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Topic: Simple closed curve

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	PlanetMath: curve
		The second notion is geometric; in this sense a curve is an arc, a 1-dimensional subset of an ambient space.
		The two notions are related: the image of a parameterized curve describes the trajectory of a moving particle.
		In algebraic geometry, the term curve is used to describe a 1-dimensional variety relative to the complex numbers or some other ground field.
	planetmath.org /encyclopedia/Curve.html (450 words)

	Curve - Wikipedia, the free encyclopedia
		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.
		From the nineteenth century there is not a separate curve theory, but rather the appearance of curves as the one-dimensional aspect of projective geometry, and differential geometry; and later topology, when for example the Jordan curve theorem was understood to lie quite deep, as well as being required in complex analysis.
	en.wikipedia.org /wiki/Curve (1204 words)

	PlanetMath: Jordan curve theorem
		is a simple closed curve in the sphere
		The two connected components mentioned in each formulation are, of course, the inside and the outside the Jordan curve, although only in the first formulation is there a clear way to say what is out and what is in.
		This is version 6 of Jordan curve theorem, born on 2002-11-11, modified 2004-02-16.
	planetmath.org /encyclopedia/JordanCurveTheorem.html (180 words)

	Curve -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		Simple examples are the (Ellipse in which the two axes are of equal length; a plane curve generated by one point moving at a constant distance from a fixed point) circle or the (A line traced by a point traveling in a constant direction; a line of zero curvature) straight line.
		A plane curve is a curve for which X is the (Click link for more info and facts about mathematical plane) mathematical plane — these are the examples first encountered — or in some cases the (Click link for more info and facts about projective plane) projective plane.
		A space curve is a curve for which X is of three dimensions, usually (A space in which Euclid's axioms and definitions apply; a metric space that is linear and finite-dimensional) Euclidean space; a skew curve is a space curve which lies in no plane.
	www.absoluteastronomy.com /encyclopedia/c/cu/curve.htm (1980 words)

	Curve Summary
		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.
		Looking at them projectively, if we have a nonsingular curve in n dimensions, we obtain a picture in the complex projective space of dimension n, which corresponds to a real manifold of dimension 2n, in which the curve is an embedded smooth and compact surface with a certain number of holes in it, the genus.
		From the nineteenth century there is not a separate curve theory, but rather the appearance of curves as the one-dimensional aspect of projective geometry, and differential geometry; and later topology, when for example the Jordan curve theorem was understood to lie quite deep, as well as being required in complex analysis.
	www.bookrags.com /Curve (4145 words)

	closed-curve geomatry
		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.
	philramble.blogspot.com (1235 words)

	SparkNotes: Polygons: Defining a Polygon
		A curve is continuous, meaning that there aren't any gaps or holes in the curve; any point on a curve can be reached from another point on the curve without leaving the curve.
		A curve whose starting point is also its endpoint is called a closed curve.
		A simple closed curve is an even more specific kind of curve: one that is closed, and doesn't intersect itself.
	www.sparknotes.com /math/geometry1/polygons/section1.html (459 words)

	Jordan Curve \| World of Mathematics
		A Jordan curve is a curve that falls in a plane and is topologically equivalent to the unit circle in that it is simple, closed, and does not cross itself.
		The curve divides the plane into exactly two parts, one part that is inside of the curve and another part that is outside of it.
		The Jordan curve theorem states that any simple closed curve divides the points of the plane not on the curve into two distinct domains, with no points in common, of which the curve is the common boundary.
	www.bookrags.com /research/jordan-curve-wom (391 words)

	The Jordan Curve Theorem
		Let's move away from these simple examples of polygons for a while and start thinking of non intersecting closed curves, as intricate or as simple as they may be.
		Any simple closed curve C divides the points of the plane not on C into two distinct domains (with no points in common) of which C is the common boundary.
		To give a concrete example, it happens that this theorem has a simple proof for the case of polygons, which are a class of simple curves that occurs very often in most important problems.
	www-cgrl.cs.mcgill.ca /~godfried/teaching/cg-projects/97/Octavian/compgeom.html (1554 words)

	Jordan Curves
		Because the unit interval is compact, whence the curve becomes the continuous image of a compact set into a hausdorff space.
		The continuous image of a compact set is compact, and since the curve lives in a hausdorff space, it is closed.
		A jordan arc is an injective path, and a jordan curve is an injective loop.
	www.mathreference.com /at-wind,intro.html (711 words)

	[No title]
		The famous Jordan curve theorem states that every simple closed plane curve divides the plane into two regions (the interior and the exterior).
		The term simple polygon is also often used to emphasize the simplicity of the polygonal curve.
		We begin by introducing a triangulation of P. triangulation of a simple polygon is a planar subdivision of (the interior of) P whose vertices are the vertices of P and whose faces are all triangles.
	www.cs.wustl.edu /~pless/546/lectures/l6.html (2841 words)

	Koch Curve
		Koch constructed his curve in 1904 as an example of a non-differentiable curve, that is, a continuous curve that does not have a tangent at any of its points.
		Moreover, the length of the curve between any two points on the curve is also infinite since there is a copy of the Koch curve between any two points.
		Three copies of the Koch curve placed around the three sides of an equilateral triangle, form a simple closed curve that form the boundary of the Koch snowflake.
	ecademy.agnesscott.edu /~lriddle/ifs/kcurve/kcurve.htm (563 words)

	The Koch Curve and Visual Resolution at Nonoscience
		It is a closed fractal curve of infinite length within a finite region of space, enclosing a finite area.
		This is the “zeroth” iteration of the Koch curve.
		It states that “A simple closed curve divides the plane into two components - the one that is inside and the one that is outside.” And loosely, a simple closed curve is one of finite arc length which doesn’t cross itself.
	www.nonoscience.info /2006/08/28/the-koch-curve-and-visual-resolution (1970 words)

	Sierpinski curve (Site not responding. Last check: 2007-10-21)
		The Sierpinski curve is a base motif fractal where the base is a square.
		The curve is the only plane locally connected one-dimensional continuum S such that the boundary of each complementary domain of S is a simple closed curve and no two of these complementary domain boundaries intersect.
		The curve is a two-dimensional generalization of the Cantor set.
	www.2dcurves.com /fractal/fractals.html (241 words)

	Jordan Curve Theorem (Site not responding. Last check: 2007-10-21)
		It is possible to start at any point on a simple closed curve and travel over every other point of the figure exactly once before returning to the starting point.
		If we shade the simple closed curve that passes through house 1, water, house 2, and electricity, we find that house 3 is inside this curve and that it has not yet been joined to gas, which is outside the curve.
		If two points on the same side of a simple closed curve are joined, the curve will be crossed an even number of times or it will not be crossed at all (zero crossings, the minimum possible).
	britton.disted.camosun.bc.ca /jbjordan.htm (671 words)

	Green's theorem - Wikipedia, the free encyclopedia
		In physics and mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C.
		Let C be a positively oriented, piecewise smooth, simple closed curve in the plane and let D be the region bounded by C.
		is used to indicate that the curve C the line integral is calculated on is closed.
	en.wikipedia.org /wiki/Green%27s_theorem (253 words)

	March 2002 solution
		Let S be a simple closed curve and let A be a point on S such that the tangent at A is defined.
		Since a portion of S' is inside the region bounded by S and a portion of S' is outside, because S is continuous and closed there must be a point, call it C, where the two curves intersect for a second time.
		The Jordan curve theorem (published by Camille Jordan in 1893) says that such a curve separates the plane into two regions, the interior (which is bounded by the curve) and the exterior (which is unbounded).
	mathcentral.uregina.ca /MP/previous2001/apr02sol.html (1188 words)

	Springer Online Reference Works
		Thus, the computation of integrals of analytic functions along closed curves (contour integrals) is reduced to the computation of residues, which is particularly simple in the case of finite poles.
		The theorem on the total sum of residues is applicable to Riemann surfaces: The sum of all residues of a meromorphic differential on a compact Riemann surface is zero.
		This theory is based on the integral theorems of Stokes and Cauchy–Poincaré, which make it possible to replace the integral of a closed form along one cycle by an integral of this form along another cycle which is homologous to the former.
	eom.springer.de /r/r081560.htm (1045 words)

	[No title]
		Examples: State whether the curve is a simple closed curve.
		Circle: A simple closed curve in which all points of the circle are equidistant from a fixed point called the center.
		Convex polygon:___________________________________ Example: Regular polygon: is a simple polygon in which all sides are congruent and all angles are congruent.
	www.math.unm.edu /courses/math112/notes10_1.doc (386 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		A polygon is a union of line segments which forms a simple closed curve.
		A curve whose beginning is its end, and doesn't intersect itself otherwise.
		A simple closed curve in the plane defines an inside (a finite region) and an outside (an infinite region.)
	www.math.csusb.edu /courses/m129/defs.html (201 words)

	[No title]
		Your curve may be written as the intersection of two elliptical cylinders with perpendicular axes: x^2+(z/0.93)^2=1 and y^2+(z-1)^2=1.
		With a designated number N of segments of the curve to be in the first quadrant, begin with any curve of length 1/4 rising from some (x0, 0, -z0) to some (0, y0, z0) through N-1 equidistant points in the first quadrant.
		The curves may be viewed as a map from the unit circle (in the complex plane) to complex n-space C^n; the optimal curve is f(z) = (z, z^2/2, z^3/3,..., z^n/n) Z.A. Melzak (Proc Amer Math Soc 11 (1960) 265-274) considered precisely the problem I did, made the same restrictive assumptions, and deduced the same answer.
	www.math.niu.edu /~rusin/known-math/97/bulky.crv (2305 words)

	[No title]
		Since the curvature can be either positive or negative (depending on whether the curve is turning clockwise or counterclockwise), some parts of the curve move outwards while others move inwards.
		That is, no matter how wildly twisting a curve is, as long as it is simple, it will "relax" to a circle and then disappear.
		The curve is embedded as the zero level set of the signed distance function.
	math.berkeley.edu /~sethian/2006/Applications/Geometry/curvecollapse.html (370 words)

	Algebraic Topology: Knots, Links, Braids
		A knot is a simple closed curve (homeomorphic image of S(1)) in Euclidean 3-space E(3).
		Schoenflies proved in 1908 that any homeomorphism from a simple closed curve in the plane E(2) onto the unit circle S(1) can be extended to a homeomorphism of the plane onto itself.
		Thus, one usually restricts knots to be tamely embedded, e.g., as a simple closed polygonal curve, and we'll do so as well.
	www.win.tue.nl /~aeb/at/algtop-5.html (1380 words)

	Fixed Point Properties of Plane Continua (Site not responding. Last check: 2007-10-21)
		maps a simple closed curve J into the continuum bounded by J fixes a point in the continuum
		closed curve J, the fixed point index of f on J is defined to be the winding number of the map g(z)
		If J is a simple closed curve such that Q Ì T(J).
	math.uc.edu /~bellh/FPPOPC/fppopc.htm (2146 words)

	simple - Definitions from Dictionary.com
		(of a verb tense) consisting of a main verb with no auxiliaries, as takes (simple present) or stood (simple past) (opposed to compound).
		Example: She is too simple to see through his lies.
		Contributions to SIMPLE IRAs are immediately 100% vested, and the IRA owner directs the investments.
	dictionary.reference.com /browse/simple (836 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Finally we use deformation of contours to see that the integral about the curve $C$ is the same as the integral about the triangle.
		However, it is true for domains surrounded by a simple closed curve.
		\gap \prob 6 {10} \sla{Calculate \sla{directly} $\dsp\int_C {dz\over z}$ when $C$ is a simple closed curve, of your choice, that encloses the origin.} \gap %\vskip 3.1 in We choose $C$ to be the unit circle.
	www.math.umn.edu /~jodeit/course/5583Test2soln (492 words)

	Informal Geometry Review
		A simple closed curve separates a plane into three disjoint sets of points: the interior, the exterior, and the curve.
		A region is the union of the interior of simple closed curve and its boundary (the curve itself.)
		Detailed definition: A polygon is a simple closed curve that is the union of three or more line segments AB, BC, CD,..., PQ such that A, B, C, D,..., P, Q are coplanar and distinct, and no three consecutively named points are collinear.
	www.csupomona.edu /~vmsmith/GeoRev.html (1391 words)

	JCT 8/29/91 (Site not responding. Last check: 2007-10-21)
		The Jordan Curve Theorem states that every simple closed curve in the Euclidean plane
		In [1] Ahlfors uses winding numbers to outline a proof that simple closed curves separate
		C and on the fact that simple closed curves are not retracts of
	math.uc.edu /~bellh/JCT/index.htm (1331 words)

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