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Topic: Plane at infinity

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	Plane at infinity
		Note that since the (real) plane at infinity is a projective plane, it is homeomorphic to the surface of a "sphere modulo antipodes", i.e.
		This spherical plane at infinity surrounds the affine 3-space.
		The points on the plane at infinity seem to have three degrees of freedom, but homogenous coordinates are equivalent up to any rescaling:
	www.brainyencyclopedia.com /encyclopedia/p/pl/plane_at_infinity.html (533 words)

	plane - definition by dict.die.net
		Plane at infinity (Geom.), a plane in which points infinitely distant are conceived as situated.
		Plane of refraction or reflection (Opt.), the plane in which lie both the incident ray and the refracted or reflected ray.
		Plane scale (Naut.), a scale for the use of navigators, on which are graduated chords, sines, tangents, secants, rhumbs, geographical miles, etc. Plane surveying, surveying in which the curvature of the earth is disregarded; ordinary field and topographical surveying of tracts of moderate extent.
	dict.die.net /plane (576 words)

	Plane at infinity -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		In (The geometry of properties that remain invariant under projection) projective geometry, the plane at infinity is a (Click link for more info and facts about projective plane) projective plane which is added to the affine 3-space in order to give it closure of (The relative frequency of occurrence of something) incidence properties.
		Note that since the (real) plane at infinity is a projective plane, it is (Click link for more info and facts about homeomorphic) homeomorphic to the surface of a "sphere modulo antipodes", i.e.
		The points on the plane at infinity seem to have three degrees of freedom, but homogenous coordinates are equivalent (Click link for more info and facts about up to) up to any rescaling:
	www.absoluteastronomy.com /encyclopedia/p/pl/plane_at_infinity.htm (605 words)

	CURVE (Lat. curvus, bent) - Online Information article about CURVE (Lat. curvus, bent) (Site not responding. Last check: 2007-10-21)
		Line at Infinity.—The foregoing notion of a point at infinity is a very important one in modern geometry; and we have also to consider the paradoxical statement that in plane geometry, or say as regards the plane, infinity is a right line.
		The theorem of duality as regards plane figures may be thus stated: two figures may correspond to each other in such manner that to each point and line in either figure there correspond in the other figure a line and point respectively.
		We may consider in relation to a curve, not only the line infinity, but also the circular points at infinity; assuming the curve to be real, these present themselves always conjointly; thus a circle is a conic passing through the two circular points, and is thereby distinguished from other conics.
	encyclopedia.jrank.org /CRE_DAH/CURVE_Lat_curvus_bent_.html (8727 words)

	CURVE - LoveToKnow Article on CURVE
		Reverting to the purely plane theory, infinity is a line, related like any other right line to the curve, and thus intersecting it mm points, real or imaginary, distinct or coincident.
		The branch, whether re-entrant or iiifinite, may have a cusp or cusps, or it may cut itself or another branch, thus having or giving rise to crunodes or double points with distinct real tangents; an acnode, or double point with imaginary tangents, is a branch by itself, it may be considered as an.
		The author considers, not only plane curves, but also cones, or, what is almost the same thing, the spherical curves which are their sections by a concentric sphere.
	94.1911encyclopedia.org /C/CU/CURVE.htm (8307 words)

	Light Field Glossary of Terms (Site not responding. Last check: 2007-10-21)
		plane, u-v: One of the two planes that is used to parameterize a slab of a light field.
		plane, s-t: One of the two planes that is used to parameterize a slab of a light field.
		Each slab is parameterized by two planes, which we call the u-v and s-t planes.
	graphics.stanford.edu /software/lightpack/doc/glossary.html (411 words)

	Chapter Plainness <i>to</i> Planisphere of P by Webster's Dictionary (1913 Edition)
		the horizontal plane upon which the object which is to be delineated, or whose place is to be determined, is supposed to stand.
		The plane on which the projection is made, corresponding to the perspective plane in perspective; — called also principal plane.
		One of the planes to which points are referred for the purpose of determining their relative position in space.
	www.bibliomania.com /2/3/257/1207/23633/4.html (338 words)

	Surfaces in 3D
		Given a family of confocal quadrics in nD Euclidean space, a line in general position is tangent to n - 1 different quadrics in the family, and the planes tangent to the quadrics at the points of tangency are pairwise orthogonal.
		Light emanating from the focal ellipse E in direction of a focal ray (i.e., towards H) is reflected by a confocal ellipsoid Q in the direction of a focal ray back to E.
		Planes: The focal set consists of a single imaginary point (taken to infinity).
	www.lems.brown.edu /vision/people/leymarie/Notes/CurvSurf/Surfaces.html (1327 words)

	Mike Brown's INFINITY I Sport plane building project (Site not responding. Last check: 2007-10-21)
		As of now, the plane is not yet started nor is it quite ready at the factory.
		The Infinity Aerospace company is slowly gathering the money through the sale of specifically designed items of the Infinity 1 that can fit other planes.
		At present there are two planes that interest me; The Infinity 1 because of its advanced design, and the Cozy Mark IV because the builder is not dependent upon the factory for parts.
	www.creation-science-prophecy.com /infinity.htm (300 words)

	Indoor Navigation in Unknown Environments
		The reason why this cue is so important is that, in a projective framework, the common intersection of the images of a set of parallel lines corresponds to the image of a point belonging to the plane at infinity.
		If four points belonging to the plane at infinity in a non-singular configuration can be reliably tracked then the resulting eight linear constraints are enough to determine the homography induced by the plane at infinity in a trivial way.
		Our idea is to use at least four additional linear constraints obtained from the knowledge of at least two sets of parallel lines, in order to constrain H to be the homography induced by the plane at infinity, and thus recover structure and motion under the assumption of rigidity, up to a uniform scaling factor.
	www.cs.rochester.edu /u/carceron/research/selfcal.html (813 words)

	Blueprint Theory (Site not responding. Last check: 2007-10-21)
		The infinity algorithm and the Argand plane of imaginary numbers is based on zero, a fictional integer of math, from the period of 876 A.D. Some are under the false impression math uses the pseudo-zero and the infinite unknown in many of its equations.
		The laws of infinity show the infinite and indivisible is a point moving everywhere at infinite speed.
		Infinity is the 5th dimension of the infinite, eternal, known, unknown.
	www.midcoast.com /~gillespi/field.htm (11746 words)

	[No title]
		Planes are spheres with infinit radius and lines are circles with infinite radius.
		The classic stereographic projection introduced by Moebius maps the Moebius plane (the x-y-plane projectively close by one point at infinity) onto the unit sphere of 3-space or -one dimension higher- the Moebius space (3-space projectively closed by one point at infinity) onto the unit hypersphere of 4-space.
		An arbitrary sphere or a plane of 3-space denote by S is mapped by stereographic projection onto the intersection of the moebius hypersphere with a hyperplane H. The generalized sterographic projection maps S onto the pole of the hyperplane H with respect to the Moebius Hypersphere.
	www.cg.tuwien.ac.at /~katja/Research/canal.html (566 words)

	[No title]
		To each straight line there is attributed a point at infinity, and each line parallell to a meets a in the point at infinity, of a.
		The points at infinity of the straight lines of a plane constitute a straight line at infinity, of this plane.
		All the points at infinity of the 3D space constitute the plane at infinity.
	www.nbi.dk /~kleppe/random/ggg/geo.html (337 words)

	GameDev.net - The Theory of Stencil Shadow Volumes (Site not responding. Last check: 2007-10-21)
		Since the far clip plane is at a finite distance from the eye position, the depth-fail technique will almost certainly produce the wrong result when the shadow volume gets clipped at the far plane.
		Once in place, the near clip plane capping of shadow volumes is done at a depth value of 0.05, which is half of the ledge.
		By projecting a far plane all the way to infinity, there is no mathematical chance of the shadow volume being clipped by the far plane when we are rendering the shadow volume.
	www.gamedev.net /reference/articles/article1873.asp (7717 words)

	Formulae For The Singularities At Infinity Of Plane Algebraic Curves (ResearchIndex)
		1.6: Polar Quotients And Singularities At Infinity Of Polynomials In..
		1 The Euler characterisic and singularities at infinity of alg..
		1 the singularities at infinity of plane algebraic curves (context) - Gwo'zdziewicz, loski
	citeseer.ist.psu.edu /gwozdziewicz00formulae.html (857 words)

	Affine Interpretations (Site not responding. Last check: 2007-10-21)
		We use the center of the sphere to project it onto the plane tangent to the sphere at the north pole.
		This plane is the plane z=1; every point on the upper hemisphere goes to a point on this plane in the obvious way.
		The point on the equator of this sphere (those points for which z=0) become the points of infinity in the plane z=1; they project onto points infinitely far away from the z-axis.
	www.geom.uiuc.edu /apps/conics/conic5.html (366 words)

	Direct Solution of Modulus Constraints (ResearchIndex)
		Abstract: The modulus constraint is a constraint on the position of the plane at infinity (1) which applies to the problem of self-calibration in the case of constant internals.
		For any pair of cameras which are known to have the same internal parameters, the classical modulus constraint is the vanishing of a certain quartic polynomial whose coefficients are determined from the cameras.
		Given a projective three-view reconstruction, it is of practical interest to recover the plane at infinity by solving...
	citeseer.ist.psu.edu /376558.html (459 words)

	index1
		Below is the new RX 912S Infinity Power Parachute in stock at Executive Sky Sports and ready to fly. If your looking for that 100 HP super quite 4 stroke power with a 1,500 hour TBO, then you've just got to see this baby.
		Below is the standard Infinity SE 582D dual seat model with the popular Rotax 582 engine. Producing a full 65 horsepower and loaded with options. This is the hot new "Orange and Black" Infinity we now have in stock.
		Plane comes completely assembled and tested, with a 3 year warranty from Infinity Powered Parachutes.
	www.executiveskysports.com (465 words)

	Appendix G - OpenGL Programming Guide (Addison-Wesley Publishing Company) (Site not responding. Last check: 2007-10-21)
		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.
		If you like to think of normal vectors as vectors instead of as the planes perpendicular to them, let v and n be vectors such that v is perpendicular to n.
	fly.srk.fer.hr /~unreal/theredbook/appendixg.html (902 words)

	Projective Geometry
		The analogy is that a, b are points at infinity of two families of parallel lines (our highways) and the line at infinity is the horizon.
		Here the world coordinate system and the image plane coordinate system overlap (their xy planes are the same).
		, the projective space, the duality is between points (thought of as the pencils of planes) and planes (thought of as the range of points) where a plane is defined by a four-tuple
	www.lems.brown.edu /~leymarie/Hakan/projective.html (1061 words)

	[No title]
		A pinhole camera consists of two planes, with a minute hole punched in the focal plane to allow rays of light through to fall upon the image plane.
		Most importantly, at this strata, transformations leave the plane at infinity unchanged, it also transforms a conic to a conic of the exact form.
		The plane that passes through the point M and both camera centres, is the epipolar plane.
	www.scs.carleton.ca /~awhitehe/PVT/tutorial (6000 words)

	Heesch's Problem
		The Heesch number of a shape in the plane is the maximum number of times that shape can be completely surrounded by copies of itself.
		More formally, if one has a tiling of the plane (a collection of disjoint connected open sets the closures of which cover the plane), the first corona of a tile is the set of all tiles that have a common boundary point with the tile, including the original tile itself.
		Aperiodic tiling seems to act as a barrier to the existence of tiling algorithms, so it is not expected that both of these problems have the same answer.
	www.ics.uci.edu /~eppstein/junkyard/heesch (753 words)

	flipcode - OpenGL/Direct3D Projection Matrix (Site not responding. Last check: 2007-10-21)
		I don't like the matrix that is produced by glFrustum(...) because I want to have the far plane at infinity.
		If you need more precision in the front, make the near clipping plane closer, thus you are loosing precision in the distance, but the far clipping plane remains at infinity.
		what i mean by impossible is you cant specify a far clipping plane of infinity in opengl.
	www.flipcode.com /cgi-bin/fcarticles.cgi?show=63832 (1734 words)

	The image of the absolute conic
		In this case both the AC and its supporting plane, the plane at infinity, are expressed through one geometric entity.
		The IAC can also be transferred from one image to another through the homography of its supporting plane (i.e.
		In fact restricting the self-calibration constraints to the epipolar geometry is equivalent to the elimination of the position of infinity from the equations.
	www.cs.unc.edu /~marc/tutorial/node87.html (464 words)

	The 222-infinity Orbifold
		Switching for convenience to the plane with infinity, they have to be at the vertices of an equilateral triangle.
		The ends of the rays should go to the middles of the sides of the triangle and complex infinity should go to the vertices of the triangle.
		The most crucial step was in using the inverse of the elliptic modular function which maps the upper half plane onto the modular triangle with angles
	www.geom.uiuc.edu /apps/pinball/poster/section3.4.html (399 words)

	ex3 (Site not responding. Last check: 2007-10-21)
		Affine 3D reconstruction is performed exactly in the same way as the projective reconstruction, only that in this case the matrix H should be the homography with respect to the plane at infinity, and not a general homography.
		The homography with respect to the plane at infinity is computed from the knowledge of parallel lines: By intersecting parallel lines in each image, one obtains a set of corresponding points which are projections of points on the plane at infinity.
		Here are a few input images that you can use to test your functions.
	www.cs.huji.ac.il /course/2002/compvis/ex3/ex3.html (501 words)

	water plane extends to infinity - GameDev.Net Discussion Forums (Site not responding. Last check: 2007-10-21)
		Far Cry's water actually doesn't extend out to infinity.
		I've seen some user made maps in which the water DOES end.
		If you do want literally infinite water though, you could make a trapezoid or quad that always has one 'side' where the camera is, perpindicular to the view direction, and another side at the cam's far plane.
	www.gamedev.net /community/forums/topic.asp?topic_id=341343 (729 words)

	Obsidian Forum Community > PPG Infinity Engine Modlist (Site not responding. Last check: 2007-10-21)
		To make things easier for me and more convenient for players everywhere, we at the Pocket Plane Group have setup a centralized database to consolidate all of these various lists.
		This new setup allows for easier categorization of all the mods, as well as allowing mod authors to submit their own entries.
		To view the full version with more information, formatting and images, please click here.
	forums.obsidianent.com /lofiversion/index.php?t4677.html (552 words)

	Notations
		For matrices bold face fonts are used (i.e.
		are used for indexing points, lines or planes.
		dual absolute conic embedded in the plane at infinity (
	www.cs.unc.edu /~marc/tutorial/node2.html (71 words)

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