Multiresolution on spherical curves

• We develop multiresolution for spherical curves directly in spherical space.
• The multiresolution scheme is of arbitrary degree.
• All constituent operations are implemented using line interpolation operations.
• The results of our subdivision generalize those of spherical Lane–Riesenfeld.

In this paper, we present an approximating multiresolution framework of arbitrary degree for curves on the surface of a sphere. Multiresolution by subdivision and reverse subdivision allows one to decrease and restore the resolution of a curve, and is typically defined by affine combinations of points in Euclidean space. While translating such combinations to spherical space is possible, ensuring perfect reconstruction of the curve remains challenging. Hence, current spherical multiresolution schemes tend to be interpolating or midpoint-interpolating, as achieving perfect reconstruction in these cases is more straightforward. We use a simple geometric construction for a non-interpolating and non-midpoint-interpolating multiresolution scheme on the sphere, which is made up of easily generalized components and based on a modified Lane–Riesenfeld algorithm.

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