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Topic: Spline (mathematics)


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  Mathematics under the Microscope
I was thinking, for some reason, about sums of two integer squares (an archetypal mathematical problem: representation of numbers by forms) and a remarkable fact that the if m and m are sums of two integer squares (let us call them bisquare numbers) then their product mn also has the same property.
Based on your impressive record of mathematical achievements, you seem to be especially well-placed to help us find people with strong academic backgrounds whose areas of specialization might be particularly relevant to the work we do.
A digitisation of the book is available from The Cornell Library Historical Mathematics Monographs website; I paged through the book: it is a quite serious text where geometry of 4-dimensional Euclidean space is developed by analogy with Euclidean stereometry, that is, 3-dimensional geometry.
www.maths.manchester.ac.uk /~avb/micromathematics   (1644 words)

  
  Curve Segments
Spline curves and polygons are collectively termed "faces", while grids and spline surfaces are termed "hulls".
It is, however, useful to understand some of the concepts that arise from the mathematics of computer-generated curves because they affect your choice of curve type when you start creating in Houdini, and they influence the way you draw that curve.
The more knots there are in one area, the smaller the spline segments and, therefore, you have a greater degree of control over the spline in that area.
www.sidefx.com /docs/content/base/model_splines.xml   (1015 words)

  
  Spline (mathematics) - Wikipedia, the free encyclopedia
Splines are popular curves in these subfields because of the simplicity of their construction, their ease and accuracy of evaluation, and their capacity to approximate complex shapes through curve fitting and interactive curve design.
Spline functions for interpolation are normally determined as the minimizers of suitable measures of roughness (for example integral squared curvature) subject to the interpolation constraints.
Smoothing splines may be viewed as generalizations of interpolation splines where the functions are determined to minimize a weighted combination of the average squared approximation error over observed data and the roughness measure.
en.wikipedia.org /wiki/Cubic_spline   (2324 words)

  
 Appendix A. Spline Representations   (Site not responding. Last check: )
Splines are piecewise polynomials where the degree of the polynomial determines the continuity and smoothness of the function approximation.
One way then to view spline fitting is in the form of an energy functional such as Equation 6.
The same type of spline is used to produce the bowls associated with the rooms as discussed in Section 3.2.1.
www-2.cs.cmu.edu /afs/cs/project/jair/pub/volume16/drummond02a-html/node26.html   (1031 words)

  
 Spline (mathematics) - Free net encyclopedia   (Site not responding. Last check: )
Image:Bezier.png In the mathematical subfield of numerical analysis a spline is a special function defined piecewise by polynomials.
Splines are a popular representation of curves in these subfields because of the simplicity of their construction, their ease and accuracy of evaluation, and their capacity to approximate complex shapes through curve fitting and interactive curve design.
The space of all natural cubic splines, for instance, is a subspace of the space of all cubic C^2 splines.
www.netipedia.com /index.php/Quadratic_spline   (2606 words)

  
 eScience Lectures Notes : Spline
Slide 14 : The Gradient of a Cubic Spline
Slide 23 : Coefficients for Cubic Bezier Splines
The mathematical equivalent is the cubic polynomial spline.
cs.anu.edu.au /escience/lecture/cg/Spline/printNotes.en.html   (4330 words)

  
 Mathematician puts the curves in fonts, computer graphics
Actually, he is a mathematician who has played a key role in developing a field called spline theory that modern computers use to reproduce fonts in a wide variety of styles and sizes.
Spline functions -- otherwise known as piecewise polynomial functions -- are simple mathematical equations that trace the shape of complex curves and surfaces.
Schumaker is not only well known for his mathematical contributions, but also because of his work with students, his role as the editor of several mathematics-related journals and as the organizer of a number of international conferences, said Manfred V. Golitschek of Würzburg University in Germany.
www.vanderbilt.edu /News/register/July3_00/story7.html   (1060 words)

  
 Appendix A. Spline Representations   (Site not responding. Last check: )
Splines are piecewise polynomials where the degree of the polynomial determines the continuity and smoothness of the function approximation.
One way then to view spline fitting is in the form of an energy functional such as Equation 6.
The same type of spline is used to produce the bowls associated with the rooms as discussed in Section 3.2.1.
www.cs.cmu.edu /afs/cs.cmu.edu/project/jair/pub/volume16/drummond02a-html/node26.html   (1031 words)

  
 SPLINE - Interpolation and Approximation of Data   (Site not responding. Last check: )
SPLINE is a library of C++ routines, using double precision arithmetic, for constructing and evaluating spline functions.
Also included are a set of routines that return the local "basis matrix", which allows the evaluation of the spline in terms of local function data.
SPLINE is also available in a FORTRAN90 version and a MATLAB version.
www.scs.fsu.edu /~burkardt/cpp_src/spline/spline.html   (659 words)

  
 Spline Macro File Tutorial
Notice also that the colour of the previewed spline changes from red at the first point to white at the last: this lets us know which direction the spline is travelling in.
The next option we can try is the spline bias value: like the spline tension this is normally zero, which means that the spline is curved equally on either side of each point in the spline.
Bezier splines have the advantage over TCB splines of allowing you to set the curvature of the spline directly for each section, so you can create some sharp corners and some smooth, and generally have more control over the spline's shape.
www.geocities.com /ccolefax/spline   (1676 words)

  
 Cubic Spline Interpolation   (Site not responding. Last check: )
Another alternative is spline interpolation, which encompasses a range of interpolation techniques that reduce the effects of overfitting.
The cubic spline, along with the three points upon which it is based, is shown in Exhibit 1.
This is the first in a series of three-day courses on financial mathematics that take participants from pre-calculus to stochastic calculus.
www.riskglossary.com /articles/cubic_spline.htm   (371 words)

  
 Master's Degree Programs in Mathematics-Graduate Catalogue 2003-2004   (Site not responding. Last check: )
To ensure that the mathematical prerequisites have been met, any student not enrolled in the mathematics degree program or the Master of Education in secondary education mathematics track must obtain permission from the Department of Mathematical Sciences to register for any graduate course offered by the department.
The bachelor’s/master’s degree program is designed to provide a student in mathematics a means to complete the requirements for both degrees in a period of five years.
Completed a minimum of seventy-five (75) and a maximum of ninety-eight (98) credit hours in their undergraduate programs in mathematics, including credits earned from advanced placement if they started at UNCW or are transfer students and have completed a minimum of two semesters as a full-time student at UNCW, a minimum of 24 hours.
www.uncwil.edu /grad_cat/math.htm   (2958 words)

  
 FontForge's math
The maximum or minimum of a spline (along either the x or y axes) may be found by taking the first derivative of that spline with respect to t.
Except where the spline is parallel to one edge of the pen, a the outer contour of a rectangular pen will be stroked by one of its end-points.
When we reach a point where the spline's slope is parallel to one edge of the pen, then on the outside path we draw a copy of that edge of of the pen, and on the inside edge we calculate a join as above.
fontforge.sourceforge.net /pfaeditmath.html   (1284 words)

  
 Singularity » Parametric Spline Class   (Site not responding. Last check: )
The Parametric Spline class is built on top of the previous Spline3 Class.
Unlike Spline3, the parameteric spline is well suited for generalized drawing with overlapping intervals.
Splines of this nature are used in mathematical typography [Knuth 1979].
www.2112fx.com /blog/pivot/entry.php?id=37   (193 words)

  
 Splines: A Perfect Fit for Signal/Image Processing (ResearchIndex)   (Site not responding. Last check: )
The first one is to provide a tutorial on splines that is geared to a signal processing audience.
The second one is to gather all their important properties, and to provide an overview of the mathematical and computational tools available (Update)
5 Bivariate cardinal interpolation by splines on a threedirect..
citeseer.ist.psu.edu /unser99splines.html   (1379 words)

  
 CubicSpline Class   (Site not responding. Last check: )
A cubic spline is a piecewise curve defined by a third degree polynomial on each interval.
A natural spline is a spline curve whose second derivative at the end points are zero.
This type of spline tends to minimize the overall curvature of the spline.
www.extremeoptimization.com /Mathematics/Reference/Extreme.Mathematics.Curves.CubicSpline.html   (195 words)

  
 Department of Mathematics - Research
Spline wavelets with respect to uniform knots on the whole real line have been constructed and thoroughly analysed in the literature.
Using the known approximation properties of a nodal spline interpolation operator, as established in a sequence of papers by JM de Villiers and CH Rohwer, the aim of this project is to develop an error analysis for the quadrature rule which results from integration of the nodal spline interpolant.
The nodal spline interpolant is also applied in the context of product integration, and its efficiency is compared to those of existing methods in the literature.
academic.sun.ac.za /maths/research/Research98.html   (2037 words)

  
 IMA:Mathematics of Surfaces VII   (Site not responding. Last check: )
Proceedings of a seventh conference on "Mathematics of Surfaces" held in Dundee in September 1996.
This reflects the wide diversity of present-day applications of surface mathematics, which include modelling parts of the human body for medical and artistic purposes as well as the representation of cars, ships, aircraft and engineering components in general.
A variety of mathematical tools may be deployed to achieve these goals and the papers in The Mathematics of Surfaces VII report developments of many of them.
www.ima.org.uk /mathematics/procsurfaces7.htm   (531 words)

  
 University of Calgary: Calendar: Courses: Applied Mathematics
Prerequisites: One of Applied Mathematics 311 or 307, and one of Mathematics 331, 353, Applied Mathematics 309, or consent of the Division.
Prerequisites: Mathematics 311 or Applied Mathematics 311; and Mathematics 353 or Applied Mathematics 309 or Mathematics 331; or consent of the Division.
Note: All graduate students in Mathematics and Statistics are required to participate in one of Applied Mathematics 621, Pure Mathematics 621, Statistics 621 each semester.
www.ucalgary.ca /pubs/calendar/current/what/courses/AMAT.htm   (1092 words)

  
 Beta Analytic - World's Largest Radiocarbon Dating Service; AMS, Mass Spectrometry, C14, and more...   (Site not responding. Last check: )
It is always important to quote the data source and mathematics with your interpretations.
The program chosen for these dendrocalibrations uses splines through the tree-ring data as calibration curves, which eliminates a large part of the statistical scatter of the actual data points and gives a better "real" approximation of the sample's calendar equivalence using the measured correlation curve.
The spline calibration allows adjustment of the average curve by a quantified closeness-of-fit parameter to the measured data points.
www.radiocarbon.com /calendar.htm   (579 words)

  
 Tom Lyche
Byung-Gook Lee, Tom Lyche, and Knut Mørken, Some Examples of Quasi-Interpolants Constructed from Local Spline Projectors, in "Mathematical Methods for curves and surfaces, Oslo 2000", T. Lyche and L. Schumaker (eds.), Vanderbilt University Press, 2001, 243-252.
Lyche, T. and K. Mørken, The sensitivity of spline functions to perturbations of the knots, BIT 39 (1999), 305-322.
Lyche, T. and K. Scherer, On the Sup-norm Condition Number of the Multivariate Triangular Bernstein Basis, in Multivariate Approximation and Splines, G. Nürnberger, J. Schmidt, and G. Walz (eds.), ISNM.125, Birkhäuser Verlag, Basel, 1997, 141-151.
www.ifi.uio.no /~tom   (604 words)

  
 Ming-Jun Lai's Publication List
A spline function s of degree d and smoothness r is a r time continuously differentiable function over [a, b] and is a polynomial of degree d when restricted to [x_i, x_{i+1}] for each i=0,..., n-1.
A spline function s of degree d and smoothness r over triangulation T is a r time continuously differentiable function over the polygonal domain U_{t\in T}t and is a polynomial of total degree d when restricted to triangle t for each t of T. One application of bivariate splines is to fit given scattered data.
A spline function s of degree d and smoothness r over T is a r time continuously differentiable function over the polygonal domain U_{t\in T}t and is a polynomial of total degree d when restricted to tetrahedron t for each t of T. A typical application is to numerically solve partial differential equations.
www.math.uga.edu /~mjlai/spline.html   (1051 words)

  
 Spline -- from Wolfram MathWorld
A piecewise polynomial function that can have a locally very simple form, yet at the same time be globally flexible and smooth.
Splines are very useful for modeling arbitrary functions, and are used extensively in computer graphics.
Bartels, R. H.; Beatty, J. C.; and Barsky, B. An Introduction to Splines for Use in Computer Graphics and Geometric Modelling.
mathworld.wolfram.com /Spline.html   (156 words)

  
 B-Splines and Geometric Design
The mathematical representation of curves and surfaces is the crux of the transition from hand lofting to computer-aided design.
One of de Boor's first contributions was to introduce the use of tensor products of splines for surface interpolation, an idea that remained in place for many years at GM for the manufacture of dies.
It was only several months later, while preparing a general talk on splines, that he recognized the value of the simple B-spline recurrence relation---because the recursion weights are positive on the interval of interest, it is the perfect tool for the practical, stable evaluation of B-splines.
users.wpi.edu /~pwdavis/sinews/spline17.htm   (1930 words)

  
 Graduate Level Mathematics Courses
Mathematical description of continuum mechanics principles, including: tensor analysis, generalized description of kinematics and motion, conservation laws for mass and momentum; invariance and symmetry principles; application to generalized formulation of constitutive expressions for various fluids and solids.
Major events in the evolution of mathematical thought from ancient times to the present, the development of various important branches of mathematics, including numeration, geometry, algebra, analysis, number theory, probability, and applied mathematics.
Optimization I. Fundamentals of mathematical analysis underlying theory of constrained optimizations for a finite number of variables, necessary and sufficient conditions for constrained extrema of equality constraint problems, sufficient conditions for fulfillment of constraint qualification, computational methods for concave programming problems and applications.
www.math.tamu.edu /teaching/graduate/courses.htm   (1653 words)

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