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Spline (mathematics)
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== Definition == {{Confusing|date=February 2009}} We begin by limiting our discussion to [[univariate|polynomials in one variable]]. In this case, a spline is a [[piecewise]] [[polynomial]] [[Function (mathematics)|function]]. This function, call it {{mvar|S}}, takes values from an interval {{math|[''a'',''b'']}} and maps them to <math>\R,</math> the set of [[real numbers]], <math display=block>S: [a,b] \to \R.</math> We want {{mvar|S}} to be piecewise defined. To accomplish this, let the interval {{math|[''a'',''b'']}} be covered by {{mvar|k}} ordered, [[Disjoint sets|disjoint]] subintervals, <math display=block>\begin{align} &[t_i, t_{i+1}], \quad i = 0,\ldots, k-1 \\[4pt] &[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) \cup [t_k] \\[4pt] &a = t_0 \le t_1 \le \cdots \le t_{k-1} \le t_k = b \end{align}</math> On each of these {{mvar|k}} "pieces" of {{math|[''a'',''b'']}}, we want to define a polynomial, call it {{mvar|P{{sub|i}}}}. <math display=block>P_i: [t_i, t_{i+1}]\to \R.</math> On the {{mvar|i}}th subinterval of {{math|[''a'',''b'']}}, {{mvar|S}} is defined by {{mvar|P{{sub|i}}}}, <math display=block>\begin{align} S(t) &= P_0 (t), && t_0 \le t < t_1, \\[2pt] S(t) &= P_1 (t), && t_1 \le t < t_2, \\ &\vdots \\ S(t) &= P_{k-1} (t), && t_{k-1} \le t \le t_k. \end{align}</math> The given {{math|''k'' + 1}} points {{mvar|t{{sub|i}}}} are called '''knots'''. The vector {{math|1='''t''' = (''t''{{sub|0}}, β¦, ''t{{sub|k}}'')}} is called a '''knot vector''' for the spline. If the knots are equidistantly distributed in the interval {{math|[''a'',''b'']}} we say the spline is '''uniform''', otherwise we say it is '''non-uniform'''. If the polynomial pieces {{mvar|P{{sub|i}}}} each have degree at most {{mvar|n}}, then the spline is said to be of '''degree''' {{math|≤ ''n''}} (or of '''order''' {{math|''n'' + 1}}). If <math>S\in C^{r_i}</math> in a neighborhood of {{mvar|t{{sub|i}}}}, then the spline is said to be of [[Smooth function|smoothness]] (at least) <math>C^{r_i}</math> at {{mvar|t{{sub|i}}}}. That is, at {{mvar|t{{sub|i}}}} the two polynomial pieces {{math|''P''<sub>''i''β1</sub>}} and {{mvar|P{{sub|i}}}} share common derivative values from the derivative of order 0 (the function value) up through the derivative of order {{mvar|r{{sub|i}}}} (in other words, the two adjacent polynomial pieces connect with '''loss of smoothness''' of at most {{math|''n'' β ''r<sub>i</sub>''}}) <math display=block>\begin{align} P_{i-1}^{(0)}(t_i) &= P_i^{(0)} (t_i), \\[2pt] P_{i-1}^{(1)}(t_i) &= P_i^{(1)} (t_i), \\ \vdots& \\ P_{i-1}^{(r_i)}(t_i) &= P_i^{(r_i)} (t_i). \end{align}</math> A vector {{math|1='''r''' = (''r''{{sub|1}}, β¦, ''r''{{sub|''k''β1}})}} such that the spline has smoothness <math>C^{r_i}</math> at {{mvar|t{{sub|i}}}} for {{math|1=''i'' = 1, β¦, ''k'' β 1}} is called a '''smoothness vector''' for the spline. Given a knot vector {{math|'''t'''}}, a degree {{mvar|n}}, and a smoothness vector {{math|'''r'''}} for {{math|'''t'''}}, one can consider the set of all splines of degree {{math|≤ ''n''}} having knot vector {{math|'''t'''}} and smoothness vector {{math|'''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 <math>S^{\mathbf r}_n(\mathbf t).</math> In the mathematical study of polynomial splines the question of what happens when two knots, say {{mvar|t{{sub|i}}}} and {{math|''t''<sub>''i''+1</sub>}}, are taken to approach one another and become coincident has an easy answer. The polynomial piece {{math|''P{{sub|i}}''(''t'')}} disappears, and the pieces {{math|''P''<sub>''i''β1</sub>(''t'')}} and {{math|''P''<sub>''i''+1</sub>(''t'')}} join with the sum of the smoothness losses for {{mvar|t<sub>i</sub>}} and {{math|''t''<sub>''i''+1</sub>}}. That is, <math display=block> S(t) \in C^{n-j_i-j_{i+1}} [t_i = t_{i+1}],</math> where {{math|1=''j{{sub|i}}'' = ''n'' β ''r{{sub|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 {{mvar|n}} and its '''extended''' knot vector <math display=block> (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) </math> where {{mvar|t{{sub|i}}}} is repeated {{mvar|j<sub>i</sub>}} times for {{math|1=''i'' = 1, β¦, ''k'' β 1}}. A [[parametric curve]] on the interval {{math|[''a'',''b'']}} <math display=block>G(t) = \bigl( X(t), Y(t) \bigr), \quad t \in [ a , b ]</math> is a '''spline curve''' if both {{mvar|X}} and {{mvar|Y}} are spline functions of the same degree with the same extended knot vectors on that interval.
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