This Brief investigates the intersections that occur between three different areas of study that normally would not touch each other: ODF, spline theory, and topology. The Least Squares Orthogonal Distance Fitting (ODF) method has become the standard technique used to develop mathematical models of the physical shapes of objects, due to the fact that it produces a fitted result that is invariant with respect to the size and orientation of the object. It is normally used to produce a single optimum fit to a specific object; this work focuses instead on the issue of whether the fit responds continuously as the shape of the object changes. The theory of splines develops user-friendly ways of manipulating six different splines to fit the shape of a simple family of epiTrochoid curves: two types of Bézier curve, two uniform B-splines, and two Beta-splines. This work will focus on issues that arise when mathematically optimizing the fit. There are typically multiple solutions to the ODF method, and the number of solutions can often change as the object changes shape, so two topological questions immediately arise: are there rules that can be applied concerning the relative number of local minima and saddle points, and are there different mechanisms available by which solutions can either merge and disappear, or cross over each other and interchange roles. The author proposes some simple rules which can be used to determine if a given set of solutions is internally consistent in the sense that it has the appropriate number of each type of solution.
The Curve Fitting Toolbox software supports these nonparametric fitting methods: -"Interpolation Methods" - Estimate values that lie between known data points.-"Smoothing Splines" - Create a smooth curve through the data. You adjust the level of smoothness by varying a parameter that changes the curve from a least-squares straight-line approximation to a cubic spline interpolant.-"Lowess Smoothing" - Create a smooth surface through the data using locally weighted linear regression to smooth data.Interpolation is a process for estimating values that lie between known data points. There are several interpolation methods: - Linear: Linear interpolation. This method fit a different linear polynomial between each pair of data points for curves, or between sets of three points for surfaces.- Nearest neighbor: Nearest neighbor interpolation. This method sets the value of an interpolated point to the value of the nearest data point. Therefore, this method does not generate any new data points.- Cubic spline: Cubic spline interpolation. This method fit a different cubic polynomial between each pair of data points for curves, or between sets of three points for surfaces.After fitting data with one or more models, you should evaluate the goodness of fit A visual examination of the fitte curve displayed in Curve Fitting app should be your firs step. Beyond that, the toolbox provides these methods to assess goodness of fi for both linear and nonlinear parametric fits-"Goodness-of-Fit Statistics" -"Residual Analysis" -"Confidence and Prediction Bounds" The Curve Fitting Toolbox spline functions are a collection of tools for creating, viewing, and analyzing spline approximations of data. Splines are smooth piecewise polynomials that can be used to represent functions over large intervals, where it would be impractical to use a single approximating polynomial. The spline functionality includes a graphical user interface (GUI) that provides easy access to functions for creating, visualizing, and manipulating splines. The toolbox also contains functions that enable you to evaluate, plot, combine, differentiate and integrate splines. Because all toolbox functions are implemented in the open MATLAB language, you can inspect the algorithms, modify the source code, and create your own custom functions. Key spline features: -GUIs that let you create, view, and manipulate splines and manage and compare spline approximations-Functions for advanced spline operations, including differentiation integration, break/knot manipulation, and optimal knot placement-Support for piecewise polynomial form (ppform) and basis form (B-form) splines-Support for tensor-product splines and rational splines (including NURBS)- Shape-preserving: Piecewise cubic Hermite interpolation (PCHIP). This method preserves monotonicity and the shape of the data. For curves only.- Biharmonic (v4): MATLAB 4 grid data method. For surfaces only.- Thin-plate spline: Thin-plate spline interpolation. This method fit smooth surfaces that also extrapolate well. For surfaces only.If your data is noisy, you might want to fit it using a smoothing spline. Alternatively, you can use one of the smoothing methods. The smoothing spline s is constructed for the specified smoothing parameter p and the specified weights wi.
The fitting of a curve or surface through a set of observational data is a very frequent problem in different disciplines (mathematics, engineering, medicine, ...) with many interesting applications. This book describes the algorithms and mathematical fundamentals of a widely used software package for data fitting with (tensor product) splines. As such it gives a survey of possibilities and benefits but also of the problems to cope with when approximating with this popular type of function. In particular it is demonstrated in detail how the properties of B-splines can be fully exploited for improving the computational efficiency and for incorporating different boundary or shape preserving constraints. Special attention is also paid to strategies for an automatic and adaptive knot selection with intent to obtain serious data reductions. The practical use of the smoothing software is illustrated with many examples, academic as well as taken from real life.
"A brief review of curve fitting terminology is presented, and the cubic spline interpolation scheme is outlined. Parametric and non-parametric curve fitting techniques are compared. The technique to fit parametric cubic splines is derived using the Euler- Lagrange formulation. Previous work on splines in tension is identified. Employing the notion of splines in tension, a method is proposed to fit a parametric curve to a set of (n + 1) points in ^-dimension space satisfying a specified set of boundary conditions. The curve fitted will not have any inflection points within any span and will be invariant with respect to coordinate translation and rotation. Using Euler-Lagrange formulation, a system of linear equations in terms of the unknown second derivatives at knots is developed. Three kinds of boundary conditions are investigated. Software is developed in VAX Fortran to fit both parametric splines in tension and parametric cubic splines. Applications where splines in tension may find use are identified. Some examples of such applications are presented and comparison to cubic spline made. Splines in tension offer a better alternative than Fourier transform in describing boundary of shape in digital image processing application. Possible extensions to the numerical scheme developed and related investigations by other workers in this field are also listed."--Abstract.
The purpose of this book is to reveal the foundations and major features of several basic methods for curve and surface fitting that are currently in use.
The Theory of Splines and Their Applications discusses spline theory, the theory of cubic splines, polynomial splines of higher degree, generalized splines, doubly cubic splines, and two-dimensional generalized splines. The book explains the equations of the spline, procedures for applications of the spline, convergence properties, equal-interval splines, and special formulas for numerical differentiation or integration. The text explores the intrinsic properties of cubic splines including the Hilbert space interpretation, transformations defined by a mesh, and some connections with space technology concerning the payload of a rocket. The book also discusses the theory of polynomial splines of odd degree which can be approached through algebraically (which depends primarily on the examination in detail of the linear system of equations defining the spline). The theory can also be approached intrinsically (which exploits the consequences of basic integral relations existing between functions and approximating spline functions). The text also considers the second integral relation, raising the order of convergence, and the limits on the order of convergence. The book will prove useful for mathematicians, physicist, engineers, or academicians in the field of technology and applied mathematics.
The purpose of this report is to demonstrate that modern computer-aided design (CAD), computer-aided manufacturing (CAM), and computer-aided engineering (CAE) systems can be used in the Department of Energy (DOE) Nuclear Weapons Complex (NWC) to design new and remodel old products, fabricate old and new parts, and reproduce legacy data within the inspection uncertainty limits. In this study, two two-dimensional splines are compared with several modern CAD curve-fitting modeling algorithms. The first curve-fitting algorithm is called the Wilson-Fowler Spline (WFS), and the second is called a parametric cubic spline (PCS). Modern CAD systems usually utilize either parametric cubic and/or B-splines.
You can work with splines in Curve Fitting Toolbox(tm) in several ways.Using the Curve Fitting app or the fit function you can:Fit cubic spline interpolants to curves or surfacesFit smoothing splines and shape-preserving cubic spline interpolants to curves (but not surfaces)Fit thin-plate splines to surfaces (but not curves)The toolbox also contains specific splines functions to allow greater control over what you can create. For example, you can use the csapi function for cubic spline interpolation. Why would you use csapi instead of the fit function 'cubicinterp' option? You might require greater flexibility to work with splines for the following reasons:You want to combine the results with other splines, You want vector-valued splines. You can use csapi with scalars, vectors, matrices, and ND-arrays. The fit function only allows scalar-valued splines.You want other types of splines such as ppform, B-form, tensor-product, rational, and stform thin-plate splines.You want to create splines without data.You want to specify breaks, optimize knot placement, and use specialized functions for spline manipulation such as differentiation and integration.If you require specialized spline functions, see the following sections for interactive and programmatic spline fitting.
