. Although the fastest quick phases of horizontal and vertical nystagmus were slower by about a factor of 1.5, those of torsional quick phases were unaffected. 7. On the basis of the electrical stimulation data and the properties revealed by the movement field analysis, we conclude that the collicular motor map is two dimensional. The reversible inactivation results suggest that the SC is not the site where three-dimensional fast phases of vestibular nystagmus are generated.(ABSTRACT TRUNCATED AT 400 WORDS)

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