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Topic: Closed curve

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	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.
		The reason for this is that such a curve encloses a region in the plane.
	www.sparknotes.com /math/geometry1/polygons/section1.html (479 words)

	PlanetMath: curve (Site not responding. Last check: 2007-10-29)
		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)

	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)

	[No title]
		A Curve is a one-dimensional geometric object usually stored as a sequence of points, with the subtype of Curve specifying the form of the interpolation between points.
		The boundary of a closed Curve is empty.
		Each Curve in the boundary of the MultiPolygon is in the boundary of exactly 1 element Polygon, and every Curve in the boundary of an element Polygon is in the boundary of the MultiPolygon.
	xml.coverpages.org /gml-rdfs0029.txt (1205 words)

	Closed Curves
		A "closed" curve is one in which the start and end points are in the same location, and at least one other point (not in that location) is included in the curve.
		The closed curve must clearly define an area that is determined as the interior of the curve, and completely separate this area from the exterior of the curve.
		A closed curve does not include any rapid links in the boundary shape; however, it may use a rapid move at the end to connect to subsequent islands, or at the begin to define a plunge location.
	www.ezcam.com /web/products/help/ezmill/curves_common/closed_curves.htm (116 words)

	Curve - Wikipedia, the free encyclopedia
		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.
		Algebraic curves are the curves considered in algebraic geometry.
		A curve may be a locus, or a path.
	en.wikipedia.org /wiki/Curve (1596 words)

	Curve Summary
		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.
		A plane algebraic curve is the locus of points f(x,y)=0, where f(x,y) is a polynomial in two variables defined over some field F. Algebraic geometry normally looks at such curves in the context of algebraically closed fields.
		We may also consider these curves to have points defined in the projective plane; if f(x, y)=0 then if x=u/w and y=v/w, and n is the total degree of f, then by expanding out w^n f(u/w, v/w)=0 we obtain g(u, v, w)=0, where g is homogeneous of degree n.
	www.bookrags.com /Curve (4145 words)

	New Headers:
		Closed curves can be generated by making the last control point the same as the first control point.
		First order continuity of a closed curve can be achieved by ensuring the tangent between the first two points and the last two points are the same.
		Closed curves are generated by specifying the first point the same as the last point.
	www.cs.indiana.edu /~yinli/PET.htm (1910 words)

	Curve through Point Cloud (3D/2D)
		Curve fairing refers to the process of shape editing to remove unwanted imperfections in a curve.
		Curves of a lesser degree are less exact and require less storage and computation time.
		Curves of a higher degree are more exact and require increased storage and computation time.
	www.vx.com /help/4368.htm (738 words)

	Curve Boundary Editing
		For closed curves, the segments corresponding to the beginning and end of the original curve are merged before keep points are considered.
		If the curve is a mold parting line, it will only extend to the border of the parameter space of the face from which it was extracted.
		Curves can be split or trimmed by each curve in the set of boundary curves.
	www.vx.com /help/0717.htm (1644 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)

	Sierpinski curve (Site not responding. Last check: 2007-10-29)
		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)

	The Topology of Complex Numbers
		We mentioned earlier that a simple closed curve is positively oriented if its interior is on the left when the curve is traversed.
		The Jordan curve theorem is a classic example of a result in mathematics that seems obvious but is very hard to demonstrate, and its proof is beyond the scope of this book.
		The difficulty lies in describing the interior and exterior of a simple closed curve analytically, and in showing that they are connected sets.
	math.fullerton.edu /mathews/n2003/ComplexPlaneTopologyMod.html (1097 words)

	What is Algebraic Topology?
		If we deform the curve, the winding number has to vary continuously but, since it is constrained to be an integer, it cannot change and must be a constant unless the curve is deformed through the origin.
		Now deform the image curve by shrinking the radius R to zero and suppose that the image curve never passes through the origin, that is to say that we never get the zero of the polynomial.
		First, it assigns to a geometric odject, the closed curve, a discrete invariant, the winding number which is an integer.
	www.math.rochester.edu /people/faculty/jnei/algtop.html (1161 words)

	Classification of Surfaces
		Recall that a closed surface is a space which in the vicinity of each point looks like the plane and which satisfies the finiteness condition: the surface can be cut up into a finite number of pieces each homeorphic to a disk.
		A key ingredient in the proof of Rado's theorem is (a strong form of) the Jordan curve theorem: any simple closed curve in the plane separates the plane into two regions, which was proved by A.
		Jordan Curve Theorem for Surfaces The maximum number of disjoint simple closed curves which can be cut from an orientable surface of genus g without disconnecting it is g.
	www.math.ohio-state.edu /~fiedorow/math655/classification.html (1023 words)

	Jordan Curve Theorem
		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)

	Jordan Curve Theorem
		The theorem is indeed obvious for smooth curves (hint: standard elementary calculus texts show how to compute the outward normal vector to such a curve) and not too difficult to extend to piecewise smooth curves.
		However his proof left open the question of whether the inside and outside of all such curves were homeomorphic to the inside and outside of the standard circle in the plane (ie.
		This strong form of the Jordan curve theorem was proved by A.
	www.math.ohio-state.edu /~fiedorow/math655/Jordan.html (470 words)

	Informal Geometry Review
		A closed curve is a curve that, when traced, has the same starting and stopping points and may cross itself at individual points.
		A simple closed curve separates a plane into three disjoint sets of points: the interior, the exterior, and the curve.
		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)

	8.2 Open/closed boundaries
		For each closed sector curve on the source surface the forward separatrix curves form a continuous separatrix surface mapping downward to negative polarity photospheric regions; this is the red curve on the right part of Figure 27
		Their footpoints trace photospheric curves which are the theoretical manifestation of coronal hole boundaries in the PSS model.
		The blue and red curves are the footprints of the upward and downward separatrices, respectively; the fl curve is the PIL.
	solarphysics.livingreviews.org /Articles/lrsp-2005-7/articlesu20.html (1954 words)

	Hippopede Coffman Deposit #12 (Site not responding. Last check: 2007-10-29)
		Each of these curves is the image of an ellipse under an inversion in a circle.
		The curve is non-convex for b < a < 2b, and convex for a > 2b when the shape is oval, but is not exactly a strictly defined ellipse.
		Each of these curves is the image of a hyperbola under an inversion in a circle.
	curvebank.calstatela.edu /hippopede/hippopede.htm (499 words)

	B-spline Curves: Closed Curves
		In the figure, control point pairs 0 and 7, 1 and 8, and 2 and 9 are placed close to each other to illustrate the construction.
		It is clear that the gap between the first and last points of the curve is closer.
		Finally, the curve becomes a closed on when control points 2 and 9 are made identical as shown in Figure (d).
	www.cs.mtu.edu /~shene/COURSES/cs3621/NOTES/spline/B-spline/bspline-curve-closed.html (332 words)

	Week8 - Basic Three-Dimensional Modeling
		If the curve is a circle, the ruled surface begins at the 0-degree quadrant point, as determined by the current X axis plus the current value of the SNAPANG system variable.
		It is the same for each curve; therefore, the distance between the vertices along the two curves differs if the curves are of different lengths.
		If both boundaries are closed, or if one is closed and the other is a point, the resulting polygon mesh is closed in the N direction and N equals SURFTAB1.
	darkwing.uoregon.edu /~arch/landcad/week8.htm (1500 words)

	MYDDAS - Spatial Extensions (Site not responding. Last check: 2007-10-29)
		A Curve is a one-dimensional geometric object usually stored as a sequence of points, with the subtype of Curve specifying the form of the interpolation between points.
		A Curve is closed if its start point is equal to its end point.
		The boundary of a closed Curve is empty.
	www.ncc.up.pt /~michel/MYDDAS/spatial/Curve.html (140 words)

	The idea behind Green's theorem*
		We could picture the microscopic circulation as a bunch of small closed curves (shown below in green), where each curve respresents the tendency for the vector field to circulate at that location (imagine that the small curves were really, really small, much smaller than pictured).
		If C is a closed curve in the plane (remember, we are talking about two dimensions), then it surrounds some region D (shown in red) in the plane.
		If C is an open curve (i.e., a curve where the endpoint isn't the same as the beginning point), please don't even think about using Green's theorem.
	www.math.umn.edu /~nykamp/m2374/readings/greensidea/index.html (732 words)

	Ideas, Concepts and Definitions (Site not responding. Last check: 2007-10-29)
		A mathematical knot is a simple closed curve.
		A link is made of more than one simple closed curve, but a link is not a simple closed curve itself.
		In map coloring, any map drawn as a simple closed curve will be two-colorable.
	www.c3.lanl.gov /mega-math/gloss/topo/clcurve.html (71 words)

	Closed timelike curve information - Search.com
		In a Lorentzian manifold, a closed timelike curve (CTC) is a worldline of a material particle in spacetime that is closed.
		This possibility was raised by Willem Jacob van Stockum in 1937 and by Kurt Gödel in 1949.
		The lower light cone is characteristic of light cones in flat space - all spacetime coordinates included in the light cone have later times.
	www.search.com /reference/Closed_timelike_curve (716 words)

	AutomaticCurve (JPT 2.4.0 API)
		then an open curve is rendered with the same end tangents as are computed automatically for the corresponding closed curve.
		One algorithm of the pair should compute the tangents for a CLOSED curve using just the vertex information and the other algorithm should compute the tangents for an OPEN curve using both the vertex and end tangent information.
		If the end tangent array is supplied, we assume that an OPEN curve is desired since that is the only case for which the end tangent array is needed; on the other hand, if the end tangent array is not supplied, we assume that a CLOSED curve is desired by default.
	www.ccs.neu.edu /jpt/jpt_2_4/docs/edu/neu/ccs/gui/AutomaticCurve.html (1241 words)

	Class figPac.fFastBezier
		The parametrized curve whose graph is the Bezier curve.
		The type determines whether the curve is plotted as an open or closed path, whether or not it is filled and so on.
		In the latter case, a bezier segment from the last vertex to the first vertex is plotted in addition to the curve itself.
	www.math.ubc.ca /~feldman/figPacDoc/figPac.fFastBezier.html (594 words)

	Closed Spline Curve (Site not responding. Last check: 2007-10-29)
		This is an applet that allows you to draw a closed spline of any degree.
		Increase the degree to the spline to see how the curve shrinks back to a point, the centroid of the control points.
		The "Finess" box control the number of line segments to be used to approximate each spline curve segment.
	www.stanford.edu /~anguyen/contrib.html (298 words)

	InterMath / Dictionary / More Information
		Closed Curve: A curve that can be traced using the same starting and stopping points, and without crossing or retracing any part of the curve.
		Closed Curve - MathWorld: Math World defines closed curve and gives other relating links.
		Closed Curve - Planet Math: A website that gives several definitions on a closed curve.
	intermath.coe.uga.edu /dictnary/info.asp?termid=72 (68 words)

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