Albedo used in astronomy are the (V-band) geometric albedo (measuring brightness when illumination comes from directly behind the observer) and the Bond albedo (measuring total proportion of electromagnetic energy reflected). Their values can differ significantly, which is a common source of confusion. In detailed studies, the directional reflectance properties of astronomical bodies are often expressed in terms of the five Hapke parameters which semi-empirically describe the variation of albedo with phase angle, including a characterization of the opposition effect of regolith surfaces. The correlation between astronomical (geometric) albedo, absolute magnitude and diameter is: formula_7, where formula_8 is the astronomical albedo, formula_9
Incompressible surface compressing "S" along "D". A non-negative complexity measure on compact surfaces without 2-sphere components is "b"("S") − "χ"("S"), where "b"("S") is the zeroth Betti number (the number of connected components) and "χ"("S") is the Euler characteristic. When compressing a compressible surface, the Euler characteristic increases by two, while "b" might remain the same or increase by 1. Thus, every properly embedded compact surface without 2-sphere components is related to an incompressible surface through a sequence of compressions. Sometimes we drop the conditions that "S" be compressible or that "D" be anything more than a disk that intersects "S" only along