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Topic: Barycentric coordinates

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	Springer Online Reference Works (Site not responding. Last check: 2007-10-20)
		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 coordinates and mesh deformation
		In a triangle, such coordinates are unique, and b_i has the geometric interpretation as the ratio of the area of the triangle formed by x and the opposite edge to vertex i over the area of the original triangle.
		Barycentric coordinates have many uses in applications such as shading, parameterization and deformation.
		Since barycentric coordinates in a general shape is not unique, it is of great interest to have a general method of building possible barycentric coordinates in the hope that we could find new and meaningful coordinates.
	www.cs.wustl.edu /~taoju/research/bary.htm (523 words)

	Barycentric coordinates
		Thus barycentric coordinates are a form of general homogeneous coordinates that are used in many branches of mathematics (and even computer graphics).
		Barycentric coordinates arise naturally whenever variable quantities have a constant sum.
		Three glass problem, where we are pouring water from one glass to another under the unrealistic assumption that in the process no drop of water is going to be spilled, is a salient example.
	www.cut-the-knot.org /triangle/barycenter.shtml (410 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-20)
		The original interpretation of the point P given by barycentric coordinates p:q:r is that P is the center of mass ("barycenter") of weights p, q and r when these are placed on the vertices A, B and C. Another interpretation is that the ratio of the areas BCP:ACP:ABP is p:q:r.
		For this reason barycentric coordinates are said to be "homogeneous." Another example of homogeneous coordinates with respect to a triangle is the notion of trilinear coordinates.
		Barycentrics of triangle centers are usually given as functions of sidelengths a, b and c and/or angles A, B and C. As an example, the incenter of a triangle ABC always has barycentrics a:b:c.
	mathforum.org /library/drmath/view/51771.html (579 words)

	DevMaster.net - Raytracing: Theory & Implementation Part 7, Kd-Trees and More Speed
		Barycentric coordinates were invented by mister Möbius (the guy that invented the famous one-sided belt).
		In this formula, P1, P2 and P3 are the coordinates of the vertices of the triangle, and a1, a2, a3 are the barycentric coordinates of a point in the triangle.
		So, if we know the barycentric coordinates of the hit point, we can accept or reject the ray with very simple tests: Point P is outside the triangle if one of the barycentric coordinates a is smaller than zero, or the summed coordinates are greater than 1.
	www.devmaster.net /articles/raytracing_series/part7.php (4037 words)

	Subject: Re: Coordinate systems
		Sometimes the name "simplicial" coordinates has been used for this, so that using n+1 coordinates for an n-space with the condition that their sum is zero would be "normalized simplicial coordinates", from which you get the unnormalized ones by letting (x+k,y+k,...) represent the same point as (x,y,...).
		Of course pentahedral coordinates would be the natural choice to use for a problem that involved 5 particular planes.
		Confocal coordinates are so called because their level surfaces are a confocal system of quadrics (or conics in 2 dimensions, where they are also called "elliptic coordinates").
	www.flowresearch.com /cs2.htm (630 words)

	On-Line Computer Graphics Notes
		These coordinates system are frequently quite useful, and the interested student will notice that they are used extensively in working with triangles.
		There is a natural way to convert the local coordinates of a frame to barycentric coordinates for a certain set of points.
		Barycentric coordinates are another important method of introducing coordinates into an affine space.
	graphics.idav.ucdavis.edu /education/GraphicsNotes/Barycentric-Coordinates/Barycentric-Coordinates.html (365 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		You can compute barycentric coordinates for points in this span which are outside the face; one or more of them will lie outside the interval [0,1].
		But note that barycentric coordinates are not particularly well suited to distance computations (unless you incorporate the distances between the vertices into your calculations).
		Thus, >each barycentric coordinate vector in K would be composed of 4 elements with >at most 3 of those elements non-zero.
	www.math.niu.edu /~rusin/known-math/99/barycentric (693 words)

	[No title]
		\medskip Prove that the equation of a line in barycentric coordinates is of the form $$ux + vy + wz = 0,$$ where $u\not= v$, or $v\not= w$, or $u\not= w$.
		In either case, let $(m, m', m'')$ be the barycentric coordinates of $M$, as explained at the beginning of the problem.
		In the affine plane $\affreal^2$, a {\it conic\/} is the set of points of coordinates $(x, y)$ such that \[ \alpha x^2 + \beta y^2 + 2\gamma xy + 2\delta x + 2\lambda y + \mu = 0, \] where $\alpha\not= 0$ or $\beta\not= 0$ or $\gamma\not= 0$.
	www.cis.upenn.edu /~cis610/cis61006hw1 (2329 words)

	Barycentric coordinates (mathematics) - Wikipedia, the free encyclopedia
		Barycentric coordinates are a form of homogeneous coordinates.
		Barycentric or area coordinates are extremely useful in engineering applications involving triangular subdomains.
		Indeed, area coordinates will also allow us to perform a linear interpolation at all points in the triangle if the values of the function is known at the vertices.
	en.wikipedia.org /wiki/Barycentric_coordinates_(mathematics) (316 words)

	Glish Solar System Client
		The new coordinates are effective until the next set_observer_coord event is issued or until the client is terminated.
		The observer's coordinates on the earth, in the IERS coordinate system, must first be set with the set_observer_coord function.
		A successive series of coordinates may be obtained by specifying the number of points requested and the time interval between the points.
	www.cv.nrao.edu /~rfisher/Glish/solar_system.html (1298 words)

	Barycentric Coordinates -- from Wolfram MathWorld
		Barycentric coordinates normalized so that they become the actual areas of the subtriangles are called homogeneous barycentric coordinates.
		so that the coordinates give the areas of the subtriangles normalized by the area of the original triangle are called areal coordinates (Coxeter 1969, p.
		In barycentric coordinates, a line has a linear homogeneous equation.
	mathworld.wolfram.com /BarycentricCoordinates.html (260 words)

	Definition of a Simplex
		Restrict attention to that region of the plane where the coordinates are nonnegative.
		Therefore the interior of the simplex, defined by positive barycentric coordinates, is, well, the inside of the simplex.
		Let x and y belong to a simplex, and pull back to the barycentric coordinates, which are all nonnegative.
	www.mathreference.com /top-sx,def.html (1215 words)

	Trilinear Coordinates -- from Wolfram MathWorld
		The homogeneous barycentric coordinates corresponding to trilinear coordinates
		, and the trilinear coordinates corresponding to homogeneous barycentric coordinates
		In trilinear coordinates, the coordinates of the vertices are 1:0:0 (
	mathworld.wolfram.com /TrilinearCoordinates.html (435 words)

	Barycentric Mapping
		Barycentric opacity functions use this barycentric space of anisotropy as their domain, assigning an opacity between
		Because of its expressive power, the barycentric space also makes sense as the domain of the color function, which assigns color to each sample point in the volume rendering according to its anisotropy.
		Volume renderings made with both barycentric opacity and color maps allow an extra dimension of information about the diffusion tensor to be represented in the volume rendering.
	www.cs.utah.edu /~gk/papers/tvcg00/node4.html (718 words)

	The Barycentric Algorithm of Schnyder
		The vertex positions are defined using a barycentric representation of the input graph G.
		A barycentric representation of the input graph is computed by first constructing a normal labeling of the angles of the faces of the input graph.
		These 3 paths leaving each interior vertex define 3 regions of the graph and the number of faces in each of these regions are the 3 barycentric coordinates of the vertex.
	www.ads.tuwien.ac.at /people/Weiskircher/gigd/node15.html (439 words)

	flipcode - Raytracing Topics & Techniques (Site not responding. Last check: 2007-10-20)
		An efficient way to calculate the intersection point of a ray and a triangle is to use barycentric coordinates.
		P = a1 P1 + a2 P2 In this formula, a1 and a2 are the barycentric coordinates of point P with respect to the end points P1 and P2.
		Using the above formulas to calculate the barycentric coordinates for the projected intersection point P the intersection code is now rather simple.
	www.flipcode.com /articles/article_raytrace07.shtml (4102 words)

	Barycentric coordinates, three jugs application
		The grid on the right represents points with all three coordinates non-negative integers.
		For example, the apex A is referred to by its coordinate string 800 which is just a shorthand for
		To account for limited capacities of the given vessels, note that under no circumstances it is possible to arrive at the water distribution represented by, say, 134.
	www.cut-the-knot.org /triangle/glasses.shtml (424 words)

	Homogeneous coordinates - Wikipedia, the free encyclopedia
		The homogeneous coordinates of a point of projective space of dimension n are usually written as (x : y : z :... : w), a row vector of length n + 1, other than (0 : 0 : 0 :... : 0).
		Therefore this system of coordinates can be explained as follows: if the projective space is constructed from a vector space V of dimension n + 1, introduce coordinates in V by choosing a basis, and use these in P(V), the equivalence classes of proportional non-zero vectors in V.
		Homogeneous coordinates are frequently used in computer graphics as they allow all affine transformations to be represented by a matrix operation.
	en.wikipedia.org /wiki/Homogeneous_coordinates (985 words)

	D3DXVec4BaryCentric (Site not responding. Last check: 2007-10-20)
		Returns a point in Barycentric coordinates, using the specified 4D vectors.
		Any point in the plane V1V2V3 can be represented by the Barycentric coordinate (f,g).The parameter f controls how much V2 gets weighted into the result and the parameter g controls how much V3 gets weighted into the result.
		Barycentric coordinates define a point inside a triangle in terms of the triangle's vertices.
	msdn.microsoft.com /library/en-us/directx9_c/D3DXVec4BaryCentric.asp?frame=true (338 words)

	Ray-triangle intersection - DmWiki
		As long as the barycentric coordinates u, v and w sum up to one, the point p is lying on the triangle.
		Before we compute the actual barycentric coordinates of the ray-plane intersection point i'd like to do a few substituions.
		term that is in the distance to plane equation, the dominat axis of the normal and the 6 factors used in the equations to compute the barycentric coordinates of the ray hitpoint.
	www.devmaster.net /wiki/Ray-triangle_intersection (612 words)

	Barycentric Coordinates (Site not responding. Last check: 2007-10-20)
		We will speak of barycentric coordinates (bcc) but suppress the detail of whether we carry n or n+1 numbers as a tactical detail that detracts from geometric presentation.
		If we omit the first component, the basis vectors for the resulting coordinate system are the edges of the simplex that emanate from the first vertex.
		These coordinates are in general oblique but there is a particularly simple way to compute the metric tensor for oblique barycentric coördinates from the edge lengths of the simplex.
	www.cap-lore.com /MathPhys/bcc.html (223 words)

	Interpolation Reference Manual (Site not responding. Last check: 2007-10-20)
		If the function is a linear function and given barycentric coordinates that allow to express the query point as the convex combination of some data points, the function can be exactly interpolated.
		Natural neighbor coordinates are defined by Sibson in 1980 and are based on the Voronoi diagram of the data points.
		Natural neighbor coordinates on closed and well-sampled surfaces can also be computed if the normal to the surface at the query point is known.
	www.cgal.org /Manual/3.2/doc_html/cgal_manual/Interpolation_ref/Chapter_intro.html (323 words)

	Barycentric Coordinates for Convex Polytopes - Joe, Tony (ResearchIndex)
		Abstract: An extension of the standard barycentric coordinate functions for simplices to arbitrary convex polytopes is described.
		The key to this extension is the construction, for a given convex polytope, of a unique polynomial associated with that polytope.
		The barycentric coordinate functions for the polytope are rational combinations of adjoints of various dual cones...
	citeseer.ist.psu.edu /257398.html (431 words)

	Journal of Graphics Tools - Papers - Generalized Barycentric Coordinates on Irregular Polygons (Site not responding. Last check: 2007-10-20)
		In this paper we present an easy computation of a generalized form of barycentric coordinates for irrecular, convex n-sided polygons.
		Triangular barycentric coordinates have had many classical applications in computer graphics, from texture mapping to ray tracing.
		A simple C++ inplementation of the barycentric coordinate calculation is available here: barycentric.cpp (as HTML text).
	jgt.akpeters.com /papers/MeyerEtAl02 (296 words)

	Image Morphing (Site not responding. Last check: 2007-10-20)
		Assuming the barycentric coordinates to remain same in the corresponding triangles in source and target images, we interpolate the intensities between the corresponding pixels in source and target images to get the intensity of the pixel under consideration.
		The barycentric coordinates of the corner corresponding I
		Assuming the barycentric coordinates to remain same in the corresponding triangles in all the N images, we interpolate the intensities between the corresponding pixels in N images (using the corresponding barycentric ordinate as the weights) to get the intensity of the pixel under consideration.
	www.cse.iitd.ernet.in /~csu02125/dip/morphing (539 words)

	[No title]
		Here are the matrices in general form: Ax means the x coordinate of A, etc. EMBED Equation.2 This allows you to quickly change the values of r, s, and t and see where the points are.
		The values r/v, s/v, and t/v are called the barycentric coordinates of P. The actual values of r, s, and t are sometimes called the barycentric proportions or the barycentric masses.
		We will spend the rest of this project studying these barycentric values and their impact on the point P. Note right away that by the way we have defined them, the sum of the coordinates will equal 1 and that all of them are non-negative.
	www.pballew.net /ceva1.doc (2675 words)

	Vector3.BaryCentric Method
		Any point in the plane v1v2v3 can be represented by the barycentric coordinates (f, g).
		Barycentric coordinates are a form of general coordinates.
		What holds true for Cartesian coordinates holds true for barycentric coordinates.
	msdn.microsoft.com /archive/en-us/directx9_m_Summer_04/directx/ref/ns/microsoft.directx/s/vector3/m/barycentric.asp?frame=true (293 words)

	Usenet Discussion of Barycentric Coordinates
		A fail-proof method is to compute the barycentric coordinates.
		Given three (non-colinear) points A,B,C, the "barycentric coordinates" of a point P with respect to A,B,C are u,v,w, such that:
		One common use for barycentric coordinates is to warp the 2D plane by computing [u,v,w] with respect to ABC, and then re-generating the point with respect to DEF.
	steve.hollasch.net /cgindex/math/barycentric.html (548 words)

	Three Jugs Problem
		Spherical coordinates are suitable for the geometry of the sphere, as are cylindrical coordinates on a cylinder.
		There are several systems of coordinates in which vertices and sides of a triangle are treated in an equitable manner.
		The most important are the barycentric and trilinear coordinates.
	www.maa.org /editorial/knot/water.html (1369 words)

	Dr. Dobb's \| Triangle Intersection Tests \| July 22, 2001 (Site not responding. Last check: 2007-10-20)
		This second step is done by dropping one of the three coordinates of the vertices and point involved, thereby turning this into a 2D problem, which we solved in the previous section.
		In addition, barycentric coordinates are generated for the intersection point.
		The y- and the z-components of w are the barycentric coordinates for the intersection point; that is, it must hold that w
	www.ddj.com /184404201 (1912 words)

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