Wiki Metodos Numericos: Definicion

Yo encontre esto pero en ingles

We begin by limiting our discussion to the univariate polynomial case. 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 $\mathbb{R}$ , the set of real numbers,

$S: [a,b]\to \mathbb{R}.$

We want S to be piecewise defined. To accomplish this, let the interval [a,b] be covered by k ordered, disjoint subintervals,

$[t_i, t_{i+1}] \mbox{ , } i = 0,\ldots, k-1$

$[a,b] = [t_0,t_1] \cup [t_1,t_2] \cup \cdots \cup [t_{k-2},t_{k-1}] \cup [t_{k-1},t_k]$

$a = t_0 \le t_1 \le \cdots \le t_{k-1} \le t_k = b$

On each of these k "pieces" of [a,b], we want to define a polynomial, call it P_i.

$P_i: [t_i, t_{i+1}]\to \mathbb{R}$ .

On the ith subinterval of [a,b], S is defined by P_i,

$S(t) = P_0 (t) \mbox{ , } t_0 \le t < t_1,$

$S(t) = P_1 (t) \mbox{ , } t_1 \le t < t_2,$

$\vdots$

$S(t) = P_{k-1} (t) \mbox{ , } t_{k-1} \le t \le t_k.$

The given k+1 points t_i are called knots. The vector ${\bold t}=(t_0, \dots, t_k)$ is called a knot vector for the spline. If the knots are equidistantly distributed in the interval [a,b] we say the spline is uniform, otherwise we say it is non-uniform.

If the polynomial pieces P_i each have degree at most n, then the spline is said to be of degree $\leq n$ (or of order n+1).

If $S\in C^{r_i}$ in a neighborhood of t_i, then the spline is said to be of smoothness (at least) $C^{r_i}$ at t_i. 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).

A vector ${\bold r}=(r_1, \dots, r_{k-1})$ such that the spline has smoothness $C^{r_i}$ at t_i for $i = 0,\ldots, k-1$ is called a smoothness vector for the spline.

Given a knot vector ${\bold t}$ , a degree n, and a smoothness vector ${\bold r}$ for ${\bold t}$ , one can consider the set of all splines of degree $\leq n$ having knot vector ${\bold t}$ and smoothness vector ${\bold r}$ . Equipped with the operation of adding two functions (pointwise addition) and taking real multiples of functions, this set becomes a real vector space. This spline space is commonly denoted by $S^{\bold r}_n({\bold t})$ .

In the mathematical study of polynomial splines the question of what happens when two knots, say t_i and t_i+1, are moved together 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 continuity losses for t_i and t_i+1. That is,

$S(t) \in C^{n-j_i-j_{i+1}} [t_i = t_{i+1}],$ where

j i = n - r i

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

$(t_0 , t_1 , \cdots , t_1 , t_2, \cdots , t_2 , t_3 , \cdots , t_{k-2} , t_{k-1} , \cdots , t_{k-1} , t_k)$

where t_i is repeated j_i times for $i = 1, \dots , k-1$ .

A parametric curve on the interval [a,b]

$G(t) = ( X(t), Y(t) ) \mbox{ , } t \in [ a , b ]$

is a spline curve if both X and Y are spline functions of the same degree with the same extended knot vectors on that interval.

Definicion

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