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Introductory Lessons in Solid Geometry

Peter Ramsay
1912 Mathematical Notes  
A sphere can be described which will pass through the four vertices of the tetrahedron.  ...  PETER RAMSAY Geometrical Construction for Refracted Ray.- Fio.2 The following variation seems an improvement on the usual method of tracing a refracted ray.  ... 
doi:10.1017/s1757748900000955 fatcat:n72bk3y5ene7zcqsi5hids7gta

Planet Map Generation by Tetrahedral Subdivision [chapter]

Torben Ægidius Mogensen
2010 Lecture Notes in Computer Science  
This ensures planet maps without discontinuities caused by mapping a flat map onto a sphere. We compare the method to other map-generator algorithms.  ...  We present a method for generating pseudo-random, zoomable planet maps for games and art. The method is based on spatial subdivision using tetrahedrons.  ...  -A normal vector of the landscape at a point on the sphere surface can be estimated from the altitude values and coordinates of the four vertices in the final tetrahedron.  ... 
doi:10.1007/978-3-642-11486-1_26 fatcat:4fwj4lsp7jfxjmrqmye4nzgzpu

Two tetraedron theorems

Henry S. White
1908 Bulletin of the American Mathematical Society  
The set of six contacts of six edges of a tetrahedron with a sphere divides naturally into threes in four ways ; for example, 12, 13, 14; 23, 34, 42, three in one face and three on edges having the opposite  ...  We wish to show, for example, that three angles at A x made by rays A X B, A X C, and A X D are equal respectively to three angles formed at B x by rays drawn to A, D, and C.  ... 
doi:10.1090/s0002-9904-1908-01598-2 fatcat:skq5njzy2bhazjknkb2vvqmsh4

Manifold surface reconstruction of an environment from sparse Structure-from-Motion data

Maxime Lhuillier, Shuda Yu
2013 Computer Vision and Image Understanding  
The previous attempts to solve this problem are currently very limited: the surface is non-manifold or has zero genus, the experiments are done on small scenes or objects using a few dozens of images.  ...  From the computational standpoint, it would be interesting to avoid dense stereo and to generate a surface directly from the sparse cloud of 3D points and their visibility information provided by Structure-from-Motion  ...  Thanks to the CRISTAL project for providing the ground truth VRML model used in the quantitative evaluation.  ... 
doi:10.1016/j.cviu.2013.08.002 fatcat:7fplvyfrgjhxxccyg26djkfkha

Tetrahedra and Relative Directions in Space, Using 2 and 3-Space Simplexes for 3-Space Localization [article]

Kevin Acres, Jan Carlo Barca
2018 arXiv   pre-print
We achieve our solution by means of a tetrahedral phased array of receivers at which we measure the phase difference, or time difference of arrival, of the signal.  ...  This is an important, currently open, problem in a number of areas, particularly in the field of swarm robotics, where rapid updates of positional information are of great importance.  ...  The following image shows the coverage of a sphere by four ±35.5° spherical segments applicable to each vertex of a regular tetrahedron.  ... 
arXiv:1810.07316v1 fatcat:4rky6mgmhfbrzh4bhem3jpilka

Simple closed geodesics on regular tetrahedra in spherical space [article]

Alexander A. Borisenko, Darya D. Sukhorebska
2021 arXiv   pre-print
On a regular tetrahedron in spherical space there exist the finite number of simple closed geodesics.  ...  with the faces angle α∈( π/3, α_1 ) there exists unique, up to the rigid motion of the tetrahedron, simple closed geodesic of type (p,q), and on a regular tetrahedron with the faces angle α∈( α_2, 2π/  ...  So if ε satisfies the condition (8.46), then on a regular tetrahedron with the faces angle α = π/3 + ε in a spherical space there exist three different simple closed geodesics of type (p, q).  ... 
arXiv:2101.07059v2 fatcat:75b4dxw6ozechd35lht5lsxf4y

Structural Principles of Giant Cells [chapter]

Sten Samson
1969 Developments in the Structural Chemistry of Alloy Phases  
The four complexes are at the vertices of a regular tetrahedron. Some of the coordination shells of ligancy 13 may enclose magnesium, and others aluminum.  ...  (say, positive) tetrahedron bounded by four hexagons and four triangles (twelve vertices); (2) the regular (negative) tetrahedron (four vertices, representing four large atoms).  ...  ACKNOWLEDGMENT Acknowledgment is made to the National Science Foundation for its financial support (Grants GP-1701 and GP-4237), which enabled me to carry out the investigations described here.  ... 
doi:10.1007/978-1-4899-5564-7_4 fatcat:urdsaey24nhp3c5rnrh5en55k4

Real-time shell space rendering of volumetric geometry

Nico Ritsche
2006 Proceedings of the 4th international conference on Computer graphics and interactive techniques in Australasia and Southeast Asia - GRAPHITE '06  
To minimize the number of ray tracing steps, a 3D distance map is used for skipping empty space in the shell volume. The shell volume is defined by a pre-computed tetrahedra mesh.  ...  The shell space is a layer of variable thickness on top or below the polygonal mesh. The technique computes view ray shell geometry intersections in a pixel shader.  ...  Right: Ray tracing using a 3D distance map. The spheres visualize the maximum distances found in the distance map at the sampling positions.  ... 
doi:10.1145/1174429.1174477 dblp:conf/graphite/Ritsche06 fatcat:nzz7l6oke5c7pf3ftpc3q327qi

Three-dimensional optical billiard chaotic scattering

David Sweet, Benjamin W. Zeff, Edward Ott, Daniel P. Lathrop
2001 Physica D : Non-linear phenomena  
point is simultaneously on the boundary of four distinct basins.  ...  We also find that depending on the configuration of the spheres, the boundary can be either a nowhere-differentiable surface or else can possess the topological Wada property whereby any basin boundary  ...  Acknowledgements This work was supported by the Office of Naval Research (Physics) and NSF DMR-9896037. Daniel P. Lathrop is a Cottrell Scholar of the Research Corporation. We thank James A.  ... 
doi:10.1016/s0167-2789(01)00235-4 fatcat:a5hxpxve7bdhvfosnahwqlbhae

Area and volume coherence for efficient visualization of 3D scalar functions

Nelson Max, Pat Hanrahan, Roger Crawfis
1990 Proceedings of the 1990 workshop on Volume visualization - VVS '90  
Ray tracing can also be used to render densities in irregular volumes, but can be expensive, and does not take advantage of coherence within the volumes.  ...  The scalar function is interpolated between values defined at the vertices, and the polyhedra are sorted in depth before compositing.  ...  Support for this research came from the Institution Research and Development funds at Livermore, and from NFS grant No. DMS-861562-01 at the University of Minnesota, Geometry Supercomputer Project .  ... 
doi:10.1145/99307.99315 fatcat:dqp35jpyazajdggdql24wq5br4

Area and volume coherence for efficient visualization of 3D scalar functions

Nelson Max, Pat Hanrahan, Roger Crawfis
1990 Computer graphics  
Ray tracing can also be used to render densities in irregular volumes, but can be expensive, and does not take advantage of coherence within the volumes.  ...  The scalar function is interpolated between values defined at the vertices, and the polyhedra are sorted in depth before compositing.  ...  Support for this research came from the Institution Research and Development funds at Livermore, and from NFS grant No. DMS-861562-01 at the University of Minnesota, Geometry Supercomputer Project .  ... 
doi:10.1145/99308.99315 fatcat:3kre3hsqgfcvlidnuzehzdoxnq

Sample-Based Cameras for Feed Forward Reflection Rendering

V. Popescu, E. Sacks, C. Mei
2006 IEEE Transactions on Visualization and Computer Graphics  
The main techniques for rendering reflections are ray tracing and environment mapping. Ray tracing searches for the scene point seen at each pixel. Purdue University. West Fig. 2.  ...  Although accurate and general, ray tracing is too slow for interactive graphics because of the inefficient search for the inverse mapping.  ...  Reflections are rendered on each reflector using the four SBCs at the vertices of the tetrahedron that contains the current viewpoint.  ... 
doi:10.1109/tvcg.2006.103 pmid:17073380 fatcat:2yrimx6ehnh6zcexlqw4snr7q4

Efficient Ray Traversal of Constrained Delaunay Tetrahedralization

Maxime Maria, Sébastien Horna, Lilian Aveneau
2017 Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications  
Acceleration structures are mandatory for ray-tracing applications, allowing to cast a large number of rays per second.  ...  This new algorithm is more efficient than the previous ones: it uses less arithmetic operations; it does not add extra thread divergence since it uses a fixed number of operation; at last, it is robust  ...  by four vertices V ⊂ X, is a Delaunay tetrahedron if it exists a circumscribed sphere S of T such as no point of X \ {V } is inside S.  ... 
doi:10.5220/0006131002360243 dblp:conf/grapp/MariaHA17 fatcat:obuvbxetdzca5nxdzruxeznq4u

Interactive 4-D Visualization of Stereographic Images From the Double Orthogonal Projection [article]

Michal Zamboj
2020 arXiv   pre-print
The double orthogonal projection of the 4-space onto two mutually perpendicular 3-spaces is a method of visualization of four-dimensional objects in a three-dimensional space.  ...  Described are synthetic constructions of stereographic images of a point, hyperspherical tetrahedron, and 2-sphere on a 3-sphere from their double orthogonal projections.  ...  In this case, we need to find the intersection A of the projecting ray N A s with the 3-sphere.  ... 
arXiv:2007.13470v1 fatcat:4zie3pjf6fbitcjbjpf6pdr4jq

Incremental Solid Modeling from Sparse and Omnidirectional Structure-from-Motion Data

Vadim Litvinov, Maxime Lhuillier
2013 Procedings of the British Machine Vision Conference 2013  
The quality and execution time of the proposed algorithm is evaluated on a large scale (2.5 km.) real sequence taken in an urban environment, and the method is quantitatively evaluated on a synthetic urban  ...  Furthermore, our method is incremental: the surface is updated for every new camera pose computed by SfM, and the update occurs in a small neighborhood of the new camera pose.  ...  Thanks to the CRISTAL project for providing the ground truth VRML model used in the quantitative evaluation.  ... 
doi:10.5244/c.27.61 dblp:conf/bmvc/LitvinovL13 fatcat:runrf6uvtfe3ndz3bci43xr7au
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