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Topic: Line at infinity

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Point at infinity

Linear perspective

Riemann sphere

Euclidean geometry

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	Line at infinity - Wikipedia, the free encyclopedia
		In geometry and topology, the line at infinity is a line which is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane.
		This is true of the line at infinity itself: it meets itself at its two endpoints (which are therefore not endpoints at all) and so it is actually cyclical.
		The analogue for the complex projective plane is a 'line' at infinity that is (naturally) a complex projective line.
	en.wikipedia.org /wiki/Line_at_infinity (777 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		The 12 interior blue lines, along with 3 blue lines at infinity, are the 15 lines corresponding to pairs of a set of 6 lines, corresponding to the 15 dimensions of Spin(6).
		Each red or green line intersects: 5 red or green lines, 4 near the outside and one in the middle; 4 interior blue lines; and 1 blue line at infinity.
		Each blue line at infinity intersects: 4 red or green lines; 4 interior blue lines, parallel to the planes of the red or green lines that intersect; and 2 blue lines at infinity, the other 2 of a triangle at infinity.
	www.valdostamuseum.org /hamsmith/27-lineComments (230 words)

	Xah: Special Plane Curves: Line
		The inclination of a line is measured by the ratio of its rise divided by its extend.
		This is the equation of a line passing {a,b} and {c,d}.
		That if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straights lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.
	xahlee.org /SpecialPlaneCurves_dir/Line_dir/line.html (1050 words)

	Simson line
		A bundle of all possible directions forms a line (at infinity), with the lines parallel to one of the directions intersecting at a single point (at infinity).
		For example, existence of Simson lines together with the Pivot theorem could be used to demonstrate the existence of Miquel's point for a 4-line.
		As P moves over the circumcircle, the point of intersection of the two lines traces a circle, the 9-point circle in the case of the (regular) Simson lines.
	www.cut-the-knot.org /ctk/SimsonLine.shtml (1388 words)

	infinity - HighBeam Encyclopedia (Site not responding. Last check: 2007-11-06)
		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.encyclopedia.com /doc/1E1-infinity.html (417 words)

	Line (mathematics) - Wikipedia, the free encyclopedia
		In two dimensions, two different lines can either be parallel, meaning they never meet, or may intersect at one and only one point.
		a, b, and c are related to the slope of the line - [a, b, c] is a vector parallel to the line, and
		The "straightness" of a line, interpreted as the property that it minimizes distances between its points, can be generalized and leads to the concept of geodesics on differentiable manifolds.
	en.wikipedia.org /wiki/Line_(mathematics) (574 words)

	CENTRAL LINES
		A line in the plane of a triangle is a central line if it passes through two triangle centers.
		The first line shown is the line at infinity.
		Among the centers on this line, the two with least indices in ETC are X(30) and X(511), so that the line is X(30)X(511), or simply 30 511.
	faculty.evansville.edu /ck6/encyclopedia/CentralLines.html (2508 words)

	4lines2cubic
		At the point where line1 meets the line at infinity, both terms are zero, so the cubic goes through this point, which because of the cube is a triple point, meaning that line1 meets the line at infinity at a point of inflection.
		At the point where line1 meets the line at infinity, both terms are zero, so that the cubic goes through this point, a double point.
		Again the point where line1 meets the line at infinity, both terms are zero, so that the cubic goes through this point.
	www.paideiaschool.org /TeacherPages/Steve_Sigur/lines2cubic.htm (621 words)

	Slopes and Infinity - SciForums.com
		The slope of a vertical line is undefined.
		It is interesting that a line with gradient of infinity looks the same as a line with gradient of negative infinity.
		It would be infinity if m was denoted as infinity (not x/0), however when there is a certain rise over zero run, it is undefined because the function still rises according to m however zero distance is travelled and appears to be infinite.
	www.sciforums.com /showthread.php?t=45378 (812 words)

	Potential vs. Completed Infinity
		There are no points for plus or minus infinity on the line, but it is natural to attach those "numbers" to the endpoints of the semicircle.
		Consequently, if we add those two "numbers" to the real line, to get the so-called "extended real line," and we equip that set with the same topology as that of the closed semicircle (i.e., the semicircle including the endpoints), then the function f is continuous everywhere on the extended real line.
		But more complicated notions such as infinity, less grounded in physical reality, are harder to explain; it is harder to be sure that we are successfully conveying a concept from the inside of one head to the inside of another.
	www.math.vanderbilt.edu /~schectex/courses/thereals/potential.html (1976 words)

	Infinity Heating and Cooling System - Carrier
		Infinity is the only system you can buy that combines Carrier technologies such as Hybrid Heat, ComfortHeat™, IdealHumidity™ and Puron® Refrigerant — allowing you to enjoy the freshest, cleanest air that Carrier offers.
		Infinity™ Control and Remote Access—The state-of-the-art control interface for the Infinity heating and cooling system, the Infinity Control is the only control you can buy that allows you to control temperature, humidity, air quality, fan speed and ventilation.
		Infinity Fan Coil—Used in conjunction with an Infinity Heat Pump or Infinity Air Conditioner when a furnace is not present.
	www.residential.carrier.com /systems/infinity.shtml (371 words)

	Infinity Cruise Review
		While considered a premium line, the Infinity had a good number of younger people on it and seemed more geared for the 40s, 50s, and 60s than the senior set one normally would expect on an Alaskan cruise.
		I was the escort and seminar facilitator for a group of 36 travel professionals traveling on Celebrity Cruise Line's Infinity sailing from Vancouver on the Inside Passage itinerary.
		Infinity had over 100 children on this particular sailing and other than pirate night, when the children visited the Trellis Restaurant to sing everyone a song, we never were aware they were on the ship.
	www.cruisereviews.com /Celebrity/Infinity29.htm (4552 words)

	Question Corner -- Understanding Projective Geometry
		As you move to infinity on the line l, the corresponding lines through the origin actually converge to a horizontal line parallel to l, so this limiting line should be included in the family.
		Notice that the intercept b does not appear in the limit: parallel lines l, l' in P (ones with the same m and different b) correspond to families of lines through the origin in 3-d space which converge to the same horizontal line through the origin.
		What all this means is that, in projective space, the "line" corresponding to l is actually a family of lines through the origin consisting of: (1) the lines that pass through l, and (2) the limiting horizontal line.
	www.math.toronto.edu /mathnet/questionCorner/projective.html (2444 words)

	Infinity Mirror Wall Displays and Infinity Mirror Tables
		We offer a full line of Infinity Light Mirror Tables and wall displays to decorate home, office and commercial interiors.
		Infinity mirrors are table and wall designs using mirrors and miniature light bulbs that line the inside walls creating the
		These infinity mirrors are made per order and are not returnable.
	www.lightenergystudio.com /infinitymain.html (456 words)

	[No title]
		Note the restriction that the line at infinity cannot be one of the n lines of the configuration.
		If all n lines are concurrent, we only get 1 projective intersection point, which may be a finite point or a point on the line at infinity, giving 0 or 1 ordinary intersection points.
		The sum of the degrees (number of lines meeting there) of these points is at most n, because each configuration line meets the line at infinity only once.
	www.ics.uci.edu /~eppstein/junkyard/how-many-intersects.html (1265 words)

	Cascade Microtech \| Press Release
		The Infinity probe’s outstanding performance on aluminum pads is achieved by optimizing the key probe factors that affect contact resistance: tip contact area, force applied, tip metallurgy and scrub (horizontal motion).
		The force delivery of the Infinity probe is such that only a small scrub is necessary to break through the aluminum oxide to make contact with the pad.
		With the addition of a high-current model to its line of Infinity probes, Cascade Microtech is again raising the bar for accuracy and repeatability in advanced device characterization and modeling.
	www.cmicro.com /index.cfm/fuseaction/pg.view/pID/1787 (637 words)

	Second Proof of Desargue's Theorem via Projective Geometry
		This is possible because we add one extra ``line'' to the usual plane, called the line at infinity.
		For every family of parallel lines in the usual plane there is a (different) point on the line at infinity
		In particular, all lines are ``created'' equal, regardless of whether they are usual lines or the ``line at infinity''.
	mathcircle.berkeley.edu /BAMA/BAMA99/node2.html (498 words)

	Pappus' Theorem
		The only possible exception is that, since S and S' are not in general parallel, there is a line on S such that the projection ray from P through this line is parallel to S', and therefore never intersects with S' (in either direction).
		Since every pair of lines intersects at a single point, we must stipulate that the intersection of any given line with the line at infinity is a single point, regardless of the direction along the line.
		Therefore, OAD and OCF are similar, so AD and CF are parallel, which proves that their point of intersection (the point H) is on the line at infinity, co-linear with the points G and I. It's worth noting that Pappus' Theorem relies on the commutivity of multiplication of lengths.
	www.mathpages.com /home/kmath542/kmath542.htm (780 words)

	Projective geometry and slopes
		In the sketch, line AC rotates around the origin, A. The rotation is controlled by point C on the unit circle.
		For the vertical lines, add the point at infinity on the line of slopes that we constructed before.
		The line at infinity intersects any regular line at the point at infinity corresponding to the regular line's slope.
	www.geom.uiuc.edu /~addingto/proj_geom/proj_slopes.html (933 words)

	infinity — Infoplease.com
		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 (
		Infinity - Infinity Director: Matthew Broderick Writer: Patricia Broderick Director of Photography: Toyomichi...
		Mea culpa, mea maxima culpa: infinity shows the contrition, still has to worry about the commission.
	www.infoplease.com /ce6/sci/A0825190.html (345 words)

	INFINITY DUO LINE
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		A distinctive two tone colouration with 20 per cent fl and 80 per cent green Duo combines low visibility with incredible abrasion resistance and superb knotting – and all from an amazingly limp line.
		IGFA rated for accurate breaking strains Supplied in outer of six bulk spools.
	www.wickfordangling.co.uk /daiwa_carp/infinity_duo_line.htm (62 words)

	Infinity Components - Line Card
		Infinity Components is one of the leading independent stocking distributors of integrated circuits and semiconductors.
		Infinity Components offers immediate attention to the most crucial shortages.
		Infinity Components is ISO compliant with ISO 9001 certification to be completed by June ‘06
	www.infinitycomponents.com /linecard.htm (118 words)

	Infinity Portal @ Infinitely.org (Site not responding. Last check: 2007-11-06)
		For example, in optics, an object which is much further away than the focal length of a lens is said to be "at infinity", as the distance of the image from the lens varies very little as the distance increases further.
		Hector Parr's Essays: Infinity - Includes a look at the concept of infinity and the logical flaws in the different contexts of infinity.
		Hotel Infinity - Includes articles on big ideas and concepts, an introduction on background ideas, activities, books to read and a page of resources as well as a vocabulary index.
	www.infinitely.org (1477 words)

	Projective Geometry
		Furthermore, as seen from its definition lines are homogeneous entities in the projective plane.
		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.
		So the conclusion is that affine transformations are those projective transformations preserving the line at infinity.
	www.lems.brown.edu /~leymarie/Hakan/projective.html (1061 words)

	Metric Fasteners: Supplied by Infinity Fasteners
		The metric fastener product line by Infinity Fasteners is a mechanical devise used for joining pallet components such as nails, staples, bolts or screws.
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		Infinity Fasteners is here to help you lower costs and still maintain high quality in all your jobs.
	www.infinityfasteners.com /metric_fastener.html (202 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 (665 words)

	Simpson Door Launches Infinity Door Line
		Blending tradition and technology, Infinity Doors are distinguished by their classic styles and a patented manufacturing process that eliminates the possibility of shrinking, splitting, and stile and rail separation.
		While most doors are assembled by conjoining separate pieces of wood, Infinity Doors are constructed from a seamless, single panel.
		Highlighted by raised panels and raised or flush moldings, the doors are primed and ready to paint.
	www.simpsondoor.com /about_us/press_room/press_releases/infinity.asp (149 words)

	Infiniti Golf sues Top-Flite Golf over its new Infinity line of golf balls
		Infiniti Golf Inc., a custom club maker, has filed a trademark infringement suit against Top-Flite Golf Co., stemming from the former Spalding Sports Worldwide unit's launch of its new line of golf ball line called Infinity.
		The complaint claims that Top-Flite's actions are likely to cause confusion "as to the source of its new line of golf balls." The suit includes allegations of violations of federal trademark law, the Illinois Consumer Fraud and Deceptive Business Practices Act, and the Illinois Deceptive Trade Practices Act.
		Infiniti seeks unspecified monetary damages, trebled under federal law, and requests an injunction prohibiting Top-Flite "from using any name or mark that includes 'Infiniti,' 'Infinity,' an infinity symbol, or any other name or mark which is likely to be contused with the Infiniti mark."
	www.pga.com /news/industry/infinitsues061603.cfm (320 words)

	Cruise Critic Reviews: Infinity
		The Constellation is Infinity's top-deck lounge and is comfortable and contemporary with fantastic views on three sides; we loved the wooden section dividers with port holes.
		Our deck 2 doll-cabin was barely large enough for two (desk drawers that can't open all the way because they are blocked by a bed, the inability to sit at the desk without leaning the coffee table onto the tiny futon couch, the minimal drawer space).
		The line also recommends tips of 75 cents for the assistant maitre d' and 50 cents for the chief housekeeper.
	www.cruisecritic.com /reviews/review.cfm?ShipID=196 (1778 words)

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