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	Spline interpolation - Wikipedia, the free encyclopedia
		In the mathematical subfield of numerical analysis spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline.
		Spline interpolation is preferred over polynomial interpolation because the interpolation error can be made small even when using low degree polynomials for the spline.
		The spline of degree n which interpolates the same data set is not uniquely defined and we have to fill in n-1 additional degrees of freedom to construct a unique spline interpolant.
	en.wikipedia.org /wiki/Spline_interpolation (443 words)

	B-spline - Wikipedia, the free encyclopedia
		In the mathematical subfield of numerical analysis a B-spline is a special spline curve.
		The spline is contained in the convex hull of its control points.
		Quadratic B-splines with uniform knot-vector is a commonly used form of B-spline.
	en.wikipedia.org /wiki/B-spline (576 words)

	POV-Ray: Documentation: 2.4.1.7 Lathe
		The splines that are used by the lathe and prism objects are a little bit difficult to understand.
		Thus quadratic splines look much smoother than linear splines but the transitions at each point are generally not smooth because the slopes on both sides of the point are different.
		If you want the spline to be smooth between segments, points 3 and 4 on one segment and points 1 and 2 on the next segment must form a straight line and point 4 of one segment must be the same as point 1 on the next segment.
	www.povray.org /documentation/view/3.6.1/281 (816 words)

	Spline interpolation -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		Spline interpolation is preferred over (Click link for more info and facts about polynomial interpolation) polynomial interpolation because the (Click link for more info and facts about interpolation error) interpolation error can be made small even when using low degree polynomials for the spline.
		The spline of degree n which interpolates the same data set is not uniquely defined and we have to fill in n-1 additional (Click link for more info and facts about degrees of freedom) degrees of freedom to construct a unique spline interpolant.
		The natural cubic spline is approximately the same curve as created by the (Click link for more info and facts about spline device) spline device.
	www.absoluteastronomy.com /encyclopedia/s/sp/spline_interpolation.htm (535 words)

	How To Draw TrueType Glyph Outlines
		Quadratic B-Splines are used by TrueType to describe the glyph outline in a TrueType font file.
		Quadratic B-Spline curves are a class of parametric curves that define the path of multiple curve segments via a few control points.
		Quadratic B-Spline curves in particular and parametric curves in general are a well-researched topic of graphics in computer science.
	support.microsoft.com /kb/243285/en-us (2089 words)

	NAME
		The spline command computes a spline fitting a set of data points (x and y vectors) and produces a vector of the interpolated images (y-coordinates) at a given set of x-coordinates.
		A spline is a device used in drafting to produce smoothed curves.
		Computes a quadratic spline from the data points represented by the vectors x and y and interpolates new points using vector sx as the x-coordinates.
	www.math.utah.edu /cgi-bin/man2html.cgi?/usr/local/man/mann/spline.n (943 words)

	ipedia.com: Spline (mathematics) Article (Site not responding. Last check: 2007-10-21)
		In the picture, the curve that passes through A, B, C, and D is an interpolating spline (specifically, a linear spline) and the curve that passes through A and D, but not B and C, is an approximating spline (specifically, a Bézier spline).
		An important characteristic of splines is that they are given by polynomials, but only piecewise: different polynomials may be used in different parts of a curve.
		The simplicity of representation and the ease with which a complex spline's shape may be computed make splines popular representations for curves in computer science, predominantly in computer graphics but also for other kinds of interpolation, such as smoothing of digital audio.
	www.ipedia.com /spline__mathematics_.html (387 words)

	POV-Ray 3.1g Documentation - Lathe (Site not responding. Last check: 2007-10-21)
		keyword creates splines overcome the transition problem of quadratic splines because they also take the fourth point into account when drawing the curve between the second and third point.
		A quadratic spline gives you n-2 segments because the last point is only used for determining the slope as explained above (thus you'll need at least three points to define a quadratic spline).
		In case of cubic or bezier splines, the Sturmian root solver is always used because a 6th order polynomial has to be solved.
	math.hws.edu /eck/cs324/s01/pov-doc/pov244.htm (849 words)

	POV-Ray: Documentation: 2.4.1.8 Prism
		These curves are defined by a set of points which are connected by linear, quadratic, cubic or bezier splines.
		The last sub-prism of a linear spline prism is automatically closed - just like the last sub-polygon in the polygon statement - if the first and last point of the sub-polygon's point sequence are not the same.
		If you want the spline to be smooth between segments, point 3 and 4 on one segment and point 1 and 2 on the next segment must form a straight line and point 4 of one segment must be the same as point one on the next segment.
	www.povray.org /documentation/view/3.6.1/282 (948 words)

	[No title]
		Quadratic and cubic splines are different in that they do not only take other points into account when connecting two points but they also look smoother and - in the case of the cubic spline - produce smoother transitions at each point.
		Cubic splines overcome the transition problem of quadratic splines because they also take the fourth point into account when drawing the curve between the second and third point.
		Quadratic spline sub-prisms need an additional control point at the beginning of each sub-prisms' point sequence to determine the slope at the start of the curve.
	www.lobos.nih.gov /Charmm/Povray3/pov30020.html (2262 words)

	Lathe Object (Site not responding. Last check: 2007-10-21)
		There are other types of splines available with the lathe, which will result in smooth curving lines, and even rounded curving points of transition, but we will get back to that in a moment.
		Therefore, when using a quadratic spline, we must remember that the first point we specify is only there so that POV-Ray can determine what curve to connect the first two points with.
		The concept of splines is a handy and necessary one, which will be seen again in the prism and polygon objects.
	www.blastwave.org /docs/povray-3.50c/povdoc_069.html (1736 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		On page 3 of the overheads, I wrote % down a system of equations for a quadratic spline based on % the three data points (2,4), (5,9), and (7,4).
		This is easy for the % linear spline (just compute the slopes of the line % segments), and straightforward for the quadratic spline % (since you will have derived the coefficients for all % of the individual quadratic polynomials).
		It plots the function and % the spline, as well as the first, second, and third derivatives % of the function and the spline.
	iris.gmu.edu /~snash/math446-s98/diary/mar19.out (629 words)

	Bezier spline curves
		Linear Bezier spline is obtained by linear interpolation between two control points P
		Quadratic Bezier spline is obtained by deCasteljau algorithm as a linear interpolation between linear interpolation between control points P
		Note that P(t) subdivides the curve in two quadratic splines (see Fig.2).
	www.ibiblio.org /e-notes/Splines/Bezier.htm (402 words)

	Quadratic Spline Collocation Methods for Systems of Elliptic PDEs (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		Optimal order approximation to the solution is obtained, in the sense that the convergence order of the QSC approximation is the same as the order of the quadratic spline interpolant.
		9 Quadratic spline collocation methods for elliptic partial di..
		4 Parallel solvers for spline collocation equations (context) - Christara - 1996 ACM
	citeseer.ist.psu.edu /500428.html (372 words)

	Numerical Analysis Technical Reports
		Optimal quadratic spline collocation (QSC) is extended to non-uniform partitions by using a mapping function from uniform to non-uniform partitions.
		In the standard formulation, the quadratic spline is computed by making the residual of the differential equations zero at a set of collocation points; the resulting error is second order, while the error associated with quadratic spline interpolation is fourth order locally at certain points and third order globally.
		Multigrid methods for quadratic spline collocation methods are not straightforward: because the basis functions used with quadratic spline collocation are not nodal basis functions, thus the design of efficient restriction and extension operators is nontrivial.
	www.cs.toronto.edu /NA/reports.html (14589 words)

	Project 1 : Phase 1d (Site not responding. Last check: 2007-10-21)
		This is a quadratic b-spline approximation curve with two subdivision iterations.
		This is a quadratic b-spline approximation curve with three subdivision iterations.
		This is a quadratic b-spline approximation curve with four subdivision iterations.
	www.cc.gatech.edu /ugrads/k/krajee/Prj1/Prj1Phase1d/Prj1Phase1d.htm (701 words)

	Quadratic Spline Galerkin Method for the Shallow Water Equations on the Sphere - Layton, Christara, Jackson ... (Site not responding. Last check: 2007-10-21)
		Quadratic Spline Galerkin Method for the Shallow Water Equations on the Sphere - Layton, Christara, Jackson (ResearchIndex)
		Quadratic Spline Galerkin Method for the Shallow Water Equations on the Sphere (2002)
		Quadratic spline Galerkin method for the shallow water equation on the sphere.
	citeseer.lcs.mit.edu /layton02quadratic.html (463 words)

	[No title]
		Additionally, for each knot in the spline there will be a BezierVertex coorsponding to it and sharing a ControlPoint with the spline.
		Removing a knot from the spline necessaryly changes the set of basis functions, and thus the same curve is no longer able to be represented by the spline.
		If the spline is very has less than 3 segments, then it is too small for the algorithm in QBSpline::solve_system().
	rioja.sangria.cs.cmu.edu /tumble/doc/classQBSpline.html (2956 words)

	Digitale Bildverarbeitung - Uni Jena (Site not responding. Last check: 2007-10-21)
		Splines work like function and may be used at (nearly) every place, where a float or vector value is expected.
		Outside the range between the lowest and highest defined argument the value of the spline is constant and equal to the value for the lowest or highest argument.
		Cubic splines can be calculated in the range between the second and last but one argument.
	pandora.inf.uni-jena.de /p/e/noo/povsp/povsp.html (678 words)

	IntroCompGeom (Site not responding. Last check: 2007-10-21)
		A quadratic B-spline curve is a piecewise parametric quadratic curve which is continuous and has a continuous tangent (first parametric derivatives).
		The number of vertices in the control polygon of a quadratic B-spline is unlimited, but there must be at least three.
		A quadratic B-spline curve is drawn by using the midpoints of two consecutive edges of its control polygon as the beginning and end control points, and the corresponding B-spline control vertex where those to edges meet, as the middle control point respectively, of a Bézier curve as illustrated in the accompanying figure.
	www.cs.umanitoba.ca /~cs219/IntroCompGeom.html (1806 words)

	Amazon.com: Spline Regression Models (Quantitative Applications in the Social Sciences): Books (Site not responding. Last check: 2007-10-21)
		Spline Regression Models shows the nuts-and-bolts of using dummy variables to formulate and estimate various spline regression models.
		For some researchers this will involve situations where the number and location of the spline knots are known in advance, while others will need to determine the number and location of spline knots as part of the estimation process.
		Although spline regression models might sound like something complicated and formidable, they are really just dummy variable models with a few simple restrictions placed on them.
	www.amazon.com /exec/obidos/tg/detail/-/0761924205?v=glance (730 words)

	Bezier spline to quadratic polynomial fragment conversion - Patent 5422990
		This method involves converting the parabola to a spline, and minimising the area between the "y" curve of the original spline, and the "y" curve of the new, parabolic spline.
		If the spline has matched the area tolerance, but fails the angle tolerance at either end, it may be possible to adjust the angles to be within tolerance while the (albeit increased) area error remains within the area tolerance.
		The parametric equation for the spline given in Equation (1) has eight degrees of freedom in two dimensions (x,y) and these degrees of freedom can often be determined by specifying the start and end points with their respective tangent directions.
	www.freepatentsonline.com /5422990.html (9420 words)

	Optimal Quadratic Spline Collocation Methods for the (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		A quadratic spline collocation method approximates the solution of a differential problem by a quadratic spline.
		In the standard formulation, the quadratic spline is computed by making the residual of the differential equations zero at a set of collocation points; the resulting error is second order, while the...
		1 A spline collocation scheme for the spherical shallow water..
	citeseer.ist.psu.edu /624800.html (353 words)

	Spline interpolation - Encyclopedia, History, Geography and Biography
		In the mathematical subfield of numerical analysis spline interpolation is a special form of interpolation where the interpolant is a piecewise polynomial called spline.
		The coefficients can be found by choosing a $z_0$ and then using the recurrence relation:
		Spline interpolation, Definition, Spline interpolant, Linear spline interpolation, Quadratic spline interpolation, Cubic spline interpolation, Interpolation using natural cubic spline, Example, Linear spline interpolation, Quadratic spline interpolation and See also.
	www.arikah.net /encyclopedia/Spline_interpolation (642 words)

	Examples (Site not responding. Last check: 2007-10-21)
		All splines are shown in two dimensional interval [0,4] x [0,1].
		The next pair of pictures represents two consecutive functions at each scale of a multiresolution-like spline wavelet dictionary spanning the same space where each scale consists of quarter-integer translates of a fixed function.
		The dimension of this space is 515 and to represent it we will use the wavelet basis given by the multiresolution analysis and the dictionary having one less scale than the basis but consisting of quarter-integer translates of wavelets and scaling functions.
	www.ncrg.aston.ac.uk /Projects/BiOrthog/examples.html (807 words)

	Fast Fourier Transform Solvers for Quadratic Spline Collocation - Constas (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		Quadratic spline collocation methods are used for the discretisation of the BVPs.
		4 cubic spline collocation method for elliptic partial differe..
		1 The use of cubic splines in the solution of two-point bounda..
	citeseer.ist.psu.edu /constas96fast.html (585 words)

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