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Topic: Barycentric subdivision

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	Springer Online Reference Works
		Barycentric coordinates were introduced by A.F. Möbius in 1827, [1], as an answer to the question about the masses to be placed at the vertices of a triangle so that a given point is the centre of gravity of these masses.
		Barycentric coordinates of a simplex are used in algebraic topology [2].
		Barycentric coordinates are used to construct the barycentric subdivision of a complex.
	eom.springer.de /B/b015280.htm (267 words)

	Barycentric Subdivision (Site not responding. Last check: 2007-10-15)
		Apply the linear map in the forward direction, which carries lines to lines, and the center is indeed coliniear with the vertex and the center of the opposite face.
		The midpoint of a segment cuts the segment in half, implementing a barycentric subdivision in 1 dimension.
		Subdivision cuts a 1-simplex in half, and the two smaller simplexes have half the diameter.
	www.mathreference.com /top-sx,subdiv.html (592 words)

	Barycentric subdivision - Wikipedia, the free encyclopedia
		In geometry, the barycentric subdivision is a standard way of dividing an arbitrary convex polygon into triangles, a convex polyhedron into tetrahedra, or, in general, a convex polytope into simplices with the same dimension, by connecting the barycenters of their faces in a specific way.
		The barycentric subdivision is chiefly used to replace an arbitrarily complicated convex polytope or topological cell complex by an assemblage of pieces, all of them of bounded complexity (simplices, in fact).
		Sometimes the term "barycentric subdivision" is improperly used for any subdivision of a polytope P into simplices that have one vertex at the centroid of P, and the opposite facet on the boundary of P.
	en.wikipedia.org /wiki/Barycentric_subdivision (992 words)

	Building Your Own Subdivision Surfaces
		In recent years, subdivision surfaces have received a lot of attention from both academics and industry professionals, people in the movie industry even apply subdivision techniques to create complex characters and produce highly detailed, smooth animation.
		Secondly, subdivision surfaces are constructed easily through recursive splitting and averaging: splitting involves creating four new faces by removing one old face, averaging involves taking a weighted average of neighboring vertices for the new vertices.
		The (a, b, g) tuple is the barycentric coordinates of v1 within the triangle T. To perform this association for each simplification step, flatten the 1-ring (as shown in Figure 5) of v1 onto a 2D plane.
	www.gamasutra.com /features/20000908/lee_pfv.htm (2543 words)

	Wavelet-based data compression - Patent 6144773
		In certain subdivision schemes (for example, subdivision using triangles) it may be desirable to associate the wavelet coefficients not with the face of a given unit but with its vertices.
		Each of the K-cells in the subdivision have K+1 0-cells on their boundary and is a K-dimensional generalization of the triangle (called a simplex, a 2-simplex is a triangle, a 3-simplex is a pyramid).
		Note that the barycentric subdivision of a triangle has 13 cells (the central 0-cell, six 1-cells from the six vertices of the subdivided boundary, and six 2-cells from the six edges of the subdivided boundary).
	www.freepatentsonline.com /6144773.html (15740 words)

	Singular Subdivision
		The barycentric subdivision of a geometric simplex can be generalized to a singular simplex v in the topological space s.
		This is not surprising, since the subdivision was built by adding w to the subdivision of the boundary.
		The subdivision operator, denoted subd(), is defined on the simplexes of s, which are the generators of c.
	www.mathreference.com /at-sh,subdiv.html (1195 words)

	Research
		Abstract: In this paper we present a subdivision scheme for mixed triangle/quad meshes that is C^2 everywhere except for isolated, extraordinary points where the surface is C^1.
		Abstract: We present a new non-stationary, interpolatory subdivision scheme capable of producing circles and surfaces of revolution and in the limit is C^1.
		This subdivision method is then extended to surfaces by generalizing each of these passes to quadrilateral meshes (including those with extraordinary vertices).
	www.cs.rice.edu /~sschaefe/research (3009 words)

	Citations: Subdivision algorithms for B'ezier triangles - Goldman (ResearchIndex) (Site not responding. Last check: 2007-10-15)
		For a polynomial p n with the given B net fa ij g on a triangle T as shown in the lower right triangle of the left one of Figure 6, one subdivision finds its B net on each of four subtriangles as shown on the middle one of Figure....
		General patch subdivision splits a patch into several patches that together have the same shape as the original one.
		Internal energy The internal energy of a surface S is the part of the total energy (1) that depends only on properties of the surface itself.
	citeseer.ist.psu.edu /context/457687/0 (754 words)

	Mesh compression/reparameterization via subdivision surfaces (Site not responding. Last check: 2007-10-15)
		Subdivision surfaces offer a very compact means for representing surfaces.
		We now have, for the “big” faces in the level 0 subdivision mesh, where each point in the face goes on the final mesh.
		At the initial levels we want to move the vertices so that the overall approximation error between the final subdivision surface and the original mesh is minimized.
	cec.wustl.edu /~cse552/Notes/MeshCompression.htm (693 words)

	[No title] (Site not responding. Last check: 2007-10-15)
		It was the following: 1) The barycentric subdivision of a semisimplicial CW complex is a regular semisimplicial CW complex.
		2) The barycentric subdivision of a regular semisimplicial CW complex is a (geometrical) simplicial complex.
		It was asserted to me by Barratt that the >> second barycentric subdivision of the realization of a simplicial >> set is a simplicial complex.
	www.lehigh.edu /~dmd1/bg13.txt (727 words)

	Subdivision.org: Re: Barycentric subdivision of a cube (Site not responding. Last check: 2007-10-15)
		In Reply to: Barycentric subdivision of a cube posted by vishal on May 06, 2002 at 11:12:53:
		We perform one round of linear subdivision on a quad mesh.
		Then, we perform one round of "centroid averaging." This means that we average all of the centroids of the quads adjacent to a vertex to get that vertex's new position.
	www.subdivision.org /wwwboard/messages/72.html (269 words)

	Amazon.com: "first barycentric subdivision": Key Phrase page (Site not responding. Last check: 2007-10-15)
		Show that the second barycentric subdivision of a A-complex is a simplicial complex.
		Namely, show that the first barycentric subdivision produces a A-complex with the property that each simplex has all its vertices distinct,...
		The first barycentric subdivision of a simplex a is the simplicial complex having all the barycenters of a as vertices.
	www.amazon.com /phrase/first-barycentric-subdivision (483 words)

	[No title] (Site not responding. Last check: 2007-10-15)
		If the simplicial set $X$ is the 2-simplex $\Delta2$ with its boundary $\partial \Delta2$ collapsed to a point, then the second Kan/normal/barycentric subdivision $Z = Sd2(X)$ still has non-degenerate 2-simplices that are not embedded, i.e., $Z$ is not a simplicial complex in the obvious way.
		Here I am assuming that by "barycentric subdivision" you mean the endofunctor of simplicial sets that is left adjoint to Kan's $Ex$.
		This was asserted in a 1956 Princeton preprint of Barratt's, titled "Simplical and semisimplicial complexes", by an argument involving two subdivisions, and some more.
	www.lehigh.edu /~dmd1/jr1229.txt (451 words)

	[No title]
		For a regular cell complex B, the simplicial complex given by F(B) is called the barycentric subdivision of B and there is a homeomor- phism B ~=kF(B)k under which every closed cell oe of B maps homeomorphi- cally to the subcomplex kF(B)oe k.
		In other words a combinatorial vector bundle (B; M) is a matroid bundle (F(B0); M) where F(B0) is the poset of cells of a PL subdivision B0 of B. Note every regular cell complex can be given the structure of a PL space via a barycentric subdivision.
		Because Gi-1 is a subcomplex of the triangulation Ti, because we have taken a barycentric subdivision, because is upper semi-continuous, and because Ti refines the stratification, (wj) (vk) for all j and k.
	www.math.purdue.edu /research/atopology/Anderson-DavisJ/MacPherson.txt (10614 words)

	Coloring booklets from finite subdivision rules
		Each booklet is based on a single finite subdivision rule, which is basically a combinatorial rule for subdividing a finite number of tiles, which are called tile types, into pieces which can be identified with tile types.
		The pentagonal subdivision rule is a good example of a conformal finite subdivision rule.
		The Lattes subdivision rule has two tile types, A and B. A and B are each a square, and each is subdivided into four subsquares.
	www.math.vt.edu /people/floyd/coloring (449 words)

	PIC 10B SA 1
		Often the subdivision must be adaptive, with the triangles in one region being smaller than those in another region.
		Each of the three "corner" triangles (i.e., the three triangles not in the middle) can then be considered separately and is further subdivided unless some criterion is satisfied.
		Your task is to use barycentric subdivision to create an applet that takes the coordinates of the three corners of a triangle as parameters and creates the following sort of pattern:
	www.math.ucla.edu /~nathan/20a.1.06s/la/la4.html (829 words)

	MCell Home : Documentation : Workshops : 1999
		Diffusing ligand molecules within each spatial subdivision only have to search for potential intersections with the surface elements that pass through the corresponding subdivision.
		If a ligand's trajectory carries it through a spatial partition, it is raymarched into the adjacent spatial subdivision and the process continues as necessary.
		The collision detection and raymarching computation within each spatial subdivision is faster than published methods.
	www.mcell.cnl.salk.edu /Documentation/Workshops/1999/raytrace.html (307 words)

	Amazon.com: "barycentric subdivision": Key Phrase page (Site not responding. Last check: 2007-10-15)
		A new complex K' called the barycentric subdivision of K is formed by introducing a new vertex at the center of each triangle and a new vertex...
		The relation between K and its barycentric subdivision K' is nonetheless rather unwieldy,...
		In the next section, the notion of 'barycentric subdivision' will be discussed; one of its crucial properties comes out of the following fact.
	www.amazon.com /phrase/barycentric-subdivision (466 words)

	Jerusalem Mathematics Colloquium
		Abstract: A large triangle can quickly be cut into small triangles by barycentric subdivision; this subdivision process, carried out an arbitrary finite number of times, is central in geometry and topology (homology and cohomology theory, simplicial complexes, classification of manifolds, etc.).
		We study more general subdivision rules arising in the differential geometric study of 3-manifolds.
		We are led to ask, "What happens asymptotically when a subdivision rule is applied infinitely often?
	www.ma.huji.ac.il /~colloq/1997-98/col10.html (140 words)

	[No title]
		The key to proving excision in turn is Proposition 2.21, which says roughly that homology does not change when it is made to be subordinate to an open cover.
		The key to this proposition is barycentric subdivision.
		John, Dawn, Samson, Aaron: Cover section (1) of the proof, on barycentric subdivision of simplices.
	noether.uoregon.edu /~dps/635 (1483 words)

	Springer Online Reference Works
		» Encyclopaedia of Mathematics »; B » Barycentric subdivision
		with a common vertex that is the barycentre of the simplex
		The formal definition of a barycentric subdivision for the case of an abstract complex is analogous.
	eom.springer.de /b/b015290.htm (108 words)

	8.5.2.1 Using interpolation for continuous state spaces
		Figure: Barycentric subdivision can be used to partition each cube into simplexes, which allows interpolation to be performed in
		If barycentric subdivision is used to decompose the cube using the midpoints of all faces, then the point-location problem can be solved in
		Examples of this decomposition are shown for two and three dimensions in Figure 8.20.
	msl.cs.uiuc.edu /planning/node407.html (484 words)

	[No title]
		This was a project for my CS 497 course.
		The screenshot shows a cube, and the "marshmallow" that you can create by using "creases" of finite sharpness, as explained in "Subdivision Surfaces for Character Animation" by Tony DeRose et al.
		This is an implementation of the fault formation algorithm for creating terrain heightfields, described by Jason Shankel in Game Programming Gems, volume I. Source not available.
	www.rajsharma.net (2084 words)

	The Associahedron and Little k-Cubes Operads
		If K_n is the associahedron governing ways to multiply n things, K_n is (n-2) dimensional.
		To do this right, it's best to apply the bar construction to the operad for semigroups, getting a barycentric subdivision of the associahedra which is a simplicial operad.
		But, we need to explain the bar construction before doing this; there may be a more lowbrow explanation.
	math.ucr.edu /home/baez/hda/associahedron.html (251 words)

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