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Embeddings of Surfaces, Curves, and Moving Points in Euclidean Space

Pankaj K. Agarwal, Sariel Har-Peled, Hai Yu
2013 SIAM journal on computing (Print)  
In this paper we show that dimensionality reduction (i.e., Johnson-Lindenstrauss lemma) preserves not only the distances between static points, but also between moving points, and more generally between  ...  low-dimensional flats, polynomial curves, curves with low winding degree, and polynomial surfaces.  ...  The authors thank Sayan Mukherjee, Piotr Indyk, Assaf Naor, Kasturi Varadarajan, and Yusu Wang for helpful discussions concerning the problems studied in this paper.  ... 
doi:10.1137/110830046 fatcat:i4wws3bdzzg2bpk3jyuwxo3mq4

Embeddings of surfaces, curves, and moving points in euclidean space

Pankaj K. Agarwal, Sariel Har-Peled, Hai Yu
2007 Proceedings of the twenty-third annual symposium on Computational geometry - SCG '07  
In this paper we show that dimensionality reduction (i.e., Johnson-Lindenstrauss lemma) preserves not only the distances between static points, but also between moving points, and more generally between  ...  low-dimensional flats, polynomial curves, curves with low winding degree, and polynomial surfaces.  ...  The authors thank Sayan Mukherjee, Piotr Indyk, Assaf Naor, Kasturi Varadarajan, and Yusu Wang for helpful discussions concerning the problems studied in this paper.  ... 
doi:10.1145/1247069.1247135 dblp:conf/compgeom/AgarwalHY07 fatcat:zxj44rw7uzfpvbd3oag63lltfm

CurvedLand: An Applet for Illustrating Curved Geometry without Embedding [article]

Gary Felder, Stephanie Erickson
2010 arXiv   pre-print
The underlying two-dimensional space has a constant, positive curvature, which causes the apparent paths and shapes of the objects in the map to appear distorted in ways that change as you view them from  ...  We have written a Java applet to illustrate the meaning of curved geometry.  ...  ACKNOWLEDGEMENTS We would like to thank the students who helped contribute to the writing of the program: Lauren Barth-Cohen, Dooshaye Moonshiram, and Nicole Brynes.  ... 
arXiv:1010.1426v1 fatcat:cxvh3coxxbembbwlf6ns3gav4y

Page 1592 of Mathematical Reviews Vol. , Issue 86d [page]

1986 Mathematical Reviews  
A motion in Euclidean 3-space is a Darboux motion if the tra- jectory of any point in the moving space is a plane curve.  ...  Motions in Euclidean kinematics such that the trajectory of any point in the moving space is a plane curve were studied by Dar- boux. The author considers the analogous problem in hyperbolic space.  ... 

Curvature Regularization to Prevent Distortion in Graph Embedding [article]

Hongbin Pei, Bingzhe Wei, Kevin Chen-Chuan Chang, Chunxu Zhang, Bo Yang
2020 arXiv   pre-print
ambient embedding Euclidean space, and hence to detect them becomes difficult for machine learning models.  ...  Recent research on graph embedding has achieved success in various applications. Most graph embedding methods preserve the proximity in a graph into a manifold in an embedding space.  ...  Recall that curve curvature is modeled as the rate of change of direction of tangent at a point that moves on the curve.  ... 
arXiv:2011.14211v1 fatcat:ng7ixm3zdvcyvl3qecm435nxa4

Page 4756 of Mathematical Reviews Vol. , Issue 95h [page]

1995 Mathematical Reviews  
and related questions, projective arc length and projective curvature); (3) Applications of the projective theory of curves (moving frame, rational curves, ruled surfaces); (4) Surfaces in space (type  ...  of conjugate nets and congruences, La- place sequences, conjugate normal congruences and connection with theory of surfaces in Euclidean space); and (6) Perspectives (tangency invariants for curves, projective  ... 

Page 6460 of Mathematical Reviews Vol. , Issue 88m [page]

1988 Mathematical Reviews  
By means of the moving frame for a point x € V, (p vectors of this frame lie in the tangent space 7, of V, in x), the author constructs for each direction in 7,, a normal direction in the space completely  ...  Let V, be a p-dimensional surface in Euclidean n-space E.,.  ... 

Differential Geometry and Relativity Theory: An Introduction

Richard L. Faber, Gregory Naber
1986 American Journal of Physics  
A parametric surface is the image of an open subset of the Euclidean plane by a continuous function, in a topological space, generally a Euclidean space of dimension at least three.  ...  space such that every point has a neighborhood which is homeomorphic to an open subset of the Euclidean plane.  ...  Of course, all these surfaces can be thought of as embedded in Euclidean space E 3 .  ... 
doi:10.1119/1.14853 fatcat:nyqiwkbinzbvhpkwind43h6nye

Page 2066 of Mathematical Reviews Vol. , Issue 93d [page]

1993 Mathematical Reviews  
be a regular /-dimensional submanifold in an n-dimensional Euclidean space E”. Taking the tangent subspace To F ! at a point Q € F; and moving this subspace to the origin, one can consider TgF!  ...  Summary: “Let ® be a nonplanar minimal surface in Euclidean space E>.  ... 

Sector models—A toolkit for teaching general relativity: I. Curved spaces and spacetimes

C Zahn, U Kraus
2014 European journal of physics  
Sector models describe curved spaces the Regge calculus way by subdivision into blocks with euclidean geometry. This procedure is similar to the approximation of a curved surface by flat triangles.  ...  We outline a workshop for high school and undergraduate students that introduces the notion of curved space by means of sector models of black holes.  ...  Gaps appear whenever the pieces of a curved surface are laid out in a plane or the blocks of a curved space are assembled in euclidean space.  ... 
doi:10.1088/0143-0807/35/5/055020 fatcat:iwfykq4hb5ejhgxu4r6hwyetbq

Embedding Spacetime via a Geodesically Equivalent Metric of Euclidean Signature

Rickard Jonsson
2001 General Relativity and Gravitation  
This geodesically equivalent, or dual, metric can be embedded in ordinary Euclidean space. On the embedded surface freely falling particles move on the shortest path.  ...  Thus one can visualize how acceleration in a gravitational field is explained by particles moving freely in a curved spacetime.  ...  When we create the embedding diagram for the space of a symmetry plane through a star, we make a mapping from our spatial plane onto a curved surface embedded in Euclidean space.  ... 
doi:10.1023/a:1012037418513 fatcat:6m3iv4wxvfgxveyiqzuvxzbgqa

Delaunay triangulations and Voronoi diagrams for Riemannian manifolds

Greg Leibon, David Letscher
2000 Proceedings of the sixteenth annual symposium on Computational geometry - SCG '00  
For a sufficiently dense set of points in any closed Riemannian manifold, we prove that a unique Delannay triangulation exists. This triangulation has the same properties as in Euclidean space.  ...  ACKNOWLEDGEMENTS We would like to thank Herbert Edelsbrunner for helpful discussions about the algorithms, Hyam Rubinstein for his suggestions for dealing with normal submanifolds of Euclidean space and  ...  SUBMANIFOLDS OF EUCLIDEAN SPACE Any manifold embedded in Euclidean space inherits a Riemannian structure where the lengths of curves on the manifold are the Euclidean lengths of the curves.  ... 
doi:10.1145/336154.336221 dblp:conf/compgeom/LeibonL00 fatcat:7z4taevduzehthikoufhif3vo4

Embedding the Reissner–Nordström spacetime in Euclidean and Minkowski spaces

Uri Jacob, Tsvi Piran
2006 Classical and quantum gravity  
These embedding diagrams serve as useful tools to visualize the geometry of the hypersurfaces and of the whole spacetime in general.  ...  We examine embedding diagrams of hypersurfaces in the Reissner-Nordstrom black hole spacetime.  ...  This research was supported by the ISF center of excellence for High Energy Astrophysics and by the Schwartzmann University Chair (TP).  ... 
doi:10.1088/0264-9381/23/12/003 fatcat:mwrymfzqmbfcbi2dghtko7xugu

Weighted averages on surfaces

Daniele Panozzo, Ilya Baran, Olga Diamanti, Olga Sorkine-Hornung
2013 ACM Transactions on Graphics  
Abstract We consider the problem of generalizing affine combinations in Euclidean spaces to triangle meshes: computing weighted averages of points on surfaces.  ...  We demonstrate that anchor points in the above applications can be added/removed and moved around on the meshes at interactive framerates, giving the user an immediate result as feedback.  ...  This work was supported in part by the ERC grant iModel (StG-2012-306877), by an SNF award 200021 137879 and a gift from Adobe Research.  ... 
doi:10.1145/2461912.2461935 fatcat:f3vy7xtgsvhhxkjkh63jd7ubga

A Riemannian Self-Organizing Map [chapter]

Dongjun Yu, Edwin R. Hancock, William A. P. Smith
2009 Lecture Notes in Computer Science  
We generalize the classic self-organizing map (SOM) in flat Euclidean space (linear manifold) onto a Riemannian manifold.  ...  We here compared the performance of the generalized SOM and the classical SOM using real 3-Dimensional facial surface normals data.  ...  Under the embedding Φ, points on M can be depicted by corresponding vectors in the Euclidean space R d .  ... 
doi:10.1007/978-3-642-04146-4_26 fatcat:6zqqgclmi5bjnjonybnoqmoaem
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