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Topic: Tangent (geometry)

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	PlanetMath: geometry
		Geometry, or literally, the measurement of land, is among the oldest and largest areas of mathematics.
		The discovery of intrinsic geometry led thoughtful geometers such as Riemann (who was a student of Gauss), Clifford, and Mach to the conclusion that a “right and natural” approach to geometry should regard surfaces as geometrical spaces in their own right on a par with Euclidean and projective space.
		Once geometric notions like tangent spaces and curvature have been defined for manifolds (we shall indicate how this is done in the next section) then one can speak of such things as the tangent space to the set of solutions of a differential equation or the curvature of a group.
	planetmath.org /encyclopedia/Geometry.html (4303 words)

	Differential geometry and topology - Wikipedia, the free encyclopedia
		Initially and up to the middle of the nineteenth century, differential geometry was studied from the extrinsic point of view: curves, surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions).
		One definition of the tangent space is as the dual space to the linear space of all functions which are zero at that point, divided by the space of functions which are zero and have a first derivative of zero at that point.
		A vector field is a function from a manifold to the disjoint union of its tangent spaces (this union is itself a manifold known as the tangent bundle), such that at each point, the value is an element of the tangent space at that point.
	en.wikipedia.org /wiki/Differential_topology (1398 words)

	tangent - Search Results - MSN Encarta
		Tangent (geometry), a line, line segment, or ray that touches a circle at one and only one point.
		Tangent (trigonometry), one of the six fundamental ratios of trigonometry.
		Arc Tangent, in trigonometry, the inverse of the tangent function.
	encarta.msn.com /encnet/refpages/search.aspx?q=tangent (184 words)

	Tangent - Wikipedia, the free encyclopedia
		In plane geometry, a straight line is tangent to a curve, at some point, if both line and curve pass through the point with the same direction; such a line is the best straight-line approximation to the curve at that point.
		It is tangent to the curve at the point indicated by the dot.
		Above, it was noted that a secant can be used to approximate a tangent; it could be said that the slope of a secant approaches the slope (or direction) of the tangent, as the secants' points of intersection approach each other.
	en.wikipedia.org /wiki/Tangent (657 words)

	Tangent - Search Results - MSN Encarta
		Tangent (geometry), a line, line segment, or ray that touches a circle at one and only one point.
		Tangent (trigonometry), one of the six fundamental ratios of trigonometry.
		Arc Tangent, in trigonometry, the inverse of the tangent function.
	ca.encarta.msn.com /Tangent.html (129 words)

	PlanetMath: tangent plane (elementary)
		Just as the tangent line is a special line which is associated to a point of a smooth curve, so too a tangent plane is a special plane which is associated to a point on a smooth surface.
		This is a generalization to three dimensions of the well-known fact of plane geometry that the tangent to a circle through a point on the circle is perpendicular to the radius through that point.
		Just as the tangent to a line may be understood as the limit of a line connecting two nearby points in the limit where the points coalesce, so too the tangent plane can be regarded as a limit.
	planetmath.org /encyclopedia/TangentPlane.html (1877 words)

	GEOMETRY - LoveToKnow Article on GEOMETRY (Site not responding. Last check: 2007-10-10)
		Pythagoras (q.v.), seeking the key of the universe in arithmetic and geometry, investigated logically the principles underlying the, known propositions; and this resulted in the formulation of definitions, axioms and postulates which, in addition to founding a science of geometry, permitted a crystallization, fractional, it is true, of the amorphous collection of material at hand.
		Pythagorean geometry was essentially a geometry of areas and solids; its goal was the regular solids the tetrahedron, cube, octahedron, dodecahedron and icosahedronwhich symbolized the five elements of Greek cosmology.
		The geometry of the circle,, previously studied in Egypt and much more seriously by Tbales, was somewhat neglected, although this curve was regarded as the most perfect of all plane figures and the sphere the most perfect of all solids.
	www.1911encyclopedia.org /G/GE/GEOMETRY.htm (21277 words)

	Line Segment (geometry) - Search Results - MSN Encarta
		Line Segment (geometry), portion of a line consisting of two points, called endpoints, and all of the points that lie between these endpoints.
		Chord (geometry), a line segment whose endpoints lie on a curve or on a circle.
		Secant (geometry), a line, line segment, or ray that intersects a circle or other curve at two and only two points.
	encarta.msn.com /Line_Segment_(geometry).html (199 words)

	Encyclopedia :: encyclopedia : Algebraic geometry (Site not responding. Last check: 2007-10-10)
		Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry.
		While projective geometry was originally established on a synthetic foundation, the use of homogeneous coordinates allowed the introduction of algebraic techniques.
		Algebraic geometry was developed largely by the Italian geometers in the early part of the 20th century.
	www.hallencyclopedia.com /Algebraic_geometry (1764 words)

	Differential geometry and topology - Gurupedia
		At every point of the manifold, there is the tangent space at that point, which consists of every possible velocity (direction and magnitude) with which it is possible to travel away from this point.
		One definition of the tangent space is as the dual space to the linear space of all functions which are zero at that point, divided by the space of functions which are zero and have a first derivative of zero at that point.
		A vector field is a function from a manifold to the disjoint union of its tangent spaces (this union is itself a manifold known as the tangent bundle), such that at each point, the value is an element of the tangent space at that point.
	www.gurupedia.com /d/di/differential_geometry.htm (938 words)

	Analytic Geometry (Site not responding. Last check: 2007-10-10)
		Tangents to conic sections: A line intersects an ellipse at exactly one point iff it is tangent to the ellipse.
		Billiards in a circle: the angle with the tangent is constant, an orbit is periodic iff this angle is a rational multiple of π, and otherwise the impact points are dense on the boundary of the billiard, Kroneckers' theorem on density of orbits of an irrational rotation.
		If a trajectory segment does not intersect the major axis segment then is tangent to an ellipse confocal with the original one and all other segments of the trajectory remain tangent to that ellipse.
	www.math.tau.ac.il /~rudnick/courses/analyticgeomsyll.html (520 words)

	tangent. The Columbia Encyclopedia, Sixth Edition. 2001-05
		For other curves and surfaces the tangent line at a given point P is defined as the limiting position, if such a limit exists, of a secant line through P and another point P´ on the curve or surface as P´ is allowed to approach P.
		The tangent plane to a surface at a point is the plane in which every line in the plane that passes through the point is a tangent line to the surface at that point.
		The study of tangent lines and planes usually requires the concepts of the calculus and is included within the scope of differential geometry.
	www.bartleby.com /65/ta/tangent.html (183 words)

	Geometry Seminar Lecture 2
		A projective geometry is a geometric structure consisting of various types of objects (points, lines, planes, etc.) and the relations between them which satisfies a set of axioms.
		It is easily seen from this construction that the points of the geometry have a natural relation to coordinates, namely we can take as coordinates of a point any non-zero vector written as an (n+1)-tuple in the rank 1 subspace corresponding to that point.
		The tangent space at P is therefore this line (which is a hyperplane in a projective plane), so the oval satisfies condition 2 and is therefore a non-degenerate quadratic set.
	www-math.cudenver.edu /~wcherowi/geom/gsln2.html (1902 words)

	Highbeam Encyclopedia - Search Results for tangent (Site not responding. Last check: 2007-10-10)
		For other curves and surfaces the tangent line at a given point P is defined as the limiting position, if such a limit exists, of a secant line through...
		tangent In trigonometry, ratio of the length of the side opposite an acute angle to the length of the side adjacent to it within a right-angle triangle.
		In geometry, a line that intersects a circle exactly once; in calculus, a line that touches a curve at one point and whose slope is equal to that of the curve at that point.
	www.encyclopedia.com /searchpool.asp?target=tangent (747 words)

	Differential geometry and topology
		Differential geometry is the study of geometry using calculus.
		The apparatus of differential geometry is that of calculus on manifolds: this includes the study of manifolds, tangent bundles, cotangent bundles, differential forms, exterior derivatives,integrals of p-forms over p-dimensional submanifolds and Stokes' theorem, wedge products, and Lie derivatives.
		Finsler geometry has the Finsler manifold as the main object of study — this is a differential manifold with a Finsler metric, i.e.
	www.knowledgefun.com /book/d/di/differential_geometry_and_topology.html (988 words)

	Geometry Software Programs Full Year Course
		Geometry help and geometry formulas are available from a clickable menu of choices.
		Geometry help is available for each geometry problem with a click on the HINT buttons.
		Geometry formulas and geometry proofs are explained and practiced with interactive geometry help.
	www.mathmedia.com /higschoolgeo.html (747 words)

	Converting Magazine - Factors to consider in choosing shear blade profiles (Site not responding. Last check: 2007-10-10)
		Wrap versus tangent slitter geometry is very important in deciding which blade profile to use.
		Tangent slitting does not restrict free deflection of the web as it passes through the slitters, because there is greater available space between adjacent lower slitter rings.
		Almost any material can be tangent slit, but especially in the case of rigid webs, a tangent slitting system is the method of choice.
	www.convertingmagazine.com /cgi-bin/searcharch.cgi?view=9_99_66 (1594 words)

	All Elementary Mathematics - Study Guide - Geometry - Tangent plane of a ball, a cylinder and a cone...
		Tangent plane of a ball, a cylinder and a cone
		The plane P, which is a tangent plane of a spherical surface (Fig.95), is perpendicular to radius OA, drawn to the tangency point A; a tangent plane of a spherical surface has only one common point with the surface – a tangency point.
		A cone is called an inscribed into a pyramid, if lateral faces of the pyramid are planes, tangent to the cone, and planes of their bases are the same.
	www.bymath.com /studyguide/geo/sec/geo19.htm (285 words)

	[No title]
		There are actually four lines tangent to the two circles, two of which you want, and two of which would result from crossing the belt into a figure-8.
		This occurs a distance L from the smaller circle centre and distance M from the point where the tangent touches the smaller circle.
		To get coordinates, compute L then sin(theta)= A/L. This would allow the angle between the radii to the tangents and the line joining the centres to be determined and hence the coordinates of the tangent points.
	www.math.niu.edu /~rusin/known-math/96/2tangents (382 words)

	Golden ratio - Wikipedia, the free encyclopedia
		The golden ratio was first studied by ancient mathematicians because of its frequent appearance in geometry.
		It has been claimed that the ancient Egyptians knew the golden ratio because ratios close to the golden ratio may be found in the positions or proportions of the Pyramids of Giza, but most likely, it was not until the Ancient Greeks that the Golden Ratio was fully understood and used.
		Laputan Logic The Cult of the Golden Ratio "The Golden Ratio, once a pristine jewel of geometrical truth and simplicity, has become a deity for a cult of hyperlinking headnodders whose chief devotional practice seems to be to handwave their way from one disconnected and unexamined falsehood to another."
	en.wikipedia.org /wiki/Golden_ratio (2572 words)

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