We study X-ray and Divergent beam transforms of Trkalian fields and their
relation with Radon transform. We make use of four basic mathematical methods
of tomography due to Grangeat, Smith, Tuy and Gelfand-Goncharov for an integral
geometric view on them. We also make use of direct approaches which provide a
faster but restricted view of the geometry of these transforms. These reduce to
well known geometric integral transforms on a sphere of the Radon or the
spherical Curl transform in Moses eigenbasis, which are members of an analytic
family of integral operators. We also discuss their inversion. The X-ray (also
Divergent beam) transform of a Trkalian field is Trkalian. Also the Trkalian
subclass of X-ray transforms yields Trkalian fields in the physical space. The
Riesz potential of a Trkalian field is proportional to the field. Hence, the
spherical mean of the X-ray (also Divergent beam) transform of a Trkalian field
over all lines passing through a point yields the field at this point. The
pivotal point is the simplification of an intricate quantity: Hilbert transform
of the derivative of Radon transform for a Trkalian field in the Moses basis.
We also define the X-ray transform of the Riesz potential (of order 2) and
Biot-Savart integrals. Then, we discuss a mini-twistor respresentation,
presenting a mini-twistor solution for the Trkalian fields equation. This is
based on a time-harmonic reduction of wave equation to Helmholtz equation. A
Trkalian field is given in terms of a null vector in C3 with an arbitrary
function and an exponential factor resulting from this reduction.Comment: 37 pages, http://dx.doi.org/10.1063/1.482610
