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Topic: Plane curve

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	Curve - Wikipedia, the free encyclopedia
		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.
	en.wikipedia.org /wiki/Curve (1204 words)

	PlanetMath: curvature (plane curve)
		The curvature of a plane curve is a quantity which measures the amount by which the curve differs from being a straight line.
		Since reparameterizing a curve by arclength is not always easy, it is useful to have a formula for curvature which is invariant under reparameterization since one could use such a formula with any parameterization.
		This is version 3 of curvature (plane curve), born on 2005-10-02, modified 2005-11-02.
	www.planetmath.org /encyclopedia/CurvaturePlaneCurve.html (439 words)

	[No title]
		An algebraic curve with degree greater than 2 is called a higher plane curve.
		The circle g(x,y) = x^2+y^2-1=0 is an example of an algebraic curve, the catenary g(x,y) = y - c cosh(x/c)=0 is an example of a nonalgebraic curve.
		sextic curve +------------------------------------------------------------ A sextic curve is an algebraic curve of degree 6.
	www.math.harvard.edu /~knill/sofia/data/curves.txt (1342 words)

	Encyclopedia: Curve (Site not responding. Last check: 2007-10-21)
		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...
		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.
		In mathematics, a cubic curve is a plane curve C defined by a cubic equation F(X,Y,Z) = 0 applied to homogeneous coordinates [X:Y:Z] for the projective plane; or the inhomogeneous version for the affine space determined by setting Z = 1 in such an equation.
	www.nationmaster.com /encyclopedia/Curve (3519 words)

	Cubic Curve (Site not responding. Last check: 2007-10-21)
		In mathematics, a cubic curve is a plane curve C defined by a cubic equation
		A cubic curve may have a singular point; in which case it has a parametrization in terms of a projective line, Otherwise a non-singular cubic curve is known to have nine points of inflection, over an algebraically closed field such as the complex numbers.
		Elliptic curves are now normally studied in some variant of Weierstrass form, defining a quadratic extension of the field of rational functions made by extracting the square root of a cubic.
	www.wikiverse.org /cubic-curve (447 words)

	Adam Coffman --- Lemniscates
		The hippopede becomes the union of a smooth curve and an isolated point (an "acnode" of the curve, where the plane is tangent).
		This curve is also called an "elliptic lemniscate of Booth," and it is the image of an ellipse under inversion with respect to its center.
		The hyperbolic lemniscate is equal to a projection of the space curve formed by the intersection of a circular paraboloid and an elliptic cone, with the same vertex and perpendicular axes.
	www.ipfw.edu /math/Coffman/pov/spiric.html (336 words)

	Xah: Special Plane Curves: Naming and Classification of Curves
		All the curves covered here are such that when you keep magnifying parts of the curve, it'll eventually looks like a line, unless you are magnifying a cusp point.
		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.
	xahlee.org /SpecialPlaneCurves_dir/Intro_dir/familyIndex.html (1043 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		To modify a curve in a manner that establishes tangent plane or curvature continuity between a selected curve and an adjacent surface or between a selected curve and a pair of construction curves.
		Two curve tangents define a plane.The first curve selected is modified to have its tangent vector lie in the plane defined by the tangent vectors of the other two curves.
		For example, if you modify a curve at its endpoint, the curvature slider will be moving the third CV from the end of the curve along a line that joins the end two CVs of the curve (for instance, along the tangent vector line).
	www.cclabs.missouri.edu /things/instruction/aw/Projecttangent.html (2415 words)

	MADCAP User's Guide - Create Menus (Site not responding. Last check: 2007-10-21)
		The red curve in the image below is an example of an elliptical arc defined with the center at the midpoint of the first end of the two lines, one axis defined by selecting the first endpoint of one of the lines, and the other axis defined by a keyed in length.
		This operation is used to generate a curve or a singular surface composed of the centroids of the constant N-lines on a surface.
		The curve or surface is then extended by adding points to the surface in the direction defined by the cell(s) at the end of the curve or surface.
	www.lerc.nasa.gov /WWW/winddocs/madcap/create_help.html (4860 words)

	Plane Curves
		A plane curve C is defined by the vanishing of a single polynomial f in one of the available ambient planes:
		Curves may also be created implicitly, such as when they arise as the images of maps.
		The base change of the curve C to a curve in the new ambient space A. The space A must be of the same type as the ambient of C and its base ring must either admit coercion from the base ring of C or have the map m between the two explicitly given.
	www.math.niu.edu /help/math/magmahelp/text982.html (1542 words)

	algebraic curve (Site not responding. Last check: 2007-10-21)
		A curve is algebraic when its defining Cartesian equation is algebraic, that is a polynomial in x and y.
		An algebraic curve is called a circular algebraic curve, when the points (±1, ± i) are on the curve.
		When a curve is not algebraic, we call the curve (and its function) transcendental.
	www.2dcurves.com /algebraic.html (184 words)

	Rudy Rucker's KappaTau Space Curve Paper (Site not responding. Last check: 2007-10-21)
		The curve is marked off in units of "arclength", where arclength is the distance measured along the curve, just as if the curve were a piece of rope that you could stretch out next to a ruler.
		In this context, the most natural way to describe a plane curve is by an equation that gives the curvature directly as a function of arclength, an equation of the form kappa = f(s), where s stands for arclength and kappa is the commonly used symbol for curvature.
		It turns out the baseball stitch curve is based on something so prosaic as a patented 1860s pen and ink drawing of a plane shape used to cut out the leather for a half of a baseball, a shape arrived at by trial and error.
	www.mathcs.sjsu.edu /faculty/rucker/kaptaudoc/ktpaper.htm (3020 words)

	[No title]
		He proved that two of the remaining surfaces, the real projective plane and the Klein bottle, could not be immersed tightly in 3-space even as topological surfaces and he conjectured that the final case, a real projective plane with one handle, could not be immersed tightly into 3-space.
		A convex curve in the plane has the TPP, whether it is smooth or polygonal or a more general topological embedding of the circle.
		If an embedded curve in the plane is not convex then there it does not coincide with the boundary of its convex hull, and there is a segment in the convex hull boundary containing points not in the curve.
	www.geom.uiuc.edu /~banchoff/WKTFB/intro.html (1502 words)

	Sierpinski curve (Site not responding. Last check: 2007-10-21)
		The curve is also known as the Sierpinski (universal plane) curve, Sierpinski square or the Sierpinski carpet.
		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)

	Curve definitions (Site not responding. Last check: 2007-10-21)
		Isoptic curve : For a given curve C consider the locus of the point P from where the tangents from P to C meet at a fixed given angle.
		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-andrews.ac.uk /~history/Curves/Definitions2.html (1325 words)

	Getzler Abstract WSU Math (Site not responding. Last check: 2007-10-21)
		Abstract: A plane curve of degree d is the set of solutions f(z_1,z_2)=0 of a polynomial of degree d in two variables.
		For example, a rational plane curve of degree 1 is a line, and it was already known to the Greeks that just one line passes through 2 points.
		Likewise, a rational plane curve of degree 2 is a conic, and it can be shown that just one conic passes through 5 given points.
	www.math.wayne.edu /~claude/abst-getzler.html (389 words)

	Plane Filling Curves
		On another occasion I mentioned that Peano's definition of his plane filling curve was entirely analytic.
		36], however, that Peano's original curve could be obtain geometrically as the limit of a sequence of curves as was the case with Hilbert's curve.
		split into pairs, in each of which the curves are obtained from each other by central symmetry with respect to the center of the unit square.
	www.cut-the-knot.com /Curriculum/Geometry/Peano.shtml (235 words)

	Lordosis: Assessment and Care
		However, the curve expressed by the tips of the spinous processes is not identical to that of the bodies of the vertebrae in one and often two ways.
		Four of the five lumbar curves shown are inconsistent with the pelvic tilt from a purely pelvic tilt/lumbar curve model.
		In assessing lumbar curve, the orientation of the thorax is not to be considered.
	www.amtamassage.org /journal/su02_journal/lordosis.html (2174 words)

	Plane Filling Curves
		Disregarding the pieces of the curve that cross small square borders we see (starting with the second stage) that those small squares contain small replicas of the curve drawn on the previous stage, i.e.
		As before, when placing curves in the smaller squares two of the staples are placed in the direction of the bigger one they replace whereas one is turned left and another right quarter turn relative to the position of the parent staple.
		The plane is known to be a complete space implying that the sequence converges to a point in the unit square which I denote f(t).
	www.cut-the-knot.org /do_you_know/hilbert.shtml (1102 words)

	Curves in Space
		Differential geometry is the investigation and characterization of the local properties of curves and surfaces in space; that is, in the neighbourhood of a point on the curve or surface.
		A tangent vector to the curve is (dx,dy) = (x'dt,y'dt).
		We are interested in finding the normal vector and tangent plane to the curve at a point on the surface, the length of curves drawn on the surface, and the curvature of the surface.
	www.du.edu /~etuttle/math/space.htm (3331 words)

	PARABOLA - Online Information article about PARABOLA (Site not responding. Last check: 2007-10-21)
		4) is a cubic curve having the equation y=ax3+bx2+cx+d.
		a2) (x —b) and the curve resembles the parabolic branch, as in the preceding case.
		Newton showed that all the five varieties of the diverging parabolas may be exhibited as plane sections of the solid of revolution of the semi-cubical parabola.
	encyclopedia.jrank.org /PAI_PAS/PARABOLA.html (1854 words)

	AllRefer.com - curve (Mathematics) - Encyclopedia
		curve, in mathematics, a line no part of which is straight; more generally, it is considered to be any one-dimensional collection of points, thus including the straight line as a special kind of curve.
		For examples of plane curves, see circle; ellipse; hyperbola; parabola.
		A twisted or skew curve is one that does not lie all in one plane, e.g., the helix, a curve having the shape of a wire spring.
	reference.allrefer.com /encyclopedia/C/curve.html (175 words)

	Curvature, Intrinsic and Extrinsic
		A plane curve can be expressed parametrically as a function of the path length s by the functions x(s), y(s). Since (ds)
		) is perpendicular to the curve. The magnitude of this vector is
		of the two principal extrinsic curvatures relative to a flat plane tangent to the surface at the point of interest. The reason this formula is so complicated is that it applies to any system of coordinates (rather than just projected tangent normal coordinates), and is based entirely on the intrinsic properties of the surface.
	www.mathpages.com /rr/s5-03/5-03.htm (2403 words)

	Curve - Xah: Special Plane Curves: Naming and Classification of Curves
		A Bezier curve in its most common form is a simple cubic equation that can be used in The Bézier curve slider demo (requires Shockwave for Director 5).
		Curves you've heard of and curves you haven't, from Astroid to the Witch of Agnesi.
		The Peano-Gosper curve is a plane-filling curve originally called a "flowsnake" by The Gosper island bounds the space that the Peano-Gosper curve fills.
	pheromone.allinfosites.com /q/pheromone-curve.htm (629 words)

	Intersect Curve with Plane
		If the curve is moved or modified the projected reference point(s) will update the next time the active part or sketch is updated.
		When the curve is an edge, a small line is referenced into the sketch and not a point.
		The length of the line has no meaning, but the direction of the line is the tangent to the intersection curve between the plane and the face neighboring the edge.
	www.vx.com /help/0836.htm (253 words)

	POV-Ray: Newsgroups: povray.binaries.animations: mathematics: animation concerning a plane curve using povray (a new ...
		POV-Ray: Newsgroups: povray.binaries.animations: mathematics: animation concerning a plane curve using povray (a new attempt to post): mathematics: animation concerning a plane curve using povray (a new attempt to post)
		The "quadratrix" is a famous curve that plays a historical role in the problem of the trisection of an angle and the quadrature of a circle.
		More details about the curve and it's use can be found on http://cage.ugent.be/~hs/quadratrix/quadratrix P.S. Some days ago I already tried to post this message, but without success.
	news.povray.org /41d9584a@news.povray.org (194 words)

	M-curves (Site not responding. Last check: 2007-10-21)
		Another way to visualize the projective plane is to imagine the disk with boundary and identify the diametrically opposite points of the boundary (this is the Poincaré disk model).
		If this system has a non-zero solution the curve is said to be singular and the solution is said to be a singular point of the curve.
		In order to describe the isotopy class of a curve we will use the coding scheme devised by Viro[6].
	pages.prodigy.net /danesmith/mcurve/mcurve.html (889 words)

	New Period Mappings For Plane Curve Singularities - Kaenders (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		For plane curve singularities we construct a mixed Hodge structure (MHS) over Zon the fundamental group of the Milnor fiber.
		The concept nearby fundamental group is introduced and we develop a theory of iterated integrals along elements of this group.
		1.3: A Note on the Hodge Theory of Curves and Periods of Iterated..
	citeseer.ist.psu.edu /kaenders98new.html (597 words)

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