# Non-geodesic Spherical Funk Transforms with One and Two Centers

## TL;DR

This paper investigates Funk-type transforms on spheres involving planes through one or two fixed points, establishing injectivity, inversion formulas, and demonstrating unique reconstruction with two centers.

## Contribution

It introduces new injectivity conditions and inversion formulas for non-geodesic Funk transforms with multiple centers, expanding understanding of spherical integral geometry.

## Key findings

- Injectivity conditions for transforms with one and two centers
- Explicit inversion formulas derived for these transforms
- Unique reconstruction possible with two distinct centers

## Abstract

We study non-geodesic Funk-type transforms associated with cross-sections of the n-sphere by k-dimensional planes passing through an arbitrary fixed point inside the sphere. The main results include injectivity conditions for these transforms, inversion formulas, and connection with geodesic Funk transforms. We also show that, unlike the case of planes through a single common center, the integrals over spherical sections by planes through two distinct centers provide the corresponding reconstruction problem a unique solution.