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Topic: Tautochrone curve

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	Curve article - Curve magazine called Curve mathematics circle straight line curves - What-Means.com (Site not responding. Last check: 2007-09-18)
		A plane curve is a curve for which X is the mathematical plane — these are the examples first encountered — or in some cases the projective plane.
		A space curve is a curve for which X is of three dimensions, usually Euclidean space; a skew curve is a space curve which lies in no plane.
		A rectifiable curve is a curve with finite length.
	www.what-means.com /encyclopedia/Curve (1377 words)

	Curve
		Cubic curve In mathematics, a cubic curve is a plane curve C defined by a cubic equation F(X,Y,Z) = 0 in X,Y, and Z. The...
		Koch curve of the Koch snowflake]] The Koch curve is a snowflake.
		Tautochrone curve A tautochrone curve is the curve for which the time taken by a particle sliding down it under uniform...
	www.brainyencyclopedia.com /topics/curve.html (1028 words)

	Encyclopedia: Curve
		A Brachistochrone curve, or curve of fastest descent, is the curve between two points that is covered in the least time by a body that starts at the first point with zero speed and passes down along the curve to the second point, under the action of constant gravity and...
		Cycloid (red) generated by a rolling circle A cycloid is the curve defined by a fixed point on a wheel as it rolls, or, more precisely, the locus of a point on the rim of a circle rolling along a straight line.
		In mathematics, a cubic curve is a plane curve C defined by a cubic equation F(X,Y,Z) = 0 applied to homogeneous coordinates [X:Y:Z] for the projective plane; or the inhomogeneous version for the affine space determined by setting Z = 1 in such an equation.
	www.nationmaster.com /encyclopedia/Curve (3519 words)

	Tautochrone curve - Encyclopedia, History, Geography and Biography
		A tautochrone curve is the curve for which the time taken by a particle sliding down it under uniform gravity to its lowest point is independent of its starting point.
		The tautochrone problem, the attempt to identify this curve, was solved by Huygens in 1659.
		This solution was later used to attack the problem of the brachistochrone curve.
	www.arikah.net /encyclopedia/Tautochrone (161 words)

	Brachistochrone curve - Wikipedia, the free encyclopedia
		A Brachistochrone curve, or curve of fastest descent, is the curve between two points that is covered in the least time by a body that starts at the first point with zero speed and passes down along the curve to the second point, under the action of constant gravity and ignoring friction.
		Hence, the brachistochrone curve is simply the trajectory of a beam of light in a medium where the speed of light increases following a constant vertical acceleration (that of gravity g).
		Hence, the brachistochrone curve is tangent to the vertical at the origin.
	en.wikipedia.org /wiki/Brachistochrone_curve (623 words)

	Brachistochrone curve (Site not responding. Last check: 2007-09-18)
		A Brachistochrone curve, or curve of fastest descent, is the curve between two points that is covered in the least time by a body that translates along it under the action of constant gravity.
		Galileo incorrectly stated in 1638 in his Discourse on two new sciences that this curve was an arc of a circle.
		The brachistochrone curve was proved to be a cycloid.
	www.uncover.us /en/wikipedia/b/br/brachistochrone_curve.html (232 words)

	Curve definitions (Site not responding. Last check: 2007-09-18)
		Pedal curve : Given a curve C then the pedal curve of C with respect to a fixed point O (called the pedal point) is the locus of the point P of intersection of the perpendicular from O to a tangent to C.
		Tautochrone : A curve down which a particle acted on by a force will traverse the distance to the lowest point in the curve in a fixed time independent of the starting position.
		Transcendental curve : A curve of the form f(x,y) = 0 where f(x,y) is not a polynomial in x and y.
	www-groups.dcs.st-and.ac.uk /~history/Curves/Definitions2.html (1325 words)

	Tautochrone curve -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-18)
		A tautochrone curve is the curve for which the time taken by a particle sliding down it under uniform (A solemn and dignified feeling) gravity to its lowest point is independent of its starting point.
		The tautochrone problem, the attempt to identify this curve, was solved by (Dutch physicist who first formulated the wave theory of light (1629-1695)) Huygens in 1659.
		He proved geometrically in his Horologium oscillatorium (The Pendulum Clock, 1673) that the curve was a (A line generated by a point on a circle rolling along a straight line) cycloid.
	www.absoluteastronomy.com /encyclopedia/T/Ta/Tautochrone_curve.htm (150 words)

	Cycloid - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-09-18)
		A cycloid is the curve defined by a fixed point on a wheel as it rolls, or, more precisely, the locus of a point on the rim of a circle rolling along a straight line.
		it is the curve of fastest descent under gravity) and the related tautochrone problem (i.e.
		In the former case the point tracing out the curve is inside the circle and in the latter case it is outside.
	www.bucyrus.us /project/wikipedia/index.php/Cycloid (451 words)

	Curves
		Curves drawn according to some rule are more interesting than arbitrary curves, and have attracted the attention of philosophers since early times.
		It is more natural to define a curve for our present purposes as the locus of a moving point than as a set of points, since the moving point emphasizes the connectivity and one-dimensional nature of a curve.
		When the curve is taken at the design speed, the resultant of gravity and centrifugal forces are normal to the track so that the curve is scarcely perceived and the movement is comfortable.
	www.du.edu /~etuttle/math/curves.htm (5870 words)

	cycloid
		Galilei (who gave the curve its name in 1699) stated in 1638 (falsely) that the brachistochrone has to be the arc of a circle.
		As a matter of fact, this curve is the opposite (mirroring in the x-axis) of the shown curve.
		The latter curve is followed by the valve of a bike.
	www.2dcurves.com /roulette/roulettec.html (1239 words)

	Tautochrone curve - Wikipedia, the free encyclopedia
		A tautochrone or isochrone curve is the curve for which the time taken by a particle sliding down it under uniform gravity to its lowest point is independent of its starting point.
		The time is equal to π times the square root of the radius over the gravitation constant.
		The tautochrone problem, the attempt to identify this curve, was solved by Christiaan Huygens in 1659.
	en.wikipedia.org /wiki/Tautochrone_curve (139 words)

	Cycloid (Site not responding. Last check: 2007-09-18)
		Mersenne gave the first proper definition of the cycloid and stated the obvious properties such as the length of the base equals the circumference of the rolling circle.
		This is the tautochrone property and was discovered by Huygens in 1673 (Horologium oscillatorium).
		In fact the evolute was studied by Huygens and because of his work on the cycloid Huygens developed a general theory of evolutes of curves.
	www-history.mcs.st-and.ac.uk /history/Curves/Cycloid.html (788 words)

	The Helen of Geometers
		The curve traced out by a point on the rim of a rolling circle is called a cycloid, and we've seen that this curve described gravitational free-fall, both in Newtonian mechanics and in general relativity (in terms of the free-falling proper time).
		Mersenne publicized the cycloid among his group of correspondents, including the young Roberval, who, by the 1630's had determined many of the major properties of the cycloid, such as the interesting fact that the area under a complete cycloidal arch is exactly three times the area of the rolling circle.
		Thus he had discovered that the cycloid is the tautochrone, i.e., the curve for which the time taken by a particle sliding from any point on the curve to the lowest point on the curve is the same, independent of the starting point.
	www.mathpages.com /rr/s8-03/8-03.htm (1364 words)

	The History of Curvature (Site not responding. Last check: 2007-09-18)
		For example, a curve that has constant curvature must be part or all of a circle (for these are the only curves that have the same amount of bending at every point).
		As he noted, curves behave like a straight line near a point of inflection (they don’t bend either way), and since the radius of curvature of a straight line is infinite, the radius of curvature at these points is infinite.
		He stated that the degree of curvature is equivalent to the second fluxion of the curve, and this is a measure of "deflection from [the] tangent" (Simpson 53).
	www.brown.edu /Students/OHJC/hm4/k.htm (3531 words)

	Euler-Lagrange Equation History Summary (Site not responding. Last check: 2007-09-18)
		curves · leonhard euler ·; joseph louis · swiss mathematician · italian mathematician · stationary · mid 1700s · initial position · partial derivative · beltrami · euler lagrange ·; lagrange equations
		During the early 1750s Lagrange began studying the tautochrone, the curve on which a weighted particle always arrives at a fixed point in the same time regardless of its initial position.
		His discoveries concerning this curve were substantial to the calculus of variations.
	www.bookrags.com /history/mathematics/euler-lagrange-equation-wom (506 words)

	Curve (Site not responding. Last check: 2007-09-18)
		A closed curve is thus a continuous mapping of the circle
		A plane curve is a curve for which X is the mathematical plane - these are the examples first encountered - or in some cases the projective plane.
		If X is a differentiable manifold, then we can define the notion of differentiable curve in X. This general idea is enough to cover many of the applications of curves in mathematics.
	www.yotor.com /wiki/en/cu/Curve.htm (1178 words)

	[No title]
		Every position on this sought after curve, was determined by a principle of change.
		The proportion between the vertical and the horizontal, dy:dx, and the resulting change in the path, dz, is a function of the rate at which the density of the medium is changing.
		For, as nature is accustomed to proceed always in the simplest fashion, so here she accomplishes two different services through one and the same curve, while under every other hypothesis two curves would be necessary the one for oscillations of equal duration the other for quickest descent.
	www.wlym.com /antidummies/part09.html (1561 words)

	Weird Words: Tautochrone
		Huygens discovered that there is one curved shape, and only one, which is perfect in this respect: the cycloid, the curve traced out when a point on the edge of a wheel rolls along a road.
		If you position a cycloidal curve like an inverted arch, and release a marble from any point on it, it will always take exactly the same time to reach the bottom, no matter where on the curve you start from.
		Huygens used this discovery to construct curved jaws from the point of support of the pendulum; these forced its string to follow the right curve no matter how large or small the swing.
	www.worldwidewords.org /weirdwords/ww-tau1.htm (268 words)

	[No title] (Site not responding. Last check: 2007-09-18)
		The initial question was whether or not it was physically possible for an object to start at any arbitrary place a long a curve and to arrive at the end of the curve at the same time.
		Moreover, if an individual is to place two identical balls on a curve at the same time they should simultaneously collide at then end of the curve.
		So, from this we can induce that there has to be a curve where the ball would have to meet the end point at an equal time no matter where the ball started.
	users.rowan.edu /~christ36/honor_cal/tautochrone.doc (1296 words)

	Brachistochrone curve -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-18)
		Given two points A and B, with A not lower than B, there is just one upside down (A line generated by a point on a circle rolling along a straight line) cycloid that passes through A with infinite slope and also passes through B.
		The problem can be solved with the tools from the (The calculus of maxima and minima of definite integrals) calculus of variations.
		(Swiss mathematician (1667-1748)) Johann Bernoulli solved the problem (by reference to the previously analysed (Click link for more info and facts about tautochrone curve) tautochrone curve) before posing it to readers of Acta Eruditorum in June 1696.
	www.absoluteastronomy.com /encyclopedia/B/Br/Brachistochrone_curve.htm (398 words)

	cycloid
		In 1658 Christopher Wren showed that the length of a cycloid is four times the diameter of its generating circle.
		The cycloid is the solution to the brachistochrone problem and the related tautochrone problem.
		The cycloid has been called "The Helen of Geometers" as it caused frequent quarrels among 17th century mathematicians.
	www.fact-library.com /cycloid.html (217 words)

	[No title]
		By the end of 1754 he had made some important discoveries on the tautochrone which would contribute substantially to the new subject of the
		Lagrange sent Euler his results on the tautochrone containing his method of maxima and minima.
		The ordinary operations of algebra suffice to resolve problems in the theory of curves.
	www.resonancepub.com /lagrangian.htm (4349 words)

	[No title] (Site not responding. Last check: 2007-09-18)
		A cycloid is a curve traced by a point on a circle of radius a as the circle rolls along some fixed line and is denoted by the following parametric equations: EMBED Equation.DSMT4 .
		The curve described as an inverted cycloid occurs when a is negative (that is, below the x-axis).
		The tautochrone property of inverted cycloids states objects placed at different points on an inverted cycloid and allowed to free fall will arrive at the lowest point of the inverted cycloid at the same time.
	members.cox.net /alsganga/Trig-MathAnalysis_files/09.07.doc (572 words)

	johbern (Site not responding. Last check: 2007-09-18)
		He studied many topics including: reflection and refraction of light, orthogonal trajectories of families of curves, quadrature of areas by a series, and the brachystochrone (curve of the quickest descent of a weighted particle moving between two points in a gravitational field).
		The method used in the calculus of variations show that the minimising curve is cycloid.
		He also studied the tautochrone (the curve on which a weighted particle will arrive at a fixed point in the same time independent of its initial position).
	www.forestcity.k12.ia.us /pages/FCHS/Site/johbern.htm (258 words)

	Parameterizations of Curves (Site not responding. Last check: 2007-09-18)
		A parameterization of a plane curve is a map x: I --> R
		The cycloid is the path traced by one point of a circle rolling on a line.
		It is also the solution to the tautochrone problem and the brachystochrone problem.
	www.brown.edu /Students/OHJC/hm4/param.htm (148 words)

	Encyclopedia: Tautochrone curve
		The time is equal to the Pi times square root of the radius over the gravitation contant.
		Gravitation is the tendency of masses to move toward each other.
		Christiaan Huygens Christiaan Huygens (pronounced in English (IPA): ; in Dutch:) (April 14, 1629– July 8, 1695), was a Dutch mathematician and physicist; born in The Hague as the son of Constantijn Huygens.
	www.nationmaster.com /encyclopedia/Tautochrone-curve (388 words)

	Tautochrone Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-09-18)
		Looking For tautochrone - Find tautochrone and more at Lycos Search.
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	www.karr.net /encyclopedia/Tautochrone (301 words)

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