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Single knots at 1/3 and 2/3 establish a spline of three cubic polynomials meeting with C 2 parametric continuity. Triple knots at both ends of the interval ensure that the curve interpolates the end points. In mathematics, a spline is a function defined piecewise by polynomials.
A First Class of Analytic Approximation Formulae". Quarterly of Applied Mathematics. 4 (2): 45– 99. doi: 10.1090/qam/15914. Schoenberg, Isaac J. (1946). "Contributions to the Problem of Approximation of Equidistant Data by Analytic Functions:Part B.—On the Problem of Osculatory Interpolation. A Second Class of Analytic Approximation Formulae".
For the third segment of the curve, plotted with parametric values of 2/3 or greater, the rightmost four control points shape the curve. In this example, multiplicity four knots resided at either end of the curve and ensures that the curve is defined over the entire parametric range of u and that the curve interpolates its end points. This is ...
An Introduction to Splines for Use in Computer Graphics and Geometric Modeling. Morgan Kaufmann. ISBN 978-1-55860-400-1. Jean Gallier (1999). Curves and Surfaces in Geometric Modeling: Theory and Algorithms. Morgan Kaufmann. Chapter 6. B-Spline Curves. This book is out of print and freely available from the author.
The second class of generalizations to multi-dimensional smoothing deals directly with this scale invariance issue using tensor product spline constructions. [ 10 ] [ 11 ] [ 12 ] Such splines have smoothing penalties with multiple smoothing parameters, which is the price that must be paid for not assuming that the same degree of smoothness is ...
Example showing non-monotone cubic interpolation (in red) and monotone cubic interpolation (in blue) of a monotone data set. Monotone interpolation can be accomplished using cubic Hermite spline with the tangents m i {\displaystyle m_{i}} modified to ensure the monotonicity of the resulting Hermite spline.
See also Subdivision surfaces, which is an emerging alternative to spline-based surfaces. Pages in category "Splines (mathematics)" The following 30 pages are in this category, out of 30 total.
A discrete spline is a piecewise polynomial such that its central differences are continuous at the knots whereas a spline is a piecewise polynomial such that its derivatives are continuous at the knots. Discrete cubic splines are discrete splines where the central differences of orders 0, 1, and 2 are required to be continuous. [1]