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Parametric Spline for 1D curves
General form: \(\displaystyle c(s) = \sum_k b_k(s) \; p_k\)
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\(b_k\): basis functions (smooth polynomial, minimal support)
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\(P_k\): control points (user defined)
Straight segment
\(c(s)=(1-s) \, p_0 + s \, p_1\)
Cubic uniform B-spline
\(\displaystyle c(s)=\sum_{k=0}^{3} b_k(s) \; p_k\)
\[ \left\{ \begin{array}{l} b_0(s) = (1-s)^3 /6 \\ b_1(s) = (3 s^3-6 s^2 +4) /6 \\ b_2(s) = (-3 s^3 +3 s^2 +3 s +1) /6 \\ b_3(s) = s^3 /6 \\ \end{array} \right. \]
