We begin by limiting our discussion to
polynomials in one variable. In this case, a spline is a
piecewise polynomial function. This function, call it , takes values from an interval and maps them to \R, the set of
real numbers, S: [a,b] \to \R. We want to be piecewise defined. To accomplish this, let the interval be covered by ordered,
disjoint subintervals, \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} On each of these "pieces" of , we want to define a polynomial, call it . P_i: [t_i, t_{i+1}]\to \R. On the th subinterval of , is defined by , \begin{align} S(t) &= P_0 (t), && t_0 \le t The given points are called
knots. The vector is called a
knot vector for the spline. If the knots are equidistantly distributed in the interval we say the spline is
uniform, otherwise we say it is
non-uniform. If the polynomial pieces each have degree at most , then the spline is said to be of
degree (or of
order ). If S\in C^{r_i} in a neighborhood of , then the spline is said to be of
smoothness (at least) C^{r_i} at . That is, at the two polynomial pieces and share common derivative values from the derivative of order 0 (the function value) up through the derivative of order (in other words, the two adjacent polynomial pieces connect with
loss of smoothness of at most ) \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} A vector such that the spline has smoothness C^{r_i} at for is called a
smoothness vector for the spline. Given a knot vector , a degree , and a smoothness vector for , one can consider the set of all splines of degree having knot vector and smoothness vector . 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^{\mathbf r}_n(\mathbf t). In the mathematical study of polynomial splines the question of what happens when two knots, say and , are taken to approach one another and become coincident has an easy answer. The polynomial piece disappears, and the pieces and join with the sum of the smoothness losses for and . That is, S(t) \in C^{n-j_i-j_{i+1}} [t_i = t_{i+1}], where . 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 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 is repeated times for . A
parametric curve on the interval G(t) = \bigl( X(t), Y(t) \bigr), \quad t \in [ a , b ] is a
spline curve if both and are spline functions of the same degree with the same extended knot vectors on that interval. ==Examples==