  We study integral transforms mapping a function on the Euclidean plane to the
family of its integration on plane curves, that is, a function of plane curves.
The plane curves we consider in the present paper are given by the graphs of
functions with a fixed axis of the independent variable, and are imposed some
symmetry with respect to the axes. These transforms contain the parabolic Radon
transform and the hyperbolic Radon transform arising from seismology. We prove
the inversion formulas for these transforms under some vanishing and symmetry
conditions of functions.
