
	Where results make sense

Topic: Brachistochrone curve

Related Topics

Brachistochrone

Tautochrone curve

Roulette (curve)

Cycloid

Calculus of variations

Heuristic (computer science)

Formal semantics

Hypercomputation

Algorithm

Mathematical logic

Timeline of mathematics

Algorithmic information theory

Computer science

Brachistochrone

In the News (Mon 21 Dec 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	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 (632 words)

	Tautochrone curve - Wikipedia, the free encyclopedia
		A tautochrone or isochrone curve is the curve for which the time taken by a frictionless particle sliding down it under uniform gravity to its lowest point is independent of its starting point.
		The curve is a cycloid, and the time is equal to π times the square root of the radius over the acceleration of gravity.
		This solution was later used to attack the problem of the brachistochrone curve.
	en.wikipedia.org /wiki/Tautochrone_curve (835 words)

	Curve definitions
		The orthoptic of a parabola is its directrix, the orthoptic of a central conic is a circle concentric with the conic which was investigated by Monge.
		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.
		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)

	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: Encyclopedia topic (Site not responding. Last check: 2007-10-12)
		The upside down cycloid is the solution to the brachistochrone problem (brachistochrone problem: a brachistochrone curve, or curve of fastest descent, is the curve between two points that...
		it is the curve of fastest descent under gravity) and the related tautochrone problem (tautochrone problem: a tautochrone curve is the curve for which the time taken by a particle sliding down it under...
		All these curves are roulettes (roulettes: A line generated by a point on one figure rolling around a second figure) with a circle rolled along a uniform curvature (curvature: The property possessed by the curving of a line or surface).
	www.absoluteastronomy.com /reference/cycloid (665 words)

	Tautochrone curve
		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.
		He proved geometrically in his Horologium oscillatorium (1673) that the curve was a cycloid.
	www.ebroadcast.com.au /lookup/encyclopedia/ta/Tautochrone_curve.html (107 words)

	Xah: Special Plane Curves: Naming and Classification of Curves
		Brachistochrone (from Greek, brakhus:short, chrone:time) means a curve that connects to two given points such that a particle sliding from the higher point to the lower point under ideal physical law (ideal gravitational force, no friction, no air-resistance, particle has no volume...etc.) will descent with the fastest time, among all possible curves.
		It's a curve that, connects two given points such that it takes the same amount of time for a particle to slide from any point on the curve to the lower point, under ideal physical law.
		Isoptic of a given curve C and a given angle α is the locus of a point P such that P is the intersection of tangents of C that meets in angle α.
	www.xahlee.org /SpecialPlaneCurves_dir/Intro_dir/familyIndex.html (1048 words)

	Brachistochrone problem
		The brachistochrone problem was posed by Johann Bernoulli in Acta Eruditorum in June 1696.
		Now Huygens had shown in 1659, prompted by Pascal's challenge about the cycloid, that the cycloid is the solution to the tautochrone problem, namely that of finding the curve for which the time taken by a particle sliding down the curve under uniform gravity to its lowest point is independent of its starting point.
		The first problem of this type [calculus of variations] which mathematicians solved was that of the brachistochrone, or the curve of fastest descent, which Johann Bernoulli proposed towards the end of the last century.
	www-gap.dcs.st-and.ac.uk /~history/HistTopics/Brachistochrone.html (1969 words)

	Brachistochrone curve: Encyclopedia topic (Site not responding. Last check: 2007-10-12)
		Given two points A and B, with A not lower than B, there is just one upside down cycloid (cycloid: A line generated by a point on a circle rolling along a straight line) that passes through A with infinite slope and also passes through B.
		Johann Bernoulli (Johann Bernoulli: Swiss mathematician (1667-1748)) solved the problem (by reference to the previously analysed tautochrone curve (tautochrone curve: a tautochrone curve is the curve for which the time taken by a particle sliding down it under...
		Tautochrone curve (Tautochrone curve: a tautochrone curve is the curve for which the time taken by a particle sliding down it under...
	www.absoluteastronomy.com /reference/brachistochrone_curve (445 words)

	cycloid (Site not responding. Last check: 2007-10-12)
		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)

	1-5.html (Site not responding. Last check: 2007-10-12)
		Determine the parametrization of the deltoid and adapt the Maple commands for the astroid to demonstrate the deltoid as it is traced by a rolling circle.
		What is the parametrization of a curve traced out by a point on the edge of a circle of radius 1 rolling around the
		Develop a Maple worksheet that demonstrates the curve in much the same way that the cycloid and astroid were presented above.
	faculty.etsu.edu /knisleyj/multicalc/Chap1/Chap1-5/1-56.html (238 words)

	Abstract (Site not responding. Last check: 2007-10-12)
		In this study we discuss the quantum dynamics of a particle, which moves classically on the brachistochrone curve corresponding to the minimization of the time functional, in a linear gravity potential.
		We derive the Lagrangian and the Hamiltonian of the particle, which moves also on the brachistochrone curve by the minimization of the action functional.
		We show that the band structure arised from Floquet theory and the problem is equivalent to the periodic delta- potential problem for the particle with positive energy in the limit of infinite potential.
	www.physik.uni-frankfurt.de /asi/posters/abs_yucel.html (140 words)

	AMCA: Brachistochrones Under Central Forces by Garry J. Tee
		Johann Bernoulli 1st proved in 1696 that the brachistochrone (curve of quickest descent) under uniform gravity is the cycloid.
		The bounded and unbounded brachistochrones are separated by a critical brachistochrone, which is bounded in radius.
		For inverse square forces the brachistochrones are constructed in terms of elliptic integrals — except that (for repulsion) the critical brachistochrone is constructed in terms of elementary functions.
	at.yorku.ca /c/a/d/l/23.htm (186 words)

	TPS Archives
		A cycloid curve is the path followed by a point on a wheel that rolls along a straight line (fig.
		The straight traverses and cycloid curves for three different distances were marked in the snow with colored dye.
		In each of the runs that produced a shorter time on the traverses rather than on the curve, we observed considerable skidding of the skis instead of the desired carving.
	www.psia.org /psia_2002/education/TPSArticles/coaching/tpsfall96cycloids.asp (1111 words)

	funiculartwo
		The curve is not a catenary because the speed of the water is greater near the middle than it is near the edges.
		The blue curve represents the rope, and the red curve the tension, which is complemented by the curve of kinetic energy.
		Here is another type of natural curve, the parabolic surface of water in a rotating container, the view of which is of course distorted by the refraction of light at the edge of the cylinder (isn't it?).
	www.brantacan.co.uk /funiculartwo.htm (6254 words)

	iqexpand.com (Site not responding. Last check: 2007-10-12)
		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.
		All these curves are roulettes with a circle rolled along a uniform curvature.
	cycloid.iqexpand.com /index.php?title=Roulette_(curve)&action=edit (636 words)

	The Brachistochrone
		The brachistochrone problem is a seventeenth century exercise in the calculus of variations.
		The brachistochrone is a cycloid, but that cycloid is not the only curve satisfying the equation.
		Neither curve can have a discontinuous derivative (which is what that angle point discussion was all about), so neither curve can slope upward until after the derivative has first reached zero.
	whistleralley.com /brachistochrone/brachistochrone.htm (2469 words)

	Visualizing the Brachistochrone Problem -- from Mathematica Information Center
		The brachistochrone problem is one of the most famous in analysis.
		First posed by Johann Bernoulli in 1696, the problem consists of finding the curve that will transport a particle most rapidly from one point to a second not directly below it, under the force of gravity only.
		The commands allow the introduction of friction, and there is a command that allows one to animate a "race" of particles dropping down two competing curves.
	library.wolfram.com /infocenter/MathSource/1897 (127 words)

	parnovsky
		Last year there was the 300th anniversary of solving the problem of curve of the fastest travel.
		On this basis a differential equation of a brachistochrone is built and solved in the next section of this article.
		(6), (8) define the brachistochrone equation in a parametrical representation.
	info.ifpan.edu.pl /firststep/aw-works/fsV/parnovsky (1846 words)

	The cycloid curve (Site not responding. Last check: 2007-10-12)
		As is well-known, the cycloid is the curve described by a point P rigidly attached to a circle C that rolls, without sliding, on a fixed line AB (fig.
		a curve of least time: given two points A, B in a vertical plane, a heavy point will take the least time to travel from A to B if it is displaced along an arc of a cycloid.
		A heavy point which travels along an arc of cycloid placed in a vertical position with the concavity pointing upwards will always take the same amount of time to reach the lowest point, independent of the point from which it was released.
	galileo.imss.firenze.it /multi/torricel/ecicloid.html (150 words)

	The Brachistochrone
		To insure a smooth transition of motion, we shall assume that these line segments are connected by means of a curved joint instead of the sharp corner where they intersect.
		Because we have no information about the vertical range of the solution curve before we do the computation, specifying the y-coordinates of the endpoints of our line segments, such as we did with this problem, is not advisable.
		Example 3: Let us approximate the shape of the brachistochrone from the point (0 meters, 2 meters) to the point (2 meters, 1 meter) by representing the path of minimum time as four line segments, but with no specifications on either the x- or y-coordinates of the endpoints of the line segments.
	www.mathcs.emory.edu /~fox/NewCCS/ModuleII/ModIIP9.html (2011 words)

	MAXIMA AND MINIMA - LoveToKnow Article on MAXIMA AND MINIMA
		Serenus of Antissa investigated the somewhat trifling problem of finding the triangle of greatest area whose sides are formed by the intersections with the base and curved surface of a right circular cone of a plane drawn through its vertex.
		The next problem on maxima and minima of which there appears to be any record occurs in a letter from Regiomontanus to Roder (July 4, 1471), and is a particular numerical example of the problem of finding the point on a given straight line at which two given points subtend a maximum angle.
		John Bernoullis famous problem of the brachistochrone, or curve of quickest descent from one point to another under the action of gravity, proposed in 1696, gave rise to a new kind of maximum and minimum problem in which we have to find a curve and not points on a given curve.
	www.1911encyclopedia.org /M/MA/MAXIMA_AND_MINIMA.htm (2366 words)

	Brachistochrone (Site not responding. Last check: 2007-10-12)
		The center track is a brachistochrone, the curve along which a particle moving under the influence of gravity alone will traverse the distance between the endpoints in minimum time.
		Another unusual property of the brachistochrone is that the time required to reach the bottom of the track is independent of the starting point on the track.
		This can be shown by releasing two balls from different positions on the brachistochrone simultaneously and observing that they reach the bottom together.
	physicsnt.clemson.edu /physdemo/1/demos/Dbrachi2.htm (126 words)

	13.4.1.1 Calculus of variations
		Lagrangian mechanics is based on the calculus of variations, which is the subject of optimization over a space of paths.
		One of the most famous variational problems involves constraining a particle to travel along a curve (imagine that the particle slides along a frictionless track).
		The problem is to find the curve for which the ball travels from one point to the other, starting at rest, and being accelerated only by gravity.
	msl.cs.uiuc.edu /planning/node698.html (594 words)

	The Brachistochrone
		The National Curve Bank also has MAPLE animations of the cycloid family of curves.
		The brachistochrone is for "the shrewdest mathematicians of all the world.
		Johnson and others suggest the Euler-Lagrange formula and boundary conditions applied to the brachistochrone will define a differential equation whose solution is similar to finding critical points (usually a maximum or minimum) in ordinary calculus.
	curvebank.calstatela.edu /brach/brach.htm (662 words)

	National Curve Bank: A Math Archive
		The National Curve Bank is a resource for students of mathematics.
		We also include geometrical, algebraic, and historical aspects of curves, the kinds of attributes that make the mathematics special and enrich classroom learning.
		Please see "Submit Your Curve" on the left for details.
	curvebank.calstatela.edu /home/home.htm (109 words)

	Project Page (Site not responding. Last check: 2007-10-12)
		This semester's project is on the fastest curve, the Brachistochrone.
		Your project is to write several pages on the Brachistochrone and it should cover the material you understand and not attempt to copy something you don't understand.
		Comparing travel time for several curves, like the straight line, circle, cycloid, a bent line.
	www.math.fsu.edu /~bellenot/class/f04/cal3/project.html (252 words)

Try your search on: Qwika (all wikis)