Concept

# Spline (mathematics)

Summary
In mathematics, a spline is a special function defined piecewise by polynomials. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low degree polynomials, while avoiding Runge's phenomenon for higher degrees. In the computer science subfields of computer-aided design and computer graphics, the term spline more frequently refers to a piecewise polynomial (parametric) curve. 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. The term spline comes from the flexible spline devices used by shipbuilders and draftsmen to draw smooth shapes. The term "spline" is used to refer to a wide class of functions that are used in applications requiring data interpolation and/or smoothing. The data may be either one-dimensional or multi-dimensional. 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. For a number of meaningful definitions of the roughness measure, the spline functions are found to be finite dimensional in nature, which is the primary reason for their utility in computations and representation. For the rest of this section, we focus entirely on one-dimensional, polynomial splines and use the term "spline" in this restricted sense. We begin by limiting our discussion to polynomials in one variable. In this case, a spline is a piecewise polynomial function. This function, call it S, takes values from an interval [a,b] and maps them to , the set of real numbers, We want S to be piecewise defined.
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related publications (3)

## Application of optimal spline subspaces for the removal of spurious outliers in isogeometric discretizations

Espen Sande

We show that isogeometric Galerkin discretizations of eigenvalue problems related to the Laplace operator subject to any standard type of homogeneous boundary conditions have no outliers in certain op
ELSEVIER SCIENCE SA2022

## One-Shot Approach for Enforcing Piecewise Linearity on Hybrid Functionals: Application to Band Gap Predictions

We present an efficient procedure for constructing nonempirical hybrid functionals to accurately predict band gaps of extended systems. We determine mixing parameters by enforcing the generalized Koop
2022

## Image registration for super resolution

Mohamed Taher Nguira

This document presents a mathematical framerwork for delay retrieval between two signals. It treats Global and Local delay cases. It's based on E-splines and Exponential reproducing kernels properties
2015
Related people

Related units

Related concepts (17)
Curve fitting
Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function is constructed that approximately fits the data. A related topic is regression analysis, which focuses more on questions of statistical inference such as how much uncertainty is present in a curve that is fit to data observed with random errors.
Spline interpolation
In the mathematical field of numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. That is, instead of fitting a single, high-degree polynomial to all of the values at once, spline interpolation fits low-degree polynomials to small subsets of the values, for example, fitting nine cubic polynomials between each of the pairs of ten points, instead of fitting a single degree-ten polynomial to all of them.
Spline (mathematics)
In mathematics, a spline is a special function defined piecewise by polynomials. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low degree polynomials, while avoiding Runge's phenomenon for higher degrees. In the computer science subfields of computer-aided design and computer graphics, the term spline more frequently refers to a piecewise polynomial (parametric) curve.
Related courses (16)
MATH-441: Robust and nonparametric statistics
In the decades from 1930 to 1950, many rank-based statistics were introduced. These methods were received with much interest, because they worked under weak conditions. Starting in the late 1950, a th
MICRO-512: Image processing II
Study of advanced image processing; mathematical imaging. Development of image-processing software and prototyping in JAVA; application to real-world examples in industrial vision and biomedical imagi
MATH-408: Regression methods
General graduate course on regression methods
Related lectures (109)
Transformations of Input or Output
Covers handling missing data, feature engineering, and output transformations in machine learning.
Bezier Curves II
Covers Bezier curves, de Casteljau Algorithm, properties, derivatives, splines, and end-points.
Piecewise Polynomial Interpolation: Splines
Covers piecewise polynomial interpolation with splines, focusing on Lagrange interpolation with Chebyshev nodes and error convergence.
Related MOOCs