tokens: and , our CDF is, analogously with Eq. ( A.2 )and using Eq. ( A.1 ), (The use of is merely a matter of convenience for the integration limits.) The basis functions, that is, any CDF caused by all objects having given fixed and , are now easy to write in closed form. They are obtained by replacing and in the integrands by the fixed and , setting , and using the integration limits to describe the inequalities between and to define the piecewise function . This replaces the integral by a sum of such basis functions each multiplied by the corresponding weight of the and bin. The resulting basis functions are given in Sect. 2.1. The above assumption of most orbits to be close to the ecliptic plane is only approximate, and one can always define populations (especially those of near-Earth asteroids) for which it is not true even approximately. Thus the validity of this assumption should be checked for the targets used. However, as we show in Sect. 4, the assumption works quite well (given the large model error budget in any case) with typical asteroid populations for which there is some concentration of viewing geometries sufficiently near the ecliptic plane. Appendix B: Ill-posedness caused by two-point variation observables As earlier, we consider the case when is (approximately) the same for the pair. Now we have, for two rotation phases and , so, with , that is, (due to symmetry, we only need to consider the interval ), we define iso- contours in the plane (for given ) by so, to have viable solutions for from , we must have , , and so exist for given only if ; that is, Denoting our CDF is thus Again, we can include the -distribution by expanding the integral in the same way as with Eq. (A.4). The basis function for a given in the two-point brightness scatter case is, from Eq. ( B.1 ), Although the -integral can be given in terms of elliptic functions, this is best computed by evaluating the double integral numerically. The maximum value of is obtained at : Our basis functions are closed-form expressions of those computed by Monte-Carlo sampling in Szabo Kiss (2008) . These can be used to determine the -distribution, although we found the accuracy inferior to the solution based on the variation observable , which was to expected. In principle, we can expand to for a -grid in the same way that were expanded to . However, a notable difference between the two-index basis functions of - or - data is that the all reach their maxima at the same point since the two-point comparison can always contain two equal brightnesses for any and . Thus the -curves of the of a given form a curve family with the same abscissae for the minimum () and maximum (); that is, members of the family can easily be mimicked by a superposition of other members unlike in the case of . A number of simulations indeed confirmed that are not usable for solving the inverse problem in practice; that is, -information is not recoverable from -data. Adding prior assumptions on the
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