Solid angle to the unit sphere with center P, which can be calculated as the surface integral: where formula_5 is the unit vector corresponding to formula_6, the position vector of an infinitesimal area of surface with respect to point P, and where formula_7 represents the unit normal vector to . Even if the projection on the unit sphere to the surface S is not isomorphic, the multiple folds are correctly considered according to the surface orientation described by the sign of the scalar product formula_8. Thus one can approximate the solid angle subtended by a small facet having flat surface area ,
Filling area conjecture onto , without increasing its volume. This implies that the replacement has not less volume than the original manifold . Filling area conjecture In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the surfaces that fill a closed curve of given length without introducing shortcuts between its points. Every smooth surface or curve in Euclidean space is a metric space, in which the (intrinsic) distance between two points of is defined as the infimum of the lengths of the curves that go from to "along" . For example, on a closed curve formula_1