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Multidimensional Minimizing Splines: Theory And Applications
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Polski English Login or register account. Multidimensional Minimizing Splines. Abstract Audience: This book would be of interest to mathematicians, geologists, engineers and, in general, researchers and post graduate students involved in spline function theory, surface fitting problems or variational methods. Authors Close. Assign yourself or invite other person as author. It allow to create list of users contirbution. In this case, a spline is a piecewise polynomial function.
We want S to be piecewise defined. To accomplish this, let the interval [ a , b ] be covered by k ordered, disjoint subintervals,.
On each of these k "pieces" of [ a , b ], we want to define a polynomial, call it P i. On the i th subinterval of [ a , b ], S is defined by P i ,. If the knots are equidistantly distributed in the interval [ a , b ] we say the spline is uniform , otherwise we say it is non-uniform. That is, at t i the two pieces P i-1 and P i share common derivative values from the derivative of order 0 the function value up through the derivative of order r i in other words, the two adjacent polynomial pieces connect with loss of smoothness of at most n - r i.
Equipped with the operation of adding two functions pointwise addition and taking real multiples of functions, this set becomes a real vector space. That is,. This leads to a more general understanding of a knot vector. The continuity loss at any point can be considered to be the result of multiple knots located at that point, and a spline type can be completely characterized by its degree n and its extended knot vector.
A parametric curve on the interval [ a , b ]. Suppose the interval [ a , b ] is [0,3] and the subintervals are [0,1], [1,2], and [2,3]. This would define a type of spline S t for which.
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Note: while the polynomial piece 2 t is not quadratic, the result is still called a quadratic spline. This demonstrates that the degree of a spline is the maximum degree of its polynomial parts.
The extended knot vector for this type of spline would be 0, 1, 2, 2, 3. The simplest spline has degree 0. It is also called a step function. The next most simple spline has degree 1. It is also called a linear spline. A closed linear spline i. A common spline is the natural cubic spline of degree 3 with continuity C 2. The word "natural" means that the second derivatives of the spline polynomials are set equal to zero at the endpoints of the interval of interpolation.
This forces the spline to be a straight line outside of the interval, while not disrupting its smoothness. It might be asked what meaning more than n multiple knots in a knot vector have, since this would lead to continuities like. By convention, any such situation indicates a simple discontinuity between the two adjacent polynomial pieces. It is commonly assumed that any knot vector defining any type of spline has been culled in this fashion.
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Another type of spline that is much used in graphics, for example in drawing programs such as Adobe Illustrator from Adobe Systems , has pieces that are cubic but has continuity only at most. This spline type is also used in PostScript as well as in the definition of some computer typographic fonts. Many computer-aided design systems that are designed for high-end graphics and animation use extended knot vectors, for example Maya from Alias.
Attea, M. Brezinski and L.
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Wuytack eds, North-Holland, Berlinet, A. Bojanov, B. Duchon, J. Eubank, R. Marcel-Dekker, New-York, Freeden, W. Green, P. A roughness penalty approach. Chapman and Hall, London, Golomb, M. Shadimetov Kh. Uzbek Math. Shadimetov, Kh. Holladay, J. Tables Aids Comput. Ignatev, M. Korneichuk, N.