Spline Interpolation and Fitting in  \(\mathbb {R}^{n}\)

This chapter unfolds a comprehensive exploration of the fitting and interpolation problem in \(\mathbb {R}^{n}\) . We present a formal definition of the fitting problem, simultaneously addressing the core challenge of accurately interpolating time-labeled data in Euclidean space. Subsequently, a thorough review of key definitions related to Bézier splines ensues, highlighting the prerequisites for achieving \(C^{m}\) continuity, ( \(m=0,1,2\) ). The chapter culminates in the introduction of an innovative method for solving the interpolation problem in \(\mathbb {R}^{n}\) through the use of \(C^{m}\) Bézier splines. This approach adeptly navigates the complexities of fitting data in multiple dimensions, ensuring the desired continuity up to the mth order and providing a nuanced and effective solution to this intricate problem.