  To probe cosmological fields beyond the Gaussian level, three-point
statistics can be used, all of which are related to the bispectrum. Hence,
measurements of CMB anisotropies, galaxy clustering, and weak gravitational
lensing alike have to rely upon an accurate theoretical background concerning
the bispectrum and its noise properties. If only small portions of the sky are
considered, it is often desirable to perform the analysis in the flat-sky
limit. We aim at a formal, detailed derivation of the bispectrum covariance in
the flat-sky approximation, focusing on a pure two-dimensional Fourier-plane
approach. We define an unbiased estimator of the bispectrum, which takes the
average over the overlap of annuli in Fourier space, and compute its full
covariance. The outcome of our formalism is compared to the flat-sky spherical
harmonic approximation in terms of the covariance, the behavior under parity
transformations, and the information content. We introduce a geometrical
interpretation of the averaging process in the estimator, thus providing an
intuitive understanding. Contrary to foregoing work, we find a difference by a
factor of two between the covariances of the Fourier-plane and the spherical
harmonic approach. We argue that this discrepancy can be explained by the
differing behavior with respect to parity. However, in an exemplary analysis it
is demonstrated that the Fisher information of both formalisms agrees to high
accuracy. Via the geometrical interpretation we are able to link the
normalization in the bispectrum estimator to the area enclosed by the triangle
configuration at consideration as well as to the Wigner symbol, which leads to
convenient approximation formulae for the covariances of both approaches.
