An Approximation to the Chapman
Grazing-Incidence Function for
Atmospheric scattering for computer graphics is the treatment of the atmosphere
as a participating medium, essentially “calculating the color of the sky.” This is
interesting for any application where the time of day, the season, or the proper-
ties of the atmosphere are not known in advance, or the viewpoint may not be
restricted to a point on the earth’s surface. It is a historically diﬃcult eﬀect to
render, especially at planetary scale.
Early attempts at atmospheric scattering can be found in [Klassen 87] and
[Nishita et al. 93]. Recent implementations with an emphasis on real time are
[Hoﬀmann and Preetham 02], [O’Neil 05], and [Bruneton and Neyret 08]. A
common theme of all these approaches is ﬁnding ways to eﬃciently evaluate or
precompute the Chapman function, Ch(x, χ). This is the density integral for a
ray in a spherically symmetric, exponentially decreasing atmosphere.
The Chapman function has been subject to extensive treatment in the physics
literature. Approximations and tabulations have been published, most of it with
a focus on precision. This article explores a diﬀerent direction for its evaluation:
an approximation so cheap that Ch(x, χ) can be considered a commodity, while
still being accurate enough for our graphics needs.
2.2 Atmospheric Scattering
This section is a brief review of atmospheric scattering and a deﬁnition of terms.
When light travels through air, it will be partly absorbed and partly scattered
into other directions. This gives rise to the phenomenon of aerial perspective. The