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Introductory Lessons in Solid Geometry

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*. ...

##
###
Planet Map Generation by Tetrahedral Subdivision
[chapter]

2010
*
Lecture Notes in Computer Science
*

This ensures planet maps without discontinuities caused by mapping

doi:10.1007/978-3-642-11486-1_26
fatcat:4fwj4lsp7jfxjmrqmye4nzgzpu
*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*. ...##
###
Two tetraedron theorems

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. ...

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

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. ...

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

2018
*
arXiv
*
pre-print

We achieve our solution by means

arXiv:1810.07316v1
fatcat:4rky6mgmhfbrzh4bhem3jpilka
*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*. ...##
###
Simple closed geodesics on regular tetrahedra in spherical space
[article]

2021
*
arXiv
*
pre-print

On

arXiv:2101.07059v2
fatcat:75b4dxw6ozechd35lht5lsxf4y
*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). ...##
###
Structural Principles of Giant Cells
[chapter]

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. ...

##
###
Real-time shell space rendering of volumetric geometry

2006
*
Proceedings of the 4th international conference on Computer graphics and interactive techniques in Australasia and Southeast Asia - GRAPHITE '06
*

To minimize

doi:10.1145/1174429.1174477
dblp:conf/graphite/Ritsche06
fatcat:nzz7l6oke5c7pf3ftpc3q327qi
*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. ...##
###
Three-dimensional optical billiard chaotic scattering

2001
*
Physica D : Non-linear phenomena
*

point is simultaneously on

doi:10.1016/s0167-2789(01)00235-4
fatcat:a5hxpxve7bdhvfosnahwqlbhae
*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*. ...##
###
Area and volume coherence for efficient visualization of 3D scalar functions

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 . ...

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

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 . ...

##
###
Sample-Based Cameras for Feed Forward Reflection Rendering

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. ...

##
###
Efficient Ray Traversal of Constrained Delaunay Tetrahedralization

2017
*
Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications
*

Acceleration structures are mandatory for

doi:10.5220/0006131002360243
dblp:conf/grapp/MariaHA17
fatcat:obuvbxetdzca5nxdzruxeznq4u
*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. ...##
###
Interactive 4-D Visualization of Stereographic Images From the Double Orthogonal Projection
[article]

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*. ...

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

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. ...

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