To parameterize an surface’s shape, it is common to further assume that we are given (or can
compute) some “local coordinate origin” point inside the surface such that the surface’s 3-coordinate shape
relative to that point is a “Strahlkörper” (literally “ray body”, or more commonly “star-shaped region”),
defined by Minkowski [138, Page 108] as
a region in -D Euclidean space containing the origin and whose surface, as seen from
the origin, exhibits only one point in any direction.
The Strahlkörper assumption is a significant restriction on the horizon’s coordinate shape (and the
choice of the local coordinate origin). For example, it rules out the coordinate shape and local
coordinate origin illustrated in Figure 1: a horizon with such a coordinate shape about the local
coordinate origin could not be found by any horizon finder which assumes a Strahlkörper surface
parameterization.
For event-horizon finding, algorithms and codes are now available which allow an arbitrary horizon topology with no Strahlkörper assumption (see the discussion in Section 5.3.3 for details). For apparent horizon finding, the flow algorithms discussed in Section 8.7 theoretically allow any surface shape, although many implementations still make the Strahlkörper assumption. Removing this assumption for other apparent horizon finding algorithms might be a fruitful area for further research.
Given the Strahlkörper assumption, the surface can be explicitly parameterized as
whereThere are two common ways to discretize a horizon shape function:
This series can then be truncated at some finite order , and the
coefficients
used to represent (discretely approximate) the horizon shape. For reasonable
accuracy,
is typically on the order of 8 to 12.
It is sometimes useful to explicitly construct a level-set function describing a given Strahlkörper. A common choice here is
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