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In the mathematical study of polynomial splines the question of what happens when two knots, say t i and t i+1, are taken to approach one another and become coincident has an easy answer. The polynomial piece P i ( t ) disappears, and the pieces P i −1 ( t ) and P i +1 ( t ) join with the sum of the smoothness losses for t i and t i +1 .
Spline curve drawn as a weighted sum of B-splines with control points/control polygon, and marked component curves. In the mathematical subfield of numerical analysis, a B-spline or basis spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition.
The key points, placed by the artist, are used by the computer algorithm to form a smooth curve either through, or near these points. For a typical example of 2-D interpolation through key points see cardinal spline. For examples which go near key points see nonuniform rational B-spline, or Bézier curve. This is extended to the forming of ...
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".
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.
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 ...
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]
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.