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Topic: Trefoil curve

	2
		The difference between the curve type is that in here, the curveís equation is first discretized before being drawn.
		curve: For the curve it is similar to the knot
		There are three styles, the first one is a real ribbon following the curve, the second one is a second curve which round about the first one, taking in account the twist value, the third one is a real ribbon where one half is of one color and the second half is of another color.
	cso.ulb.ac.be /cso/povweb/functions.html (2914 words)

	Trefoil knot - Wikipedia, the free encyclopedia
		In knot theory, the trefoil knot is the simplest nontrivial knot.
		The trefoil knot is chiral, meaning it is not equivalent to its mirror image.
		It is not a slice knot, meaning that it does not bound a smooth 2-dimensional disk in the 4-dimensional ball; one way to prove this is to note that its signature is not zero.
	en.wikipedia.org /wiki/Trefoil_knot (269 words)

	YourArt.com >> Encyclopedia >> curve (Site not responding. Last check: 2007-10-27)
		A plane curve is a curve for which X is the Euclidean 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.
		Important examples of algebraic curves are the conics, which are nonsingular curves of degree two and genus zero, and elliptic curves, which are nonsingular curves of genus one studied in number theory and which have important applications to cryptography.
	www.yourart.com /research/encyclopedia.cgi?subject=/curve (1503 words)

	Oct8.html
		We see that the curve lives on a torus of inner radius 3 and outer radius 5 (tube radius 1 and meridian radius 4).
		The trefoil knot is a basic example in a highly developed field of mathematics called
		By comparing the equations for the trefoil knot and the toroidal spiral, we might guess that the tefoil knot also lives on a torus.
	www.uwec.edu /smithaj/fall2001/216/lectures/Oct8.html (241 words)

	American Mathematical Society :: Feature Column
		Recall that a knot is simply a closed curve in space with no double point, and that one says that two knots are the same, or are equivalent, if one can go from the first to the second through a continuous deformation without creating intersection points (see for instance [6], [7], [8]).
		Hence the space of lattices of area 1 is identified with the complement of a trefoil knot in the 3-sphere, which, after deleting one point, is the complement of a trefoil knot in the usual 3-space.
		Recall that the trefoil knot is the boundary of a Seifert surface where the argument of Δ is equal to some fixed value, for instance 0 (which corresponds to Δ being a positive real number).
	www.ams.org /featurecolumn/archive/lorenz.html (4847 words)

	birdsfoottrefoil
		Birdsfoot trefoil is a deep rooted plant that is short lived.
		Birdsfoot trefoil is utilized best as pasture in which it is a high quality forage.
		Crown and root rots are serious diseases of birdsfoot trefoil that reduces stands.
	southerforages.homestead.com /birdsfoottrefoil.html (406 words)

	Visualizing Quaternions: Quaternion Maps Documentation
		Quaternion Maps are plots of the quaternion values corresponding to 3D orientation frames attached to a curve, surface, or volume.
		The main-window displays the curve, while the quaternion map appears in the smaller thumbnail view in the auxiliary window.
		Otherwise, an 'Un-knot' or a loop is generated that is not a knotted curve, i.e.
	www.cs.indiana.edu /~hanson/quatvis/Quaternion-Maps/index.html (1161 words)

	Differential Geometry\\ Math 424, Spring, 2003 (Site not responding. Last check: 2007-10-27)
		I. These curves do not have ``angles'' like in the example, which follows from the fact that the tangent line to the curve is well-defined at each point.
		There is a whole class of such curves, called torus knots because they lie on the torus and are knotted, which means they cannot be homotoped to a standard circle while staying embedded throughout the the homotopy - that is, it can't be unraveled without crossing itself.
		For one thing, the curvature of a space curve is always positive (for a plane curve considered as a space curve, the space-curve curvature is simply the absolute value of its plane-curve curvature).
	www.lehigh.edu /dlj0/courses/424sp03-1.html (3476 words)

	Polycut Multiple Universes (Site not responding. Last check: 2007-10-27)
		However, the gluing is done in such a way that there are actually two separate branch curves in the covering space: one is of order 2 and one is of order 1 (note the orders total to 3, which is the number of sheets).
		All that is needed are functions computing the branch curves as functions of one parameter and some structures describing the gluing of the the cuts and some other data.
		Each branch curve may be an open or closed curve, which may be knotted or not.
	www.susqu.edu /facstaff/b/brakke/polycut/polycut.htm (4841 words)

	Glossary
		The weapon is extremely similar to the messer, with the exception that the messer is usually made with a slight curve, whereas the backsword is almost always straight.
		Tiara is the papal crown, a costly covering for the head, ornamented with precious stones and pearls, which is shaped like a bee-hive, has a small cross at its highest point, and is also equipped with three royal diadems.
		Tressure A diminutive of the orle appearing as a narrow band near the edge of a coat of arms, often ornamented with fleurs-de-lis, as in the Scottish Royal Arms.
	www.ceu.hu /medstud/manual/SRM/gloss.htm (6563 words)

	The Optiverse: script (Site not responding. Last check: 2007-10-27)
		One approach is to associate an energy to every closed curve in space, and evolve the curve to reduce this energy.
		Since our starting curve was an unknot, though quite tangled, it evolves to the round circle, which has the minimum possible energy for any curve.
		The curve actually has a continuous Möbius symmetry: it is the orbit of a point under a rigid rotation of the three-sphere.
	torus.math.uiuc.edu /jms/Videos/ke/script.html (378 words)

	Sculpture Maths - How we made the pictures
		This file allows the specification of a base 3-dimensional curve, called here the space curve; in all the five examples here, the space curve is a circle, given by the formula [4cos(t),4sin(t),0], for a range of values of t.
		The plane curve is the circle [cosu + 2/…3, sinu] centre ((2/…3), 0) and radius 1, and the twist is by 8t/3.
		We found it difficult to model the sculpture exactly, since the trefoil curve as used by John was not described by the simple mathematical formulae we tried.
	www.popmath.org.uk /sculpmath/pagesm/how.html (872 words)

	SpaceStudent.html
		The use of vectors to model three dimensional curves (paths) is critical to the sciences.
		Space curves and vector-valued functions require their domains to be specified for the parametric/component functions.
		Determine the parametric curve for the top-half of the intersection.
	www.mc.maricopa.edu /~dschultz/SpaceStudent.html (761 words)

	John M. Sullivan: Annotated Bibliography
		In particular, we show that the distortion of a knotted curve (the maximum arc/chord length ratio) is at least pi, twice that of a round circle.
		We define the second hull of a space curve, consisting of those points which are doubly enclosed by the curve in a certain sense.
		Curves and surfaces which are optimal for geometric energies often have aesthetically pleasing shapes.
	torus.math.uiuc.edu /jms/Papers/annotbib/subj.shtml (4606 words)

	Lorenz and modular flows: a visual introduction
		These circles are only topological circles; in fact they are themselves trefoil knots with the exceptions of the two special cases that we mentioned which are shorter (they close after π/2 and π/3 instead of π) and which turn out to be round circles.
		The set of lattices of area 1 for which Δ has a given argument defines a surface in the 3-sphere whose boundary is the trefoil.
		Recall that the trefoil knot is the boundary of a Seifert surface where the argument of Δ is equal to some fixed value, for instance 0 (which corresponds to Δ being a positive real number).
	www.josleys.com /articles/ams_article/Lorenz3.htm (4718 words)

	Hyperseeing, Hypersculptures and Space Curves
		Two diagrams of the so-called trefoil knot are shown in Figure 12 (a) and (b).
		The space curve in Figure 14 is based on the trefoil knot.
		A space curve based on the figure eight knot is shown in Figure 16.
	members.tripod.com /vismath5/friedman (3076 words)

	Space Curves, Frenet Frames, and Torsion
		Notice also that for a plane curve, the binormal is identically perpendicular to the plane in which the curve lies and the torsion is 0.Thus we have the Frenet-Serret formulae:
		Notice also that for a plane curve, the binormal is identically perpendicular to the plane in which the curve lies, and the torsion is 0.
		This plots the curve in blue together with normal vectors in red and binormal vectors in green.
	www.math.umd.edu /~jmr/241/curves2.htm (862 words)

	A Geometry View
		The Fundamental Theorem of Plane Curves states that plane curves are classified, up to a rotation and translation, by their curvature---which is the rate of change of angular direction per unit arc length s.
		Torsion measures the departure of a curve from a plane.
		The next two curves are again encased in ruled tubes to make them clearer as two knots: the trefoil and the figure eight knot.
	www.maths.manchester.ac.uk /~kd/geomview/geometry.html (1087 words)

	A lumpy knot (Site not responding. Last check: 2007-10-27)
		We conceive of this surface as the union of the circles with centers on the curve and with radius t.
		The plane of the circle is perpendicular to the curve’s tangent.
		The normal to this surface a linear combination of stk and the unit tangent to the curve.
	cap-lore.com /code/OpenGL/ScreenSaver (239 words)

	Talk:Curve - InfoSearchPoint.com (Site not responding. Last check: 2007-10-27)
		A curve is not a subset of R^n, it's a continuous mapping of an interval of real numbers into a topological space.
		The definition of elliptic curve isn't really a true generalisation of a curve as a continuous map from an interval of real numbers, in the case over finite fields, this certainly isn't the case, as the set of all (x,y) would be finite and certainly not the continuous image of an interval of real numbers.
		Your attempt at a definition of a topological curve includes the idea of open sets as subsets of C, but does not give any clue what space C is embedded in; thus, any closed subset of any space is a curve by your definition.
	www.infosearchpoint.com /wiki.php?title=_Talk:Curve&printable=yes (4689 words)

	Carving for Mathematical Understanding of Surfaces
		The twisted trefoil knot surface shown was created on a 3D Systems stereolithographic machine along with a wood carving of the new collapsed surface.
		Returning briefly to the spanning surface of the trefoil knot, it is clear that this surface is not a minimal surface as it bulges out.
		The eight knot basis for this non-orientable sculpture was formed using piecewise special Bezier quartic curves [4] defined as an interpolating curve for the eight points shown here, which stem from lines drawn from the center to the vertices of a regular tetrahedron.
	www.wcnet.org /~clong/carving/carving.html (841 words)

	Harmonic Knots (Site not responding. Last check: 2007-10-27)
		Definition 2.3 The crossing number of a knot K, denoted c[K], is the smallest crossing number of all diagrams representing knots from the knot class.
		Example 2.4 The crossing number of the trefoil is 3 and the crossing number of the figure eight knot is 4.
		The bridge index of the trefoil is 2.
	www2.carthage.edu /~trautwn/hknots.html (1142 words)

	Ivars Peterson's MathTrek - The Tangled Task of Distinguishing Knots
		In this context, a knot is a one-dimensional curve that winds through itself in three-dimensional space, finally catching its tail to form a closed loop.
		Mathematicians can ask the same questions about a knotted curve that sailors or boy scouts may ask about a knotted rope.
		The simplest possible true knot is the overhand, or trefoil, knot, which is really just a circle that winds through itself.
	www.maa.org /mathland/mathtrek_02_24_03.html (1038 words)

	Geometry of Curves
		We enter the cycloid as a three dimensional curve in order to be able to use the cross-product later on.
		This behaviour is not apparent from the plot, which makes it look as though the curve becomes straight near the cusp.
		Notice also that for a plane curve, the binormal is identically perpendicular to the plane in which the curve lies, and thus the torsion is 0.
	www.math.umd.edu /users/jmr/241/curves.html (1580 words)

	Trefoil Knot (Site not responding. Last check: 2007-10-27)
		Maple's tubeplot command draws a "tube" around a parametric curve--in this case, the curve for a trefoil knot.
		You can experiment with other numbers of frames, as well as with various parametric curves and various values for the radius of the tube.
		An animation editor was used to slow down the display of the frames, as well as to add a 3-second display time to the final frame.
	www.jcu.edu /math/ictcm99/animations/knot.htm (117 words)

	Knot Theory
		Figure 5.6: Planar diagrams of knots: (a) the trivial or unknot ; (b) figure-eight knot ; (c) left-handed trefoil ; (d) right-handed trefoil; (e) square knot ; (f) granny knot.
		A mathematician's knot is a non-self-intersecting smooth closed curve (a string) embedded in three-space.
		For instance, Figure 5.8 shows two topologically equivalent planar diagrams for the trefoil knot and a sequence of ``moves'' showing their equivalence.
	cnls.lanl.gov /People/nbt/Book/node141.html (552 words)

	Stewart Calculus ET 5e 0534393217 (Site not responding. Last check: 2007-10-27)
		So the curve lies entirely in the first quadrant.
		Thus the polar equation of the curve is
		We show a computer-drawn graph of the curve from above, as well as views from the front and from the right side.
	math.berkeley.edu /~wang/2005summer/homeworks/hk-0705.html (369 words)

	KNOTS
		The trefoil knot shown in Figure 2 is an example of such a closed knotted loop.
		This means that each curve is equipped with a directional arrow, and we keep track of the direction of the arrow when the curve is deformed by the Reidemeister moves.
		This proves that the trefoil is knotted, since an unknotted trefoil would have a simple circle among its diagrams, and the simple circle can be colored with only one color.
	www2.math.uic.edu /~kauffman/Tots/Knots.htm (16146 words)

	All Tied Up In Knots
		The curve may be a simple loop or it may have a number of twists and crossings.
		(Hint: Compare the crossings.) Trefoil #2 also has three crossings, but it is not the same knot as trefoil #1 because the corresponding crossings in trefoil #2 go under where in #1 they go over, and vice versa.
		Trefoil #2 is the mirror image of trefoil #1.
	www.riverdeep.net /current/2002/09/091602_knots.jhtml (1383 words)

	Gray's Anatomy of the Human Body - The Muscles of the Thorax - Yahoo! Education (Site not responding. Last check: 2007-10-27)
		The fibers arising from the xiphoid process are very short, and occasionally aponeurotic; those from the medial and lateral lumbocostal arches, and more especially those from the ribs and their cartilages, are longer, and describe marked curves as they ascend and converge to their insertion.
		It is shaped somewhat like a trefoil leaf, consisting of three divisions or leaflets separated from one another by slight indentations.
		The anterior abdominal muscles come into action so that the umbilicus is drawn upward and backward, but this allows the diaphragm to exert a more powerful influence on the lower ribs; the transverse diameter of the upper part of the abdomen is greatly increased and the subcostal angle opened out.
	messenger.yahooligans.com /reference/gray/subjects/subject?id=117 (3101 words)

	[No title] (Site not responding. Last check: 2007-10-27)
		Certainly a (tame) simple closed curve in R^3 always bounds both orientable and non-orientable surfaces.
		> Sorry for the delay, but here's the trefoil knot (use constant-spaced font here), with the Seifert surface shown (one side is., other side is o).
		The curve itself is the dashed line, and the over/under crossings are indicated by a break in the line.
	www.math.niu.edu /~rusin/known-math/00_incoming/seifert (305 words)

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