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Topic: Epicycloid

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	Epicycloid - LoveToKnow 1911
		The epicycloid was so named by Ole Romer in 1674, who also demonstrated that cog-wheels having epicycloidal teeth revolved with minimum friction (see Mechanics: Applied); this was also proved by Girard Desargues, Philippe de la Hire and Charles Stephen Louis Camus.
		Epicycloids also received attention at the hands of Edmund Halley, Sir Isaac Newton and others; spherical epicycloids, in which the moving circle is inclined at a constant angle to the plane of the fixed circle, were studied by the Bernoullis, Pierre Louis M. de Maupertuis, Francois Nicole, Alexis Claude Clairault and others.
		The tangential polar equation to the epicycloid, as given above, is p= (a+2b) sin (a a+2b),I', while the intrinsic equation is s=4(bla)(a+b) cos (ala+2b)>G and the pedal equation is r2=a2+ (4b.a+b)p 2 l(a+2b).
	www.1911encyclopedia.org /Epicycloid (708 words)

	Caustic - LoveToKnow 1911
		an epicycloid in which the radii of the fixed and rolling circles are equal.
		remaining circular, the question can be similarly treated, and it is found that the caustic is an epicycloid in which the radius of the fixed circle is twice that of the rolling circle (fig.
		In both cases the curves are epicycloids; in the first case the radii of the rolling and the fixed circles are a(2n - I) /4n and a/2n, and in the second, an/(2n+ I) and a/(2n4-I), where a is the radius of the mirror and n the number of reflections.
	www.1911encyclopedia.org /Caustic (861 words)

	Epicycloid and Hypocycloid
		Epicycloid is a special case of epitrochoid, and hypocycloid is a special case of hypotrochoid.
		We define the vertexes of the epicycloid to be points on the curve that coincides with a circumscribed circle.
		The epicycloid with rolling circle radius b is equal to the hypocycloid with rolling circle radius b+1.
	www.xahlee.org /SpecialPlaneCurves_dir/EpiHypocycloid_dir/epiHypocycloid.html (1001 words)

	The Epicycloid
		The epicycloid is like a cycloid on the circumference of a circle.
		It has as special cases the cardioid (when b = a in the definition below) and the nephroid (when b = a/2).
		Geometrically, the epicycloid is traced by a point on the radius of a small circle (of radius b) rolling on the edge of a larger circle (of radius a), as shown below.
	www.math.hmc.edu /~gu/curves_and_surfaces/curves/epicycloid.html (74 words)

	Vehicle tire including tread portion defined by cycloid curve or epicycloid curve - Patent 6575214
		Preferably, the tread profile is defined by a curve which is a part of the locus of an equation such as elliptic equation, cycloid equation, epicycloid equation, involute equation and the like, which equation is differentiatable in the range of variables corresponding to the range from the tire equator to each tread edge.
		5 is the locus of an epicycloid equation for the tread curve.
		The epicycloid is the locus of a point N set on the circumference of a circle (d) rolling on the circumference of a fixed base circle (c) of which center is placed on the origin O of x-y coordinates.
	www.freepatentsonline.com /6575214.html (3225 words)

	Epicycloid and Hypocycloid
		Epicycloid is a special case of epitrochoid, and hypocycloid is a special case of hypotrochoid.
		We define the vertexes of the epicycloid to be points on the curve that coincides with a circumscribed circle.
		The epicycloid with rolling circle radius b is equal to the hypocycloid with rolling circle radius b+1.
	xahlee.org /SpecialPlaneCurves_dir/EpiHypocycloid_dir/epiHypocycloid.html (1003 words)

	Epicycloid - Search Results - MSN Encarta
		Epicycloid, in geometry, a curve resembling a series of arches traced out by a point on the circumference of a circle that rolls around another...
		An epicycloid is therefore an epitrochoid with h==b.
		In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point of a circle — called epicycle — which rolls without slipping around a fixed circle.
	encarta.msn.com /Epicycloid.html (144 words)

	NationMaster - Encyclopedia: Cardioid
		That is, a cardioid is a curve that can be produced as a locus — by tracing the path of a chosen point of a circle which rolls without slipping around another circle which is fixed but which has the same radius as the rolling circle.
		In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point of a circle â€”; called epicycle â€” which rolls around without slipping around a fixed circle.
		Since the cardioid is an epicycloid with one cusp, its parametric equations are In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point of a circle â€”; called epicycle â€” which rolls around without slipping around a fixed circle.
	www.nationmaster.com /encyclopedia/Cardioid (997 words)

	THE EPICYCLOID
		The epicycloid is the special plane curve defined as the path traced by a point P on a circle that rolls around a fixed circle without slipping.
		The next known practical use envisioned for epicycloids was in the working of mechanical gears, although there is some debate about who first thought of this.
		Some of these special epicycloids have names of their own, such as the one-cusp cardioid, the two-cusp nephroid and the five-cusp ranunculoid.
	online.redwoods.cc.ca.us /instruct/darnold/CalcProj/sp05/astley/EpicycloidReport.htm (537 words)

	Springer Online Reference Works (Site not responding. Last check: )
		When the point is not situated on the rolling circle, but lies in its exterior (or interior) region, then the curve is called an elongated (respectively, shortened) epicycloid or epitrochoid (see Trochoid).
		Epicycloids belong to the so-called cycloidal curves (cf.
		Epicycloids and, more generally, trochoids are important for kinematical constructions, cf.
	eom.springer.de /e/e035860.htm (193 words)

	epicycloid
		The isoptic of the ordinary epicycloid is an epitrochoid.
		The epicycloid curves have been studied by a lot of mathematicians around the 17th century: Dürer (1515), Desargues (1640), Huygens (1679), Leibniz, Newton (1686), de L'Hôpital (1690), Jakob Bernoulli (1690), la Hire (1694), Johann Bernoulli (1695), Danilel Bernoulli (1725) and Euler (1745, 1781).
		The variable b is the ratio of the distance from the starting point to the center of the rolling circle, and the radius of that circle.
	www.2dcurves.com /roulette/roulettee.html (656 words)

	epicycloid
		The epicycloid curve is of special interest to astronomers, who find it in various coronas.
		All of these curves are traced by a point P on a circle of radius a, differing only in the location of the point P and whether circle "b" rolls on the inside or outside of circle "a".
		The pedal curve of the epicycloid, providing the pedal point is the center, is called the rhodonea curve.
	online.redwoods.cc.ca.us /instruct/darnold/CalcProj/Sp99/LindaL/epicycloid.html (556 words)

	Trochoids, Etc.
		The important curves here are epicycloids and hypocycloids, and the term trochoid does not appear to be used for them, though it would seem appropriate.
		The pin at P has just run down epicycloid c, and is about to be pushed by epicycloid c' as rotation continues.
		When practical pins are used, some of the tooth is cut away parallel to the epicycloid to accommodate the radius of the tooth, and a semicircular recess is made between each tooth into the pitch circle of the driver for the same reason.
	www.du.edu /~jcalvert/tech/trochoid.htm (4092 words)

	Little Gallery of Roulettes
		An epicycloid (Du¨rer 1525) is the locus of a point on the circumference of which is rolling around the outside of a fixed circle.
		First, here's a 7:1 epicycloid colored in lavender with three purple epiroulettes line drawn in the same picture, but with their drawing points beyond the radius of the rolling circle, so part of these curves lie inside the fixed circle.
		Next, here's a 2:1 epicycloid (that is, a nephroid) colored gold with a red epiroulette where the drawing point is within the radius of the rolling circle.
	www.math.clarku.edu /~djoyce/roulettes/roulettes.html (616 words)

	Fluid displacement apparatus with improved helical rotor structure - Patent 6244844
		According to yet another aspect of the present invention, a rotor for use in a fluid displacement apparatus includes a cylindrical body portion having a helical groove therein, and a helical tooth portion disposed adjacent the helical groove and extending radially from the cylindrical body portion.
		A third tooth surface 380 is disposed between the first and second tooth surfaces 330, 340, and is configured to confront the inner surface 111 of the housing 110 illustrated in FIGS.
		In radial cross section, the second tooth surface 340 defines an epicycloid curve 360 that extends radially from the pitch circle 311.
	www.freepatentsonline.com /6244844.html (4838 words)

	epicycloid - definition by dict.die.net
		epicycloid n : a line generated by a point on a circle rolling around another circle
		['e]picyclo["i]de.] (Geom.) A curve traced by a point in the circumference of a circle which rolls on the convex side of a fixed circle.
		Note: Any point rigidly connected with the rolling circle, but not in its circumference, traces a curve called an epitrochoid.
	dict.die.net /epicycloid (125 words)

	NSDL Metadata Record -- Thumb-shaped Teeth, Rack and Pinion
		Gear teeth profiles in the 19th century used both epicycloids and involute (evolute) curves as described in the abstracts of models Q3 and Q4.
		From Reuleaux we quote: ?By combining the evolute and epicycloid, using the two curves for opposite sides of the same tooth a profile of great strength is obtained.
		The epicycloid curves are on the faces the contact the right hand sides of the rack teeth.
	www.nsdl.org /mr/710703 (146 words)

	R Series: R1-7. Spherical Cycloids and Involutes
		Involutes, Cycloids, Epicycloids, Hypocycloids, and Trochoids on Sphere
		A circle on a spherical surface forms a cone from the center of the sphere; in the case of a great circle this cone is actually a planar disk.
		R03 is a small circle rolling on the outside of a larger circle and the resulting spherical epicycloid are marked with wire.
	kmoddl.library.cornell.edu /tutorials/14 (527 words)

	Going around under and through circles: mathematics and computer art; part 3.
		In both cases, we started with a large circle of radius A and a smaller circle of radius B. To form a hypocycloid, we traced out the path traversed by a fixed point on the smaller circle as it rolls around the inside of the large circle.
		In this article, we will look at several further possibilities relating to epicycloids and hypocycloids and see some more mathematical patterns as well as the artistic shapes that result.
		The common factor of 2 in the radii 20 and 6 essentially is removed when forming the shape of the epicycloid.
	www.atarimagazines.com /creative/v10n7/141_Going_around_under_and_th.php (1743 words)

	EPICYCLOID - Online Information article about EPICYCLOID
		Newton and others; spherical epicycloids, in which the moving circle is inclined at a See also:
		The epicycloid when the radii of the circles are equal is the See also:
		Epicycloids are also examples of certain caustics (q.v.).
	encyclopedia.jrank.org /EMS_EUD/EPICYCLOID.html (960 words)

	Reference.com/Encyclopedia/Epicycloid
		In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point of a circle — called epicycle — which rolls without slipping around a fixed circle.
		If k is an irrational number, then the curve never closes, and fills the space between the larger circle and a circle of radius R+2r.
		The epicycloid is a special kind of epitrochoid.
	www.reference.com /browse/wiki/Epicycloid (214 words)

	3DLDF User and Reference Manual
		From each of these new positions, an epicycloid is drawn.
		The number of epicycloids drawn upon each iteration.
		pointed to by the pointers on this vector are used for drawing the epicycloids.
	www.gnu.org /software/3dldf/manual/user_ref/3DLDF/Epicycloids.html (245 words)

	Cycloids
		The epicycloid has been studied by such luminaries as Leibniz, Euler, Halley, Newton and the Bernoullis.
		The epicycloid curve is of special interest to astronomers and the design of cog-wheels with minimum friction.
		For the epicycloid I usually use two rolls of tape with different radii, identify a point on the outer one with a magic marker, and then roll it around the other and ask the students what they think the path would look like.
	mathdemos.gcsu.edu /mathdemos/cycloid-demo/index.html (999 words)

	Epicycloid-A-Day
		After almost a year on hiatus, the Epicycloid blog will be making a comeback.
		I have kind of adapted myself to using the term epicycloid to collectively describe what most folks call Spirographs.
		Truthfully, there are several kinds of mathematical curves that can be used to even more carefully note which is which amidst these special kinds of curves.
	pinkfrog.net /automata/epicycloid (177 words)

	August's Conformal Projection of the Sphere on a Two-Cusped Epicycloid (Site not responding. Last check: )
		The first step is to take the world, and project it using Lagrange's projection, so that the world is conformally mapped to a circle.
		Then, the circle is conformally mapped to the inside of the epicycloid that forms the boundary of the map using complex numbers.
		Functions over the complex numbers are an inexhaustible source of conformal mappings, since functions are extended to complex numbers in the fashion that permits them to be differentiable over the complex plane, and the condition for differentiability and for forming a conformal mapping are one and the same.
	www.quadibloc.com /maps/mcf0702.htm (1353 words)

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