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Topic: Cubic curve

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Plane curve

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	Paths - SVG 1.1 - 20030114
		Alternate forms of curve are available to optimize the special cases where some of the control points on the current segment can be determined automatically from the control points on the previous segment.
		A cubic Bézier segment is defined by a start point, an end point, and two control points.
		Draws a cubic Bézier curve from the current point to (x,y) using (x1,y1) as the control point at the beginning of the curve and (x2,y2) as the control point at the end of the curve.
	www.w3.org /TR/SVG/paths.html (5558 words)

	Search ScienceWorld
		Informally, an elliptic curve is a type of cubic curve whose solutions are confined to a region of space that is topologically equivalent to a torus.
		An algebraic curve over a field K is an equation f(X,Y)==0, where f(X,Y) is a polynomial in X and Y with coefficients in K, and the degree of f is the maximum degree of each of its terms (monomials).
		The pedal of a curve C with respect to a point O is the locus of the foot of the perpendicular from O to the tangent to the curve.
	scienceworld.wolfram.com /search/index.cgi?num=&q=Curves (532 words)

	Curve - LoveToKnow 1911 (Site not responding. Last check: 2007-10-29)
		A quartic curve has 28 double tangents, their points of contact determined as the intersections of the curve by a curve II = o of the order 14, the equation of which in a very elegant form was first obtained by Hesse (1849).
		We have in like manner, as derivatives of a given curve, the caustic, catacaustic or diacaustic as the case may be, and the secondary caustic, or curve cutting at right angles the reflected or refracted rays.
		And it then appears that there are two kinds of non-singular cubic cones, viz, the simplex, consisting of a single sheet, and the complex, consisting of a single sheet and a twin-pair sheet; and we thence obtain (as for cubic curves) the crunodal, the acnodal and the cuspidal kinds of cubic cones.
	www.1911encyclopedia.org /Curve (7677 words)

	Cubic Bezier Curves on the Canvas (Site not responding. Last check: 2007-10-29)
		segment curve, where in the first case the first knot of the first segment is reused as the last knot in the last segment, and in the second case the first knot and control point in the first segment are reused as the last control point and knot in the last segment respectively.
		Cubic Bezier curves, being for example the native curve format in Postscript and its descendants, is probably the most common format for smooth curves in computing today.
		curves of the Tk canvas are splines (of degree
	www.tcl.tk /cgi-bin/tct/tip/168.xml (1126 words)

	CSC 418: Parametric Curves
		Cubic curves are commonly used in graphics because curves of lower order commonly have too little flexibility, while curves of higher order are usually considered unnecessarily complex and make it easy to introduce undesired wiggles.
		The curves for y(t) and z(t) are contructed in an analogous fashion to that for x(t).
		Hermite curve segments can be connected in a continuous fashion by ensuring that the end-point of one curve is the same as the starting point of another, as well as ensuring that the tangent vectors for this point have the same direction.
	www.dgp.toronto.edu /~ah/csc418/fall_2001/notes/curves.html (1222 words)

	Cubic Curve -- from Wolfram MathWorld
		Newton's classification of cubic curves appeared in the chapter "Curves" in Lexicon Technicum by John Harris published in London in 1710.
		This is the hardest case and includes the serpentine curve as one of the subcases.
		Newton's classification of cubics was criticized by Euler because it lacked generality.
	mathworld.wolfram.com /CubicCurve.html (330 words)

	Implementing Curved Surface Geometry
		The downside of curves and curved surfaces is that they are perhaps the most difficult of the three methods to learn and understand.
		Unfortunately, second-degree curves will always lie in a plane, and we're working in three dimensions, so it would be better to have a space curve, a curve that isn't confined to two dimensions or less.
		A Hermite curve is a cubic curve described by its endpoints p0 and p1 and the tangent vectors at the endpoints, v0 and v1.
	www.gamasutra.com /features/20000530/sharp_pfv.htm (4658 words)

	A Catalog of Cubic Plane Curves
		This cubic plane curve is asymptotic to the semicubical parabola.
		This cubic curve is asymptotic to a line, and crosses itself at a node located at the origin.
		This cubic curve is asymptotic to a line, and has a cusp at the origin.
	staff.jccc.net /swilson/planecurves/cubics.htm (905 words)

	Gamasutra - Features - "An In-Depth Look at Bi-Cubic Bezier Surfaces" [10.27.99]
		The bounding curves of the surface are Bézier curves dictated purely by the control points at the edge.
		Bicubic Bézier surfaces are bounded by cubic Bézier curves.
		These curves are defined by four points, the simplest form of curve which is not constrained within a plane.
	www.gamasutra.com /features/19991027/deloura_01.htm (535 words)

	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 (1018 words)

	Mathcad Library
		Cubic splines are a common method of approximating a curve.
		A cubic polynomial is completely determined if its value and the value of its derivative is specified at two distinct points.
		This property of cubic polynomials is used for creating "good-looking", that is, continuous, smooth and well behaved interpolation curves.
	www.mathcad.com /library/LibraryContent/MathML/complex_splines.htm (1486 words)

	Surface Construction Schemes
		Now we have two curves, both are cubic, and they have exactly the same knots, and exactly the same number of control vertices.
		A sweep surface of this curve should contain all the curves generated by rotating this curve, and its horizontal cross sections should always be circles.
		At each control vertex of the curve, we form a network similar to the trajectory circle except that we set their radii to be the Y-coordinate of each control vertex, connect the network vertically.
	www.math.hmc.edu /~gu/math142/mellon/Application_to_CAGD/Surface_Construction_Schem.html (2564 words)

	Hermite Help Page
		A piecewise cubic hermite curve is a curve that is represented with four degrees of freedom.
		Pictorially this is displayed on the "Curve Editor:" canvas with the beginning control point of the hermite curve displayed in red, and the end control point displayed in green.
		The point in time of this parametric curve is displayed by a fl dot on the "Curve Editor" canvas and by a fl vertical line on the hermite basis functions canvas.
	www.cs.utah.edu /~dav/curve_ed/hermite_help.html (261 words)

	Piecewise Cubic Bézier curves
		The curve is made continuous by the setting the tangents the same at the join.
		If the curve is being used for animation steps then the strength also controls the velocity, note the samples shown in red are further apart for the long tangent vectors.
		A common application for these curves in computer graphics is the creation of a smooth flight path that passes through keyframe points in space.
	local.wasp.uwa.edu.au /~pbourke/surfaces_curves/bezier/cubicbezier.html (698 words)

	On-Line Geometric Modeling Notes
		The Bézier curve representation is one that is utilized most frequently in computer graphics and geometric modeling.
		The curve is continuous, infinitely differentiable, and the second derivatives are continuous (automatic for a polynomial curve).
		quadratic curve - the primary difference is that we have four control points and must proceed one additional level in the recursion to get a point on the curve.
	graphics.idav.ucdavis.edu /education/CAGDNotes/Cubic-Bezier-Curves/Cubic-Bezier-Curves.html (358 words)

	Cubic plane curve - Wikipedia, the free encyclopedia
		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's elliptic functions, defining a quadratic extension of the field of rational functions made by extracting the square root of a cubic.
	en.wikipedia.org /wiki/Cubic_curve (366 words)

	Dr. Dobb's \| Forward Difference Calculation of Bezier Curves \| April 15, 2003
		Another property of the Bezier curve is that it leaves its first control point on a tangent to the line segment between the first and second control points and enters the final control point on a tangent to the line segment connecting the last and next-to-last control point.
		Since a Bezier curve is invariant under such transformations, transforming the control points of a curve has the same effect as transforming the curve itself.
		A cubic Bezier curve is described by four points: the first and fourth describe the endpoints of the curve; the second and third control the tangents of the curve at the endpoints, as well as the magnitude of the curvature.
	www.ddj.com /184403417?pgno=2 (2213 words)

	CMPT 361: Curves
		When the curve to be approximated is smooth, it may take a large number of points to achieve a reasonable degree of accuracy, and interactive manipulation of these points is tedious.
		Each curve segment is given by 3 functions, x,y, and z, which are cubic polynomials in the parameter t.
		Bezier curves specify the tangent vectors for the endpoints indirectly with two points that are not on the curve.
	www.cs.sfu.ca /~torsten/Teaching/Cmpt361/LectureNotes/HTML/08_2Dcurves (2121 words)

	cubic curve
		An algebraic curve described by a polynomial equation of the general form:
		One of Isaac Newton's many accomplishments was the classification of the cubic curves.
		Newton found 72 different species of curve; later investigators found six more, and it is now known that there are precisely 78 different types of cubic curves.
	www.daviddarling.info /encyclopedia/C/cubic_curve.html (177 words)

	Paths
		A cubic Bézier curve shall be drawn from the current point to (x,y) using (x1,y1) as the control point at the beginning of the curve and (x2,y2) as the control point at the end of the curve.
		A cubic Bézier curve shall be drawn from the current point to (x,y).
		A quadratic Bézier curve is drawn from the current point to (x,y) using (x1,y1) as the control point.
	www.w3.org /TR/SVGMobile12/paths.html (2245 words)

	CubicCurve2D (Java 2 Platform SE v1.4.2)
		Returns the square of the flatness of the cubic curve specified by the controlpoints stored in the indicated array at the indicated index.
		Returns the square of the flatness of the cubic curve specified by the indicated controlpoints.
		The cubic solved is represented by the equation: eqn = {c, b, a, d} dx^3 + ax^2 + bx + c = 0 A return value of -1 is used to distinguish a constant equation, which may be always 0 or never 0, from an equation which has no zeroes.
	java.sun.com /j2se/1.4.2/docs/api/java/awt/geom/CubicCurve2D.html (2221 words)

	Approximating Cubic Bezier Curves in Flash MX
		It is possible to ensure continuity of two bezier curves by making sure that the tangent to the last control point of the first curve is the same as the tangent to the first control point of the second curve.
		Indeed we don't calculate any point on the cubic curve through a polynomial equation, we don't have to derive any new control points as they are already available to us, the only thing we need to do is to calculate the middle of a segment, which is straightforward.
		Furthermore, we know that the curve must be continuous, two adjacent quadratic curves in the approximation must therefore have the same tangents at their common point.
	timotheegroleau.com /Flash/articles/cubic_bezier_in_flash.htm (4207 words)

	Cubic-Bézier Approximation Of Side-View Curve Of Helix (Site not responding. Last check: 2007-10-29)
		It is well known that scaling a Bézier curve scales in the same way its control polygon and vice versa, and since the initial untilted spiral curve is first scaled only along the y-axis by cos(φ), its control point y-values will be scaled the same way.
		Furthermore, in cases where the same parameter set is used to derive both approximations, a Bézier approximation of the sum of two true curves is the same as the sum of their respective Béziers approximations.
		Our two added curves are the scaled untilted spiral and an ellipse which is a circle of radius R, centered at the origin, and scaled by sin(φ) along the y-axis.
	members.aol.com /mszlazak/helix.html (2598 words)

	Approximating Cubic Bezier Curves in Flash MX
		It is possible to ensure continuity of two bezier curves by making sure that the tangent to the last control point of the first curve is the same as the tangent to the first control point of the second curve.
		Indeed we don't calculate any point on the cubic curve through a polynomial equation, we don't have to derive any new control points as they are already available to us, the only thing we need to do is to calculate the middle of a segment, which is straightforward.
		Since that point is on the cubic curve and since we know the formula for the cubic bezier, we can calculate the exact tangent to the bezier curve.
	www.timotheegroleau.com /Flash/articles/cubic_bezier_in_flash.htm (4207 words)

	TIP #168: Cubic Bezier Curves on the Canvas (Site not responding. Last check: 2007-10-29)
		A spline is a curve that passes through a set of given points (the knots of the curve) in a given order, satisfies some smoothness condition, and in some sense is best possible under these conditions.
		The -smooth 1 curves of the Tk canvas are splines (of degree 2) in this sense, even though the points used for defining them are (with the exception for endpoints) not the knots of the spline.
		It would also be possible to use the semantics of the Tkspline package [1] -smooth method providing cubic Bezier curves, but that package reverts to using the -smooth true method of smoothing if the number of points is not 3N+1, which doesn't seem useful and may cause bugs to be less visible.
	www.tcl.tk /cgi-bin/tct/tip/168.html (1030 words)

	New Headers:
		The curve is always contained within the convex hull of the control points, it never oscillates wildly away from the control points.
		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.
		The curve always passes through the end points and is tangent to the line between the last two and first two control points.
	www.cs.indiana.edu /~yinli/PET.htm (1910 words)

	4lines2cubic
		A general property of a cubic curve is that a line intersects it three times, counting multiplicities and complex roots.
		Since the cubic is formed with the product of the functions representing three lines being –1, only the regions where the signs of the three lines multiply to a negative are occupied.
		At the point where line1 meets the line at infinity, both terms are zero, so the cubic goes through this point, which because of the cube is a triple point, meaning that line1 meets the line at infinity at a point of inflection.
	www.paideiaschool.org /TeacherPages/Steve_Sigur/lines2cubic.htm (621 words)

	CubicCurve2D (Java Platform SE 6)
		Returns the flatness of the cubic curve specified by the control points stored in the indicated array at the indicated index.
		Returns the square of the flatness of the cubic curve specified by the control points stored in the indicated array at the indicated index.
		Returns the square of the flatness of the cubic curve specified by the indicated control points.
	java.sun.com /javase/6/docs/api/java/awt/geom/CubicCurve2D.html (2712 words)

	Computer curve construction system II - Patent 7038682
		The peak point is the point on the curve that is farthest away from the chord sometimes referred to as the shoulder point of the curve.
		In this particular embodiment the whole curve component c is symmetric with respect to the axis that is perpendicular to the chord and goes through the center of the chord.
		This curve c is already drawn when the mouse button (input device) is pressed for the end point, and when the mouse is dragged, the end point is dragged to a new location, and the curve c is changed, the final shape of which is drawn when the mouse is released.
	www.freepatentsonline.com /7038682.html (6810 words)

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