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Topic: Parallel (geometry)

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algebraic geometry

differential geometry

Euclidean geometry

triangle geometry

non Euclidean geometries

square geometry

projective geometry

analytic geometry

cube geometry

Riemannian geometry

Prism geometry

hyperbolic geometry

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	Parallel (geometry) - Wikipedia, the free encyclopedia
		Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these.
		When embedded in Euclidean space a dimension higher, parallels of latitude can be generated by the intersection of the sphere with a plane parallel to a plane through the center.
		The angle of parallelism depends on the distance of the point to the line with respect to the curvature of the space.
	en.wikipedia.org /wiki/Parallel_(geometry) (1037 words)

	Euclidean geometry - Wikipedia, the free encyclopedia
		In hyperbolic geometry the sum of the three angles are always less than 180° and can approach zero, while in elliptic geometry the sum is greater than 180°.
		Absolute geometry, formed by removing the parallel postulate, is also a consistent theory, as is non-Euclidean geometry, formed by alterations of the parallel postulate.
		For example, geometry on the surface of a sphere is a model of an elliptical geometry, carried out within a self-contained subset of a three-dimensional Euclidean space.
	en.wikipedia.org /wiki/Euclidean_geometry (2331 words)

	Parallel - Wikipedia, the free encyclopedia
		Parallel transport, in mathematics using vectors and smooth curves
		Parallel key, the minor (or major) key of a major (or minor) key with the same tonic
		Parallel (manga) (ぱられる) is a shōnen manga by Toshihiko Kobayashi
	en.wikipedia.org /wiki/Parallel (159 words)

	Learn more about Non-Euclidean geometry in the online encyclopedia. (Site not responding. Last check: 2007-10-21)
		In Euclidean geometry, however, the lines remain at a constant distance, while in hyperbolic geometry they "curve away" from each other, increasing their distance as one moves farther from the point of intersection with the common perpendicular.
		While Euclidean geometry (named for the Greek mathematician Euclid) includes some of the oldest known mathematics, non-Euclidean geometries were not widely accepted as legitimate until the 19th century.
		Euclidean geometry is modelled by our notion of a "flat plane." The simplest model for elliptic geometry is a sphere, where lines are "great circles" (such as the equator or the meridians on a globe), and points opposite each other are identified (considered to be the same).
	www.onlineencyclopedia.org /n/no/non_euclidean_geometry_1.html (1089 words)

	Geometry Session 4, Part B: Parallel Lines
		Parallel lines are two lines in the same plane that never intersect.
		Another way to think about parallel lines is that they are "everywhere equidistant." No matter where you measure, the perpendicular distance between two parallel lines is constant.
		With dynamic geometry software, you can draw two lines that look parallel, but you can't be sure that they are parallel unless you construct them to be parallel.
	www.learner.org /channel/courses/learningmath/geometry/session4/part_b/index.html (304 words)

	geometry. The Columbia Encyclopedia, Sixth Edition. 2001-05
		Another type of geometry, called affine geometry, includes Euclid’s parallel postulate but disregards two other postulates concerning circles and angle measurement; the propositions of affine geometry are also valid in the four-dimensional geometry of space-time used in the theory of relativity.
		For example, Euclidean geometry is the study of properties unchanged by similarity transformations, affine geometry is concerned with properties invariant under the linear transformations (affine collineations) that preserve parallelism, and projective geometry studies invariants under the more general projective transformations (collineations and correlations).
		Topology, perhaps the most general type of geometry although often considered a separate branch of mathematics, is concerned with properties invariant under continuous transformations, which carry neighborhoods of points into neighborhoods of their images.
	www.bartleby.com /65/ge/geometry.html (681 words)

	NonEuclid: Parallel Lines (Site not responding. Last check: 2007-10-21)
		Parallel lines are infinite lines in the same plane that do not intersect.
		This is a theorem in Euclidean Geometry, yet in Hyperbolic Geometry it is proved false by the above counter example (Both BA and BC are parallel to DE, yet BA is not parallel to BC).
		In Euclidean Geometry, lines that do not have an end (infinite lines), also do not have a boundary (a point that they are headed toward, yet never reach).
	www.cs.unm.edu /~joel/NonEuclid/parallel.html (534 words)

	SparkNotes: Constructions: Parallel Lines
		Although parallel lines are usually thought of in pairs, an infinite number of lines can be parallel to one another.
		The Parallel Postulate states that there exists one line through C which is parallel to line AB.
		Whenever you encounter three lines, and only two of them are parallel, the third line, known as a transversal, will intersect with each of the parallel lines.
	www.sparknotes.com /math/geometry1/constructions/section3.rhtml (509 words)

	Geometry: Parallel Lines - Math for Morons Like Us
		Parallel lines seem rather innocent, but are used in some complex geometry situations to help you solve problems.
		If two lines are cut by a transversal, and the corresponding angles are congruent (congruent angles have the same measure), the lines are parallel.
		If two lines are cut by a transversal so that interior angles on the same side of the transversal are supplementary, the lines are parallel.
	library.thinkquest.org /20991/geo/parallel.html (762 words)

	Xah: Introduction to Real Projective Plane
		A special case of Desargues's Theorem is when the corresponding sides are all parallel, we say they met at infinity, and their intersections are collinear on the line at infinity.
		In affine (or Euclidean) geometry, the line p (through O) parallel to o would be exceptional, for it would have no corresponding point on o; but when we have extended the affine plane to the projective plane, the corresponding point P is just the point at infinity on o.
		In affine geometry the point X makes an infinite jump; but in projective geometry its motion, through the single point at infinity, is continuous.
	xahlee.org /projective_geometry/projective_geometry.html (6397 words)

	Geometry - Search Results - MSN Encarta
		Geometry, branch of mathematics that deals with shapes and sizes.
		Geometry may be thought of as the science of space.
		- kind of geometry: a particular system or class of geometry, e.g.
	ca.encarta.msn.com /Geometry.html (152 words)

	Parallel (geometry) - Search Results - MSN Encarta
		Parallel (geometry) - Search Results - MSN Encarta
		Parallel (geometry), in Euclidean geometry, lines that remain the same distance apart, however far they are extended in either direction.
		Prism (geometry), in geometry, three-dimensional solid, of which the bases are two parallel planes.
	uk.encarta.msn.com /Parallel_(geometry).html (104 words)

	Basic Terms
		We may think of a line as a "straight" line that we might draw with a ruler on a piece of paper, except that in geometry, a line extends forever in both directions.
		A ray is one of the basic terms in geometry.
		We say that two line segments are parallel if the lines that they lie on are parallel.
	www.mathleague.com /help/geometry/basicterms.htm (572 words)

	NonEuclid: Postulates and Proofs
		A-5S (Spherical Geometry Parallel Axiom): Given a line l and a point not on l, no lines exist that contain the point, and are parallel to l.
		In Euclidean Geometry, for any triangle ABC, there exists a unique parallel to BC that passes through point A. Additionally, it is a theorem in Euclidean Geometry that when two parallel lines are cut by a transversal, then the opposite interior angles are congruent; therefore, ∠NAB ≅ ∠ABC and ∠MAC ≅ ∠ACB.
		In Hyperbolic Geometry, however, there are an infinite number of lines that are parallel to BC and pass through point A, yet there does not exist any line such that both: ∠NAB ≅ ∠ABC and ∠MAC ≅ ∠ACB.
	www.cs.unm.edu /~joel/NonEuclid/proof.html (1392 words)

	[No title]
		Geometry offers students a means of describing, analyzing, and understanding aspects of their world.
		Geometry also focuses on the development of reasoning and proof, using definitions and axioms.
		Parallel lines have the same slope. Use properties, postulates, and theorems to determine whether two lines are parallel.
	www.pen.k12.va.us /VDOE/Instruction/Math/geometrycf.doc (2263 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-21)
		Date: 10/22/2001 at 21:44:19 From: Yj Subject: Non-Euclidean geometry I have read some of the questions and answers in the archives about non-Euclidean geometry, but can't understand all of them.
		Date: 10/23/2001 at 00:20:02 From: Doctor Schwa Subject: Re: Non-Euclidean geometry Euclidean geometry is based on several assumptions about space, one of which is: in any given plane, if you choose a line l, and a point P not on l, there is exactly one line through P that's parallel to l.
		This is usually called the "parallel postulate." Non-Euclidean geometry is based on changing one or more of the assumptions, most commonly this parallel postulate.
	mathforum.org /library/drmath/view/55372.html (139 words)

	Math Tools Browse
		Students discover relationships among the angles formed when parallel lines are intersected by a transversal.
		Euclid's proposition states that when a transversal crosses two parallel lines it makes the alternate angles equal, the alternate equal to the opposite and interior angle, and the sum of the interior...
		Euclid's proposition states that when two straight lines are parallel to the same straight line, then they are parallel to each other.
	mathforum.org /mathtools/cell/g,10.5,ALL,ALL (665 words)

	Parallel Curves, Surfaces, and Evolutes -- from Mathematica Information Center
		If gamma is a parametrically defined curve, and N is its normal, then the parallel curves are (gamma + r N) where r is a scalar.
		The evolute is the set of all the cusps in all the parallel curves.
		It is also the envelope of the lines normal to the curve.
	library.wolfram.com /infocenter/MathSource/1903 (108 words)

	GEOMETRY LINE PARALLEL [new_l] old_l dirn [dist] [refcs] [csname] [div]
		This command is used to create a parallel copy of an existing line.
		Will create a new line called LPAR23 parallel to L23, the distance between them is 1.293 in the X axis of the current coordinate system.
		Will create a line parallel to L11 offset by 3.95 in the negative Z direction of coordinate system RECT11.
	www.princeton.edu /~dynaflow/femgv/manuals/userman/node153.htm (358 words)

	intersect
		The term Parallel has a number of important meanings: Parallel (geometry) occurs in geometry.
		In Euclidean geometry if two lines or planes are parallel, then every point on one is located exactly the same minimum distance from the other line or plane.
		Parallel computing is the simultaneous execution of the same task (split up and specially adapted) on multiple processors in order to obt
	www.experiencefestival.com /intersect (669 words)

	Elliptic Geometry (Site not responding. Last check: 2007-10-21)
		This geometry then satisfies all Euclid's postulates except the 5th.
		In this document, we will examine some properties of triangles in elliptic geometry, which for our purposes will be equivalent to geometry on a...
		Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there...
	www.ellipticgeometry.info (258 words)

	Parallel Geometry (Site not responding. Last check: 2007-10-21)
		The simplest geometry, parallel, was used in first generation scanners.
		As mentioned above, the focal length is not used in this simple geometry.
		For optimal scanning in this geometry, the scan diameter should be equal to the phantom diameter.
	www.ctsim.org /manual/ctsim22.html (80 words)

	Question Corner -- Do Parallel Lines Meet At Infinity? (Site not responding. Last check: 2007-10-21)
		In a geometry like this, all lines intersect at infinity, in addition to any finite point where they might happen to meet.
		Then you can consider two parallel lines to meet at the extra point corresponding to their common direction, whereas two non-parellel lines do not intersect at infinity but intersect only at the usual finite intersection point.
		This is called projective geometry, and is described in more detail in the answer to another question.
	www.math.toronto.edu /mathnet/plain/questionCorner/infinity.html (486 words)

	Building Blocks - Words - First Glance (Site not responding. Last check: 2007-10-21)
		Geometry is about the shape and size of things.
		Like algebra, geometry has its own special vocabulary.
		These pictures show the four basic concepts on which the rest of geometry is built.
	www.math.com /school/subject3/lessons/S3U1L1GL.html (128 words)

	TN:ED:GEOMETRY
		Course Description: Geometry is a course that uses problem situations, physical models, and appropriate technology to investigate geometric concepts, relationships, and systems.
		The concepts/topics emphasized in the course include measurement, geometric patterns, coordinate geometry, two- and three-dimensional figures, transformational geometry, congruence, and similarity.
		On one side of a sheet of plain paper give a short history of the building including when it was built, who designed it, and where it is located.
	www.state.tn.us /education/ci/cigateendofcourse/geometry2.htm (2026 words)

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