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Topic: Cubic Hermite curve

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	Hermite Help Page
		A piecewise cubic hermite curve is a curve that is represented with four degrees of freedom.
		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.
		The curve is called a piecewise curve because it is created/drawn one piece at a time.
	www.cs.utah.edu /~dav/curve_ed/hermite_help.html (261 words)

	RENDERMAN - RICURVES BASICS
		In the case of curves generated by Maya and mtor the curve type is "b-spline" ie.
		The simpliest curve type to use is a "b-spline" because it can be defined by any number of cv's - greater than 4.
		The only time when the restriction on the number of cv's can be ignored is when a "periodic" curve is produced, in which case, the end of the curve wraps around to coincide with the beginning of the curve.
	www.fundza.com /rman_helper/ri_curves_basics (503 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)

	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.cs.helsinki.fi /group/goa/mallinnus/curves/curves.html (1474 words)

	[No title] (Site not responding. Last check: )
		Hermite and spline curves can be generated in a similar manner.
		A Hermite piecewise polynomial would be appropriate if the derivatives (dx/dt and dy/dt) at the endpoints of each of the sections were known.
		The Bezier curve requires 4 points (the two end points of the curve and two guidepoints which are taken to be on the tangents to the endpoints).
	www.cs.uregina.ca /~norma/cs261-3.5.html (678 words)

	Encyclopedia: Number theory (Site not responding. Last check: )
		This sense of the term arithmetic should not be confused either with elementary arithmetic, or with the branch of logic which studies Peano arithmetic as a formal system.
		Algebraic geometry, especially the theory of elliptic curves, may also be employed.
		In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections.
	www.nationmaster.com /encyclopedia/Number-theory (4687 words)

	Cubic Hermite spline - Wikipedia, the free encyclopedia
		In the mathematical subfield of numerical analysis a cubic Hermite spline, named in honor of Charles Hermite (Hermite is pronounced air MIT), is a third-degree spline with each polynomial of the spline in Hermite form.
		The Hermite form consists of two control points and two control tangents on each for each polynomial.
		Since each subinterval must share tangents with neighboring subintervals, many techniques exist to determine values for shared tangents.
	en.wikipedia.org /wiki/Cubic_Hermite_spline (131 words)

	Parametric Cubic Curves (Site not responding. Last check: )
		In a slope-intercept form function, the slope of a curve is defined by the infamous three words: "rise over run." When the "run" is zero, a dimensional vortex opens and sucks you away to the outer limits.
		The simplest of the cubic curves, the Hermite form, is defined by two endpoints and the tangent vectors at these endpoints.
		, parameterizes the endpoints and tangent vectors of the Hermite curve.
	ript.net /~spec/curves (2373 words)

	U.S. Treasury - Treasury Yield Curve Methodology
		Because the on-the-run securities typically trade close to par, those securities are designated as the knot points in the quasi-cubic hermite spline algorithm and the resulting yield curve is considered a par curve.
		To reduce volatility in the 1-year CMT rate, and due to the fact that there is no on-the-run issue between 6-months and 2-years, Treasury uses an additional input to insure that the 1-year rate is consistent with on-the-run yields on either side of it’s maturity range.
		Yield curve rates are normally available at Treasury’s interest rate web sites as early as 5:00 PM and usually no later than 6:00 PM each trading day.
	www.ustreas.gov /offices/domestic-finance/debt-management/interest-rate/yieldmethod.html (498 words)

	Hermite Curve Interpolation
		We'll lose some of the flexibility of the hermite curves, but as a tradeoff the curves will be much easier to use.
		It's damn difficult, but when they are derived from hermite curves the cardinal splines turn out to be very easy to understand.
		They share one thing with the hermite curves: They are still cubic polynomials, but the way they are calculated is different.
	cubic.org /docs/hermite.htm (1073 words)

	Computer Graphics : Last Lecture : 10 / 24 : Spline
		Quadratic curves are not flexible enough and going above degree 3 gives rises to complications and so the choice of cubics is the best compromise for most computer graphics applications.
		The tangent vectors to the curve at the end points are coincident with the first and last edge of the control point polygon.
		The curve is transformed by applying any affine transformation (that is, any combination of linear transformations) to its control point representation.
	escience.anu.edu.au /lecture/cg/Revisal/spline.en.html (447 words)

	ceg477 midterm 2
		For two curve segments to join smoothly at a point, is it required that their first derivatives at the point be the same or their second derivative?
		The conrol points in a rational Bezier curve have weights associated with them, which may be varied to produce different curves even with the same control points.
		Find the equation of a Hermite cubic curve that will pass through points (0,0) and (1,1) and will be tangent to the x axis at (0,0) and have slope 1 at (1,1).
	www.cs.wright.edu /people/faculty/agoshtas/ceg477mid2qa.html (510 words)

	Parametric Cubic Curves (Site not responding. Last check: )
		You can choose which curve to show with the buttons under the applet and reset the applet with the reset button.
		There is not a big visual difference between the Hermite and the Bézier curves, but Hermite curves use the tangent vector of the endpoint as a geometric constraint of the curve whereas Bézier uses another point relative the origin as constraint.
		This makes splines suitable for curves consisting of large amount of points since only the four segments that is affected by a point has to be recalculated when a control point is moved.
	www.sm.luth.se /%7Epeppar/presentations/bibdc961114/misc_applets/ParamCurve (248 words)

	Home · (Site not responding. Last check: )
		The stretching and curving forces are provided by actual edges/gradients within the image which seek to attract a nearby part of the curve.
		The more the curve is deformed by the edge forces, the greater is its internal 'reaction'.
		The principal idea of providing the curve with its own internal properties is to curtail the extent to which noisy and spurious edge information will distort it.
	www.thisismad.freeserve.co.uk /Snake-exe.html (616 words)

	Hermite Spline
		The Hermite Spline command creates an interpolation curve called a Hermite cubic spline, in which the tangent direction can be defined at each interpolation point.
		The shape of the Hermite spline is determined by the direction of these tangents and by a shared value, called the stretch factor.
		For each input point to be interpolated, you must specify the direction of the tangent to the interpolating curve at that point.
	thinkcare.think3.com /thinkcare/english/curves/cv_04.htm (378 words)

	The Bezier Curve
		The Hermite polynomial is referred to as a "clamped cubic," where "clamped" refers to the slope at the endpoints being fixed.
		Bézier curves are the basis of the entire Adobe PostScript drawing model which is used in the software products: Adobe Illustrator, Macromedia Freehand and Fontographer.
		The construction of a Bézier curve using Bernstein polynomials is more appealing mathematically because the coefficients in the linear combination are just the coordinates of the given four points.
	math.fullerton.edu /mathews/n2003/BezierCurveMod.html (396 words)

	Hermite Curve Interpolation
		Hermite curves are very easy to calculate but also very powerful.
		Understanding the mathematical background of hermite curves will help you to understand the entire family of splines.
		Maybe you have some experience with 3D programming and have already used them without knowing that (the so called kb-splines, curves with control over tension, continuity and bias are just a special form of the hermite curves).
	www.cubic.org /docs/hermite.htm (1073 words)

	CS284 Fall2000 Project Ideas
		A geodesic line on a surface is one for which its normal vector at every point is in line with the normal vector of the surface; i.e., the curve bends only as necessary to follow the surface, but does not bend unnecessarily within the surface.
		Find a good heuristic approximation to move the control points of a curve in such a way as to minimize some functional over the arclength of the curve, e.g., curvature squared, square of curvature derivative, torsion squared, or combinations of such terms.
		Build a wavelet-based curve editor that allows global optimization of 2D or perhaps 3D curves at interactive speeds.
	www.cs.berkeley.edu /~sequin/CS284/LECT/ProjectIdeas.html (1111 words)

	Re: hermite curve -- Jeremy Drouin -- 2003/05/01
		Well, I am trying to animate for motion paths along curves.
		On Thu, 1 May 2003 08:08:16 +0100, Andy Hayes wrote: > I must admit that studying curves is on my "to-do" list but I thought > that > hermite curves were a representation of cubic curves where the symmetry > of > the curve is preserved, i.e.
		with this in mind if all the tangency handles on the curves > where the same length and parallel with the horizontal axis you would > have a > hermite curve, following the (3-2x)* x^2 interpolation equation form?
	www.softimage.com /community/xsi/discuss/archives/xsi.archive.0305/msg00008.htm (468 words)

	[No title] (Site not responding. Last check: )
		This deformation technique will be used to simulate squash and stretch by constraining the orientation of the object so that it's medial axis is in the same direction as the object's velocity.
		The deformed medial axis will be represented by an interpolating cubic Hermite curve.
		Every point on the medial axis (Hermite curve) not only has a position but also a local frame defined by the three normal vectors T, N, B, where T is in the direction of the derivative of the curve.
	www.cs.unc.edu /~reb/classes/comp259/project/update.html (527 words)

	CS284 Lecture Summary and Links (Site not responding. Last check: )
		Generate a piecewise cubic Hermite interpolation curve through these points, using non-uniform knot spacing.
		You should not use build-in functions in Matlab, Mathematica, or other packages for generating cubic Hermite interpolation.
		The first part of every function should contain a block of comments that give a detailed description of the code's purpose, usage, and its input and output variables.
	www-inst.eecs.berkeley.edu /~cs284/program1.2004.html (203 words)

	Cubic Hermite spline - Encyclopedia, History, Geography and Biography (Site not responding. Last check: )
		Cubic Hermite spline - Encyclopedia, History, Geography and Biography
		In the mathematical subfield of numerical analysis a cubic Hermite spline, named in honor of Charles Hermite (pronounced air MEET), is a third-degree spline with each polynomial of the spline in Hermite form.
		This encyclopedia, history, geography and biography article about Cubic Hermite spline contains research on
	www.arikah.net /encyclopedia/Hermite_curve (160 words)

	Xplode introduction
		Basehermite returns the graph of the cubic Hermite basis polynomials
		Beziercurve generator of coordinate functions of B´ezier curves of arbitrary degree.
		Hermite generator of the coordinate functions of a cubic Hermite curve
	www.dia.uniroma3.it /~paoluzzi/plasm/cplasm/docs/curves.html (311 words)

	[No title] (Site not responding. Last check: )
		This worksheet will illustrate how to plot cubic Hermite and Bezier curves once the corresponding x(t) and y(t) equations have been found.
		The following screen shot illustrates how to plot the Bezier curve that fits the same data used to construct the cubic Hermite curve in the first example.
		Notice again that the equations and range are included inside the brackets and that the range is from 0 to 1.
	www.math.byu.edu /math311/mparplot.html (230 words)

	GENIE++ DOCUMENT
		In order to blend two curves, you must define the new blended curve to equal the sum of the points of the two curves to be blended.
		For example, if the two curves to be blended have an I varying index from 1 to 5, then the new (blended) curve must be defined with an I varying index of 1 to 10.
		For example, a curve defined from 1 10 1 1 1 1 with the first point at 0,0,0 and the last point at 1,1,0, will be redefined with the first point being 1,1,0 at index 1 1 1 and the second point being 0,0,0 at index 10 1 1.
	www.eng.uab.edu /me/Faculty/bsoni/Research/genie++/menu21.html (7688 words)

	Re: Hermite to Bezier problem
		The original representation is a Cubic Hermite Spline with Basis: > >[ 2.0 1.0 -2.0 1.0 ] >[ -3.0 -2.0 3.0 -1.0 ] >[ 0.0 1.0 0.0 0.0 ] >[ 1.0 0.0 0.0 0.0 ] > >It is a one dimensional spline, even though control points have two >coordinates (x,y).
		The y ordinate is interpolated using the hermite >basis and a proprietary (simple) algorithm to produce tangents (is that >the right term?
		There was a trick once for this; use non-uniform splines and take the square root of the distance between points to scale things.
	www.usenet.com /newsgroups/comp.graphics.algorithms/msg00748.html (409 words)

	Hermite Approximation for Offset Curve Computation (ResearchIndex)
		The present paper proposes a new method for calculating the G 1 - continuous o#set curve to a cubic B#ezier curve, based on the Hermite approximation technique.
		Introduction Cubic B#ezier curves are widely used in CAGD applications such as...
		7 A geometric characterization of parametric cubic curves (context) - Stone, DeRose - 1989 ACM DBLP
	citeseer.ist.psu.edu /294274.html (300 words)

	List of Curve Topics Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: )
		Looking For list of curve topics - Find list of curve topics and more at Lycos Search.
		Find list of curve topics - Your relevant result is a click away!
		Look for list of curve topics - Find list of curve topics at one of the best sites the Internet has to offer!
	www.karr.net /search/encyclopedia/List_of_curve_topics (249 words)

	AUFLIC --- Accelerated Unsteady Flow Line Integral Convolution (Site not responding. Last check: )
		Also, AUFLIC employs a fourth-order Runge-Kutta integrator with adaptive step size and error control in combination with cubic Hermite polynomial curve interpolation to achieve faster, more accurate pathline advection, and faster line convolution (due to evenly sampling the texture) than the Euler method.
		Each of these seeds travels through a different-length part of the same curve during the first time step of the SCAP, but they synchronously run though the same trace over the remaining time steps.
		Pathline reuse is an inter-SCAP operation by which the position a pathline passes through within a fractional time into the second time step of the previous SCAP is used to release a new seed at exactly the same global time, but in the first time step of the current SCAP.
	www.erc.msstate.edu /~zhanping/Research/FlowVis/AUFLIC/AUFLIC.htm (374 words)

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