Spline Interpolation on the Sphere \(\mathbb {S}^{n}\)

Spherical splines have applications across various domains, including cardiology, computer vision, geophysicsGeophysics, biology, astronomy, animation, robotics, and motion planning utilizing quaternions [1–4]. One noteworthy example occurs in vector cardiograms, where the electrical activity of the heart during a heartbeat is represented as a nearly planar orbit in \(\mathbb {R}^3\) .In this context, spherical splines offer an effective approach for modeling and analyzing the intricate dynamics of cardiac electrical signals. These splines provide a flexible and efficient framework to capture the complexities of the heart’s electrical behavior, facilitating the interpretation and diagnosis of cardiac conditions [1, 2]. Another significant application arises in computer graphics and animation, specifically with spherical splines on the unit sphere \(\mathbb {S}^3\) . This utilization enables the smooth representation of orientations of solid bodies, as quaternions can be interpreted as pairs of antipodal points on \(\mathbb {S}^3\) . Spherical spline curves, in this case, offer a means to specify a smooth transitions of solid orientations, contributing to the creation of visually appealing and realistic animations in computer graphics [3–5].