  We consider two families of Funk-type transforms that assign to a function on
the unit sphere the integrals of that function over spherical sections by
planes of fixed dimension. Transforms of the first kind are generated by planes
passing through a fixed center outside the sphere. Similar transforms with
interior center and with center on the sphere itself we studied in previous
publications. Transforms of the second kind, or the parallel slice transforms,
correspond to planes that are parallel to a fixed direction. We show that the
Funk-type transforms with exterior center express through the parallel slice
transforms and the latter are intimately related to the Radon-John d-plane
transforms on the Euclidean ball. These results allow us to investigate
injectivity of our transforms and obtain inversion formulas for them. We also
establish connection between the Funk-type transforms of different dimensions
with arbitrary center.
