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	1.4 Homogeneous Coordinates in the Plane
		The connection between the point in space with cartesian coordinates (x,y,t) and the point in the plane with homogeneous coordinates (x:y:t) becomes apparent when we consider the plane t=1 in space, with cartesian coordinates given by the first two coordinates x, y of space (Figure 1).
		The plane cartesian coordinates of Q are (x/t, y/t), and (x:y:t) is one set of homogeneous coordinates for Q.
		Projective coordinates are useful for several reasons, one the most important being that they allow one to unify all symmetries of the plane (as well as other transformations) under a single umbrella.
	www.geom.uiuc.edu /docs/reference/CRC-formulas/node6.html (343 words)

	Homogeneous coordinates - Wikipedia, the free encyclopedia
		In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius, allow affine transformations to be easily represented by a matrix.
		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).
		Homogeneous coordinates are frequently used in computer graphics as they allow all affine transformation to be represented by a matrix operation.
	en.wikipedia.org /wiki/Homogeneous_co-ordinates (998 words)

	Homogeneous (Site not responding. Last check: 2007-10-16)
		A homogeneous differential equation is usually one of the form Lf = 0, where L is a differential operator, the corresponding inhomogeneous equation being Lf = g with g a given function; the word homogeneous is also used of equations in the form Dy = f''(''y''/''x).
		A homogeneous space for a Lie group G, or more general transformation group, is a space X on which G acts transitively and continuously - so equivalently a coset space G/H where H is a closed subgroup.
		Given a colouring of the edges of a complete graph, the term homogeneous applies to a subset of vertices such that all edge connecting two of the subset have the same colour; and in much greater generality in Ramsey theory for colourings of k-element subsets.
	www.serebella.com /encyclopedia/article-Homogeneous.html (869 words)

	Triangle Scan Conversion using 2D Homogeneous Coordinates
		Homogeneous coordinates are commonly used for transformations in 3D graphics.
		While homogeneous coordinates are used for 3D transformations, points are converted back to true 3D after hither clipping.
		The coefficients are derived directly from the homogeneous coordinates of the vertices.
	www.cs.unc.edu /~olano/papers/2dh-tri (4648 words)

	Incidence (geometry) - Wikipedia, the free encyclopedia
		Given line L and point P in a projective plane, and both expressed in homogeneous coordinates, then P⊂L iff the dual of the line is perpendicular to the point (so that their dot product is zero); that is, if
		in a projective plane and expressed in homogeneous coordinates are concurrent iff their scalar triple product is zero, viz.
		iff the determinant of the homogeneous coordinates of the points is equal to zero.
	en.wikipedia.org /wiki/Incidence_%28geometry%29 (526 words)

	PlanetMath: projective space (Site not responding. Last check: 2007-10-16)
		Homogeneous coordinates differ from ordinary coordinate systems in that a given element of projective space is labelled by multiple homogeneous “coordinates”.
		Projective space also admits a more conventional type of coordinate system, called affine coordinates.
		It must be noted that a single system of affine coordinates does not cover all of projective space.
	www.planetmath.org /encyclopedia/HomogeneousCoordinates.html (329 words)

	Homogeneous co-ordinates: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-10-16)
		The homogeneous co-ordinates of a point of projective space of dimension n are usually written as (x : y : z :...
		This distinction between brackets and parentheses means that addition of points in homogeneous coordinates will be defined in two different ways, EHandler: no quick summary.
		In mathematics, barycentric coordinates are coordinates defined by the vertices of a simplex....
	www.absoluteastronomy.com /encyclopedia/h/ho/homogeneous_co-ordinates.htm (1517 words)

	LEDA Guide: Cartesian and Homogeneous Coordinates (Site not responding. Last check: 2007-10-16)
		The homogeneous coordinates of a point in the plane are a triple (x,y,w) with w!=0.
		The Cartesian coordinates of a point are of type double in the floating point kernel and of type rational in the rational kernel.
		The homogeneous coordinates of a point in the rational kernel are of type integer.
	www.algorithmic-solutions.info /leda_guide/geometry/cart_hom_coord.html (198 words)

	Translation (geometry) - Wikipedia, the free encyclopedia
		It can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system.
		Since a translation is an affine transformation but not a linear transformation, homogeneous coordinates are normally used to represent the translation operator by a matrix.
		To translate an object by a vector v, each homogeneous vector p (written in homogeneous coordinates) would need to be multiplied by this translation matrix:
	www.wikipedia.org /wiki/Translation_(geometry) (295 words)

	Homogeneous co-ordinates (Site not responding. Last check: 2007-10-16)
		In mathematics homogeneous co-ordinates introduced by August Ferdinand Möbius make possible in projective space just as Cartesian co-ordinates do in Euclidean space.
		Therefore this of co-ordinates can be explained as follows: the projective space is constructed from a space V of dimension n+1 introduce co-ordinates in V by choosing a basis and use in P(V) the equivalence classes of proportional non-zero in V.
		To add homogeneous coordinates it is that the denominator be common.
	www.freeglossary.com /Homogenous_coordinates (593 words)

	VB Helper Tutorial: Beyond Flatland
		Homogeneous coordinates allow a program to represent various three-dimensional transformations such as rotation and translation in a homogeneous (uniform) way.
		One of the most important features of homogeneous coordinates is that you can combine transformations by multiplying their matrices.
		Using the coordinates of the point of view, it creates two simpler rotation transformations to move the point of view into the Z axis.
	www.vb-helper.com /tut4.htm (1879 words)

	[No title] (Site not responding. Last check: 2007-10-16)
		Plücker coordinates are a representation of directed[4] lines in 3-space.
		When solving problems using Plücker coordinates, sometimes you end up with a line in Plücker space, and you need to find the intersection of that line with the Grassman manifold to find the Plücker points that correspond to the real solutions to the problem.
		Homogeneous coordinates are useful for many geometric tasks, and show up in graphics and geometry calculations on a regular basis.
	www.flipcode.com /articles/article_pluecker-pf.shtml (916 words)

	Homogeneous Coordinates (Site not responding. Last check: 2007-10-16)
		One of the many purposes of using homogeneous coordinates is to capture the concept of infinity.
		Given a homogeneous coordinate (x,y,w) of a point in the xy-plane, let us consider (x,y,w) to be a point in space whose coordinate values are x, y and w for the x-, y- and w- axes, respectively.
		Therefore, as a homogeneous point moves on a curve defined by homogeneous polynomial f(x,y,w)=0, its corresponding point moves in three-dimensional space, which, in turn, is projected to the plane w=1.
	www.cs.mtu.edu /~shene/COURSES/cs3621/NOTES/geometry/homo-coor.html (1133 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).
		In the red triangles, two coordinates are less than 1/4 while the third is between 1/2 and 3/4.
		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)

	Homogeneous coordinates (Site not responding. Last check: 2007-10-16)
		In mathematics, homogeneous co-ordinates, introduced by August Ferdinand Möbius, make calculations possiblein projective space just as Cartesian co-ordinates do in Euclidean space.
		The homogeneous co-ordinates of a point of projective spaceof dimension n are usually written as (x:y:z:... :w), a row vector of lengthn+1, other than (0:0:0:... :0).
		Therefore this system of co-ordinates can be explained as follows: if theprojective space is constructed from a vector space V of dimension n+1, introduce co-ordinates in V bychoosing a basis, and use these in P(V), the equivalence classes of proportional non-zero vectors in V.
	www.therfcc.org /homogeneous-coordinates-214317.html (482 words)

	Homogeneous Coordinates
		This equivalence is plain when one considers the transformation from homogeneous coordinates back to image plane coordinates, which is carried out by reversing the process for converting into homogeneous coordinates - scale the homogeneous vector such that the third component is equal to one and then discard the third component.
		In addition to the linearity which they bring to equations, a further important benefit of using homogeneous coordinates is that they make it possible to represent points which are at infinity on the image plane.
		In non-homogeneous coordinates, there is no numerical representation for a point at infinity; in homogeneous coordinates, such a point has its third component equal to zero i.e.
	www.homepages.informatics.ed.ac.uk /rbf/CVonline/LOCAL_COPIES/BEARDSLEY/node1.html (392 words)

	Homogeneous coordinates and points at infinity (in a plane).
		Note that each non-zero multiple of these homogeneous coordinates is also a triple of homogeneous coordinates of the same ideal point.
		A homogeneous equation is such condition for the homogeneous coordinates of the point.
		G(x,y,z) = 0 is a homogeneous equation of c.
	www.ping.be /~ping1339/homog.htm (1919 words)

	Homogeneous Coordinates
		In this case we need three ``homogeneous coordinates'' instead of two ``inhomogeneous'' ones to represent each ray.
		In fact, in homogeneous coordinates, any polynomial can be re-expressed as a homogeneous one.
		Translate this into homogeneous coordinates and show that the line at infinity is tangent to it.
	www.cse.iitd.ernet.in /~vaibhav/vision/tutorial/node7.html (575 words)

	CITIDEL: Viewing 'Clipping using homogeneous coordinates' (Site not responding. Last check: 2007-10-16)
		Homogeneous coordinates are a convenient mathematical device for representing and transforming objects.
		The space represented by homogeneous coordinates is not, however, a simple Euclidean 3-space.
		The clipping problem is usually solved without consideration for the differences between Euclidean space and the space represented by homogeneous coordinates.
	www.citidel.org /?op=getobj&identifier=oai:ACMDL:articles.807398 (236 words)

	Homogenous and inhomogenous coordinates
		So the point whose inhomogeneous coordinate is a number x, rather than ∞, has homogeneous coordinates (x,1), while the point whose inhomogeneous coordinate is ∞, has homogeneous coordinates (1,0).
		A point with homogeneous coordinates (x,y,0) is on the line at infinity.
		The easiest way to do this is write them in inhomogeneous coordinates, and then use their first coordinates (the projection on the x-axis) or their second coordinates (the projection on the y-axis).
	www.math.fau.edu /Richman/geometry/coords.htm (1180 words)

	Homogeneous coordinates and points at infinity (in a plane).
		Homogeneous coordinates and points at infinity (in a plane).
		Note that each non-zero multiple of these homogeneous coordinates is also a triple of homogeneous coordinates of the same ideal point.
		A homogeneous equation is such condition for the homogeneous coordinates of the point.
	www.ddart.net /science/mathmatics/mathtutorial/homog.htm (1919 words)

	Homogeneous coordinates
		Because scaling is unimportant, the coordinates (X,Y,W) are called the homogeneous coordinates of the point.
		The concepts of homogeneous coordinates are summarized in Figure 2.
		In homogeneous coordinates the line becomes Y=0 which yields the solution (X,0,0), the ideal point associated with the horizontal direction.
	ai.stanford.edu /~birch/projective/node4.html (515 words)

	Geometry Kernels
		Homogeneous representation can be seen as a divison-free representation of Cartesian coordinates.
		These are all parameterized by a number type, which is used for storing the coordinates and the arithmetic in the corresponding primitives and predicates.
		While in the internal homogeneous representation, an integral number type is sufficient, rational numbers must sometimes be used outside the internal representation, for example, when the squared length of a vector is computed.
	www.cgal.org /DManual/html/Developers_manual/Chapter_kernels.html (1114 words)

	1.4 Homogeneous Coordinates in the Plane (Site not responding. Last check: 2007-10-16)
		The connection between the point in space with cartesian coordinates (x,y,t) and the point in the plane with homogeneous coordinates (x:y:t) becomes apparent when we consider the plane t=1 in space, with cartesian coordinates given by the first two coordinates x, y of space (Figure 1).
		The plane cartesian coordinates of Q are (x/t, y/t), and (x:y:t) is one set of homogeneous coordinates for Q.
		Projective coordinates are useful for several reasons, one the most important being that they allow one to unify all symmetries of the plane (as well as other transformations) under a single umbrella.
	geom.math.uiuc.edu /docs/reference/CRC-formulas/node6.html (343 words)

	Homogeneous coordinates
		Because scaling is unimportant, the coordinates (X,Y,W) are called the homogeneous coordinates of the point.
		The concepts of homogeneous coordinates are summarized in Figure 2.
		In homogeneous coordinates the line becomes Y=0 which yields the solution (X,0,0), the ideal point associated with the horizontal direction.
	vision.stanford.edu /~birch/projective/node4.html (515 words)

	[No title]
		The main strategy for performing these transformations is to perform them on each of the coordinates of the primitives (the vertices, in the case of a line or polygon).
		One advantage of representing points in homogeneous coordinates is that we can transform the points in an elegent way using a 3x3 matrix.
		Just as in 2D, we can define homogeneous coordinates of a point (x,y,z) in 3D to be (w x, w y, w z, w) where w is non-zero.
	www.cim.mcgill.ca /~langer/557B/Jan18.html (1316 words)

	Appendix G - OpenGL Programming Guide (Addison-Wesley Publishing Company)
		In the discussion that follows, the term homogeneous coordinates always means three-dimensional homogeneous coordinates, although projective geometries exist for all dimensions.
		A homogeneous plane is denoted by the row vector (a, b, c, d), where at least one of a, b, c, or d is nonzero.
		If p is a homogeneous plane and v is a homogeneous vertex, then the statement "v lies on plane p" is written mathematically as pv = 0, where pv is normal matrix multiplication.
	www.kmit.sk /manuals/gl-spec/appendixg.html (902 words)

	Reference Manual: (Site not responding. Last check: 2007-10-16)
		Rather, we use homogeneous coordinates to avoid division operations, since the additional coordinate can serve as a common denominator.
		As we mentioned before, homogeneous coordinates permit to avoid division operations in numerical computations, since the additional coordinate can serve as a common denominator.
		Since the homogeneous representation allows one to avoid the divisions, the number type associated with a homogeneous representation class must be a model for the weaker concept ring type only.
	graphics.stanford.edu /courses/cs368/CGAL/ref-manual1/Chapter_Preliminaries.html (1190 words)

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