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Topic: Point at infinity

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Line at infinity

Linear perspective

	Point at infinity - Wikipedia, the free encyclopedia
		The point at infinity, also called ideal point, is a point which when added to the real number line yields a closed curve called the real projective line,
		Since the lines are parallel, they intersect at a point at infinity which lies on
		When a pair of projective lines are parallel they intersect at their common point at infinity.
	www.wikipedia.org /wiki/Point_at_infinity (230 words)

	DIAMETERS AND AXES OF - Online Information article about DIAMETERS AND AXES OF (Site not responding. Last check: 2007-10-21)
		on the polar to the point at infinity on the parallel chords.
		The tangents at the end points of a diameter are parallel, and are parallel to the chords bisected by the diameter.
		Hence by § 77 (7) the points are in in-volution.
	encyclopedia.jrank.org /DEM_DIO/DIAMETERS_AND_AXES_OF.html (3320 words)

	INFINITY. The Columbia Encyclopedia: Sixth Edition. 2000 (Site not responding. Last check: 2007-10-21)
		The term infinity is used in a somewhat different sense to refer to a collection of objects that does not contain a finite number of objects.
		The German mathematician Georg Cantor showed that there are different orders of infinity, the infinity of points on a line being of a greater order than that of prime numbers (see transfinite number).
		In geometry one may define a point at infinity, or ideal point, as the point of intersection of two parallel lines, and similarly the line at infinity is the locus of all such points; if homogeneous coordinates (x
	www.bartleby.com /aol/65/in/infinity.html (180 words)

	Various Geometries
		Points on that plane are associated with straight lines through point O. (Incidentally, the set of all lines through a given point is called a pencil of lines.
		For x being the point at infinity, f(x) = a/c (with the same caveat.) For c = 0, f(x) turns out to be a simpler, linear transformation f(x) = ax + b which maps the point at infinity on itself.
		Since, for n=2, one ideal point is associated with a pencil of parallel lines, this point is reached regardless which direction one moves on the line.
	www.cut-the-knot.org /triangle/pythpar/Geometries.shtml (2183 words)

	Simson line
		By definition, points on the circumcircle are conjugate to points at infinity, one point at infinity per direction.
		The feet of the perpendiculars from points outside the circumcircle to the sides of the triangle are not colinear.
		Remarkably, the point of intersection of the two lines lies on the 9-point circle of the given triangle.
	www.cut-the-knot.org /ctk/SimsonLine.shtml (1129 words)

	Point at infinity -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		The point at infinity can also be added to the (Click link for more info and facts about complex plane) complex plane,, thereby turning it into a closed surface known as the complex projective line,, a.k.a.
		Since the lines are (Something having the property of being analogous to something else) parallel, they intersect at a point at infinity which lies on 's (Click link for more info and facts about line at infinity) line at infinity.
		Moreover, each of the two lines is, in, a projective line: each one has its own point at infinity.
	www.absoluteastronomy.com /encyclopedia/p/po/point_at_infinity.htm (288 words)

	Question Corner -- Understanding Projective Geometry
		The remaining points in projective space are horizontal lines through the origin in 3-d space; these are the "points at infinity".
		On a sphere, the shortest path between two points is an arc that is part of a "great circle" on the sphere (one whose centre is the centre of the sphere).
		The points on the equator are the "points at infinity".
	www.math.toronto.edu /mathnet/questionCorner/projective.html (2444 words)

	[No title]
		If that means nothing to you, think of adding some points to space to "close it up", in the sense that a circle can be thought of a copy of the real line (shrunk to an open interval) with both ends approaching an additional point (the "point at infinity").
		So that becomes the condition that the zero-locus of Q includes points at infinity, and thus also becomes the condition that the zero-locus of P is unbounded.
		There is one point at infinity in Q=0, though: x0=x1=0 (with x2 arbitrary) describes a line through the origin where Q vanishes.
	www.math.niu.edu /~rusin/known-math/99/proj_alggeo (708 words)

	Comments on 3893 \| Ask MetaFilter (Site not responding. Last check: 2007-10-21)
		And these points, whether they are in an 'infintely straight' line, or just a normally straight one, won't all touch the circle, because the circle will curve away from one of them.
		This is thus defined to be the point at infinity, and all infinities are the same.
		Actually, three points in arbitrary real 3D space should be just as good for forming a circle, because three points define a plane, and yer thus reverting to describing a circle on the 2D plane defined by those points.
	ask.metafilter.com /mefi/3893 (1927 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		All points in the plane are then mapped onto the surface of the sphere by connecting the plane points to the North Pole of the sphere.
		As the point in the plane that is being mapped moves further from the origin, its mapping A’ on the surface of the sphere moves closer to the North Pole.
		Points A, B, and C in the plane are mapped to A’, B’, C’ on the sphere, respectively.
	dana.ucc.nau.edu /~wl/William.Hermann.Stereo.Projection.htm (585 words)

	Fractal of the Day (FotD) by Jim Muth
		In today's case, I was searching for the point where the entire fractal expands to infinity and returns from the opposite direction, and once I had found this point with the evolver, I did not bother with the rounding.
		It is only a single point, but as the lines are rotated, this point of intersection moves off in one direction, reaches infinity, then returns from the opposite direction.
		Points that go off in any one direction and pass beyond infinity return from the opposite direction.
	home.att.net /~Fractals_3/FotD_03-02-08.html (589 words)

	desarg.html
		Note that the "finite point" [7,-5,4] does not lie on the line K, In the exercises we find that the "finite point" with coordinates given by one of the names for the point at infinity on a line will also be on the line if and only if the line goes through the origin.
		Then the line of intersection of the two planes must contain all points of intersection of corresponding sides since each such point is the intersection of a line in one of the planes with a line in the other.
		If these lines have two points in common or are parallel and have a point in common then they are the same line and since each lies in either the plane of ABC or that of A'B'C' they would have to lie in both and hence be the line of intersection (the green line).
	www.ms.uky.edu /~carl/vps98/desarg1.html (3761 words)

	Incredible Technology used in Infinity Retail Point of Sale (POS) software
		Whether you want to keep a month, a year or a decade's worth of sales, Infinity can be used to build a complete data warehouse for on-line analytical processing.
		Infinity has a patented Linker(tm) technology, which allows for a genuine distributed database set-up.
		Infinity's Linker(tm) technology can be configured for minimal latency ensuring that POS can be seamlessly updated with the latest Head Office configurations within seconds.
	www.cypresspos.com /Infinity_retail_pos_technology.htm (473 words)

	Canon Digital Photography Forums - What is Infinity -- Photographically, not Philosophically?
		Infinity is the distance where you lose the parallax movement that you get when you move your camera.
		Optically, infinity is any distance where the focusing error (when focused at inifinity) is small enough to not be noticable.
		While there is theoretically nothing past infinity, such lenses used to be made to accomodate expansion and contraction of the lens materials due to extreme cold and heat, which cause a shift in the infinity focusing point.
	photography-on-the.net /forum/printthread.php?t=46520 (1521 words)

	Projective geometry and slopes
		Just as a point at infinity was added to the number line, add a point at infinity to this line.
		A point at infinity is on every line that has the slope it corresponds to.
		Connect the ends of the diameter with a point at infinity corresponding to the slope of the diameter.
	www.geom.uiuc.edu /~addingto/proj_geom/proj_slopes.html (933 words)

	Math Forum: Ask Dr. Math FAQ: Large Numbers and Infinity
		Infinity is not a number; it is the name for a concept.
		There are no numbers bigger than infinity, but that does not mean that infinity is the biggest number, because it's not a number at all.
		Even though infinity is not a number, it is possible for one infinite set to contain more things than another infinite set.
	mathforum.org /dr.math/faq/faq.large.numbers.html (679 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		It gives a parametrization of the curve (1) everywhere except the cusp; t=0 maps to the point at infinity with homogeneous coordinates (0,1,0).
		The three points are regarded as adding up to zero in the group law, and lo and behold they correspond to a triple of t values which also adds to zero, because the sum of the three roots of the cubic is minus the coefficient of t^2 (0) divided by the coefficient of t^3.
		So the group law then (picking any point away from the cusp as the origin) works out to be just the same as addition.
	www.math.niu.edu /~rusin/known-math/00_incoming/singular (409 words)

	Construction by Compasses Alone
		(It also can't be a point in the plane.) In this diagram, the sphere is tangent to the plane at the point S (south pole), O is the center of the sphere, and N (north pole) is the projection point.
		A point A on the sphere corresponds to a point B on the plane when the three points N, A, and B are collinear.
		Thus, we have a correspondence between the plane with a point of infinity with the sphere.
	aleph0.clarku.edu /~djoyce/java/compass/compass4.html (827 words)

	Involutions
		Deduce that on the projective line, all involutions with one fixed point have two fixed points, and thence that they map points to their harmonic conjugates with respect to these two fixed points.
		Given the invariance of the cross ratio under complex collineations, show that all involutions of the projective line map points to their harmonic conjugates with respect to the two fixed points of the involution.
		Also, any line through any two points of the figure meets the line through the two opposite points on the axis: hence the axis is easy to find given a few correspondences.
	homepages.inf.ed.ac.uk /rbf/CVonline/LOCAL_COPIES/MOHR_TRIGGS/node36.html (362 words)

	Fractal eXtreme: x86 and Infinity
		An infinity records the fact that the result is really big, possibly infinitely big, in a manner that a really large finite number just can't.
		Their non-standard format causes difficulty for many floating point units, and was the cause of much wrangling when IEEE 754 was being worked on.
		Infinity plus infinity is equal to infinity, but a Pentium 4 takes 850 times as long to figure that out as it takes to figure out that 1234.1341234874 + 843.09821341234 = 2077.2323368997399.
	www.cygnus-software.com /papers/x86andinfinity.html (1235 words)

	Infinity Retail Point of Sale (POS) Software from Cypress Solutions
		Infinity’s excellence is in part due to its scalability.
		The critical difference with Infinity is that it is entirely built on MS SQL Server Desktop Engine.
		It also means that there is only one entry point to the Infinity system (at head office) as replication automatically distributes customer and inventory information to all stores or network nodes.
	www.cypresspos.com /Infinity_retail_pos_software.htm (449 words)

	Simson line
		The point of concurrency Q is defined as the isogonal conjugate of F. By symmetry, if Q is the isogonal conjugate of F, then P is the isogonal conjugate of Q. Whose isogonal conjugates are points on the circumcircle?
		It becomes a straight line - the simson of point F. Simson lines have many interesting properties and are also useful in establishing other facts.
		To obtain the Simson line of point P on the circumcircle, three lines are drawn from F that make angles of 90
	www.maa.org /editorial/knot/SimsonLine.html (1119 words)

	The Existence of God
		If this point is not granted, then we might as well not even have this discussion.
		No matter where you put the point on that line, there is an infinite amount of time before and after it.
		By definition, now, the first and second points are in the exact same location, because we have infinity going in not one but two directions.
	www.carmical.net /articles/existence.html (1047 words)

	Complex Analysis
		It has the property that a nonzero point is mapped onto the point Z such that
		Any point of unit modulus is mapped onto itself.
		The usual way to visualize the point at infinity is by using what we call the stereographic projection and is attributed to Riemann.
	math.fullerton.edu /mathews/c2002/ca0206.html (363 words)

	EPIG: EPIG::FillRule class Reference
		After counting the crossings, if the count is odd, the point is inside; if even, the point is outside.
		This rule determines the "insideness" of a point on the canvas by drawing a ray from that point to infinity in any direction and counting the number of path segments from the given shape that the ray crosses.
		This rule determines the "insideness" of a point on the canvas by drawing a ray from that point to infinity in any direction and then examining the places where a segment of the shape crosses the ray.
	www.gel.ulaval.ca /~epig/1.1/epig-doc/classEPIG_1_1FillRule.html (275 words)

	Impressive features in Infinity Retail Point of Sale (POS) Software
		Infinity will hold all items at all stores, some items at some stores or individual items at individual stores.
		And because Infinity's inventory management is branch related, stores know exactly who has what stock where at the touch of a button.
		Infinity maintains suppliers and debtors, tracks payments and outstanding bills, produces statements and aged trial balances and much more.
	www.cypresspos.com /Infinty_retail_pos_features.htm (538 words)

	Connectedness Proof, Mu-Ency at MROB
		If there is no other critical point in the basin of our superattractive orbit (for example, if our map is quadratic hence doesn't have any more critical points), then the Botkher coordinate extends to the entire immediate basin of attraction, mapping it into the unit disc.
		Now as noted before, infinity is a superattractive fixed point, and since 0 is not in its basin, its immediate basin of attraction (which is by definition the complement of the filled-in Julia set) contains no critical points besides infinity.
		What this means is that a map with an order n superattractive fixed point has the same local geometry and dynamics as the map z->z^n near the origin (or near infinity, for that matter.) This new local coordinate near the fixed point is called the Botkher coordinate.
	www.mrob.com /pub/muency/connectednessproof.html (2274 words)

	Infinity--Projective Geometry #2
		If we're developing the geometry of the eye to describe what you see, then these "apparent points" should be regarded as just as real (really seen, that is) as the "real points" on the lines, at least so far as the eye is concerned.
		In fact any other line which hits this "front" point at infinity will be parallel to our line and will also hit the same "back" point at infinity.
		This may be great from a spiritual point of view, but it's not so good from a geometry point of view.
	www.math.lsa.umich.edu /mmss/coursesONLINE/infinity/Geometry/Lesson2.shtml (787 words)

	INSTITUTE OF PHYSICS PUBLISHING Physiol. Meas. 22 (2001) 693711 PHYSIOLOGICAL MEASUREMENT PII: S0967-3334(01)24107-0 ... (Site not responding. Last check: 2007-10-21)
		In this paper, a method is proposedto approximately standardize the reference of scalp EEG recordings to a point at infinity.
		This method is based on the fact that the use of scalp potentials to determine the neural electrical activities or their equivalent sources does not depend on the reference, so we may approximately reconstruct the equivalent sources from scalp EEG recordings with a scalp point or average reference.
		As a point at infinity is far from all the possible neural sources, this method may be considered as a reference...
	smealsearch.psu.edu /92880.html (250 words)

	Point At Infinity Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-10-21)
		Point At Infinity Encyclopedia Article, Definition, History, Biography
		Looking For point at infinity - Find point at infinity and more at Lycos Search.
		Find point at infinity - Your relevant result is a click away!
	www.artisticnudity.com /encyclopedia/Point_at_infinity (185 words)

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