This paper addresses a statistical challenge underlying the precise analysis of curves in \(\mathbb {R}^3\) from discrete and noisy observations. A significant part of curve or functional data analysis relies heavily on understanding the shapes of curves. While Frenet curvature parameters are recognized for their ability to capture the full geometric characteristics of curves, their application in functional data analysis has been limited due to the complexities of their statistical estimation. We address this gap by providing a comprehensive review of existing methods for estimating Frenet curvatures, which often depend on latent variable estimation during preprocessing. Additionally, we introduce a new unified intrinsic approach that leverages the Lie group SE(3) to offer a more robust and reliable estimation of Frenet parameters, aiming to advance the precision of shape-based curve analysis.
