  The method of common lines is a well-established reconstruction technique in
cryogenic electron microscopy (cryo-EM), which can be used to extract the
relative orientations of an object given tomographic projection images from
different directions.
  In this paper, we deal with an analogous problem in optical diffraction
tomography. Based on the Fourier diffraction theorem, we show that rigid
motions of the object, i.e., rotations and translations, can be determined by
detecting common circles in the Fourier-transformed data. We introduce two
methods to identify common circles. The first one is motivated by the common
line approach for projection images and detects the relative orientation by
parameterizing the common circles in the two images. The second one assumes a
smooth motion over time and calculates the angular velocity of the rotational
motion via an infinitesimal version of the common circle method. Interestingly,
using the stereographic projection, both methods can be reformulated as common
line methods, but these lines are, in contrast to those used in cryo-EM, not
confined to pass through the origin and allow for a full reconstruction of the
relative orientations. Numerical proof-of-the-concept examples demonstrate the
performance of our reconstruction methods.
