Fractional derivative for interpolation in [formula omitted] and SO(n) applications in functionally graded materials and rigid body transformations

Abstract

An integration of fractional calculus, data interpolation, and Lie group leads to new treatments of interpolation problems. A local and centered fractional derivative is introduced, which enables the fractional interpolant to create a Cα-smooth (α∈R+) curve by extending integer order cost functions to fractional ones. The fractional interpolant fills the gaps between the traditional linear interpolating segments, cubic splines, and higher-order splines. Interpolating rigid body transformations has enormous applications in mechanics, computer graphics, and robotics. Previous approaches have provided the re-parametrization methods, the recursive methods (e.g., de Casteljau’s algorithm), and the gradient methods. To approximate the correct interpolating curve on SO(3), a closed-form model that merges the tangent spaces of knots is proposed. The model first constructs a family of candidate curves and then combines them into an improved approximation. Applications in modeling functionally graded materials and rigid body transformations are discussed.