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The general setting is as follows. We would like to construct a curve whose shape is described by a sequence of ''p'' points <math>\mathbf{d}_0, \mathbf{d}_1, \dots, \mathbf{d}_{p-1}</math>, which plays the role of a ''control polygon''. The curve can be described as a function <math> \mathbf{s}(x)</math> of one parameter ''x''. To pass through the sequence of points, the curve must satisfy <math>\mathbf{s}(u_0)=\mathbf{d}_0, \dots,
\mathbf{s}(u_{p-1})=\mathbf{d}_{p-1}</math> for a given knot sequence <math>u_0<u_1<\ldots<u_{p-1}</math>. But this is not quite the case: in general we
One approach to solve this problem is by [[spline (mathematics)|spline]]s. A spline is a curve that is a piecewise ''n<sup>th</sup>'' degree polynomial. This means that, on any interval ''<nowiki>[</nowiki>u<sub>i</sub>, u<sub>i+1</sub>)'', the curve must be equal to a polynomial of degree at most ''n''. It may be equal to different polynomials on different intervals. The polynomials must be ''synchronized'': when the polynomials from intervals ''<nowiki>[</nowiki>u<sub>i-1</sub>, u<sub>i</sub>)'' and ''<nowiki>[</nowiki>u<sub>i</sub>, u<sub>i+1</sub>)'' meet at the point ''u<sub>i</sub>'', they must have the same value at this point and their derivatives must be equal up to order <math>n-1</math> (to ensure that the curve is as smooth as possible without restricting <math>s(x)</math> to a plain polynomial within <math>[u_{i-1},u_{i+1})</math>).
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