This page is for testing things before placing in the main book.
We now develop the mathematical tools needed to specify directions and angles in threedimensional space. These mathematical concepts are fundamental to the specification of how much light there is and what direction it is traveling.
Directions
We have frequent need to specify directions. In order to do this in Euclidean threedimensional space, let
,
, and
be three mutually perpendicular unit vectors that define a righthanded Cartesian coordinate system. We choose
to be in the direction that the wind is blowing over the ocean surface, (i.e.
points downwind), and
points downward, perpendicular to the mean position of the water surface;
is then in the direction given by the cross (or vector) product
=
. Our choice of a "windbased" coordinate system simplifies the mathematical specification of seasurface wave spectra, as must be done when we discuss radiative transfer across a windblown sea surface. The choice of
pointing downward is natural in oceanography, where depths are customarily measured as positive downward from an origin at mean sea level.
With our choice of
,
, and
, an arbitrary direction can be specified as follows. Let
denote a unit vector pointing in the desired direction. The vector
has components
,
, and
in the
,
, and
directions, respectively. We can therefore write
=
+
+
, or just
= (
) for notational convenience. Note that because
is of unit length, its components satisfy
An alternative description of
is given by the polar coordinates
and
, defined as shown in Fig. . The nadir angle
is measured from the nadir direction
and the azimuthal angle
is measured positive counterclockwise from
, when looking toward the origin along
(i.e. when looking in the
direction). The connection between
= (
) and
= (
) is obtained by inspection of Fig. :
where
and
lie in the ranges
and
. The inverse transformation is
The polar coordinate form of
could be written as
=
but since the length of
is 1, we drop the radial coordinate for brevity.
Figure:
Definition of the polar coordinates (
) and of the upward (
) and downward (
) hemispheres of directions.
is an element of solid angle centered on
.

Another useful description of
is obtained using the cosine parameter

(3) 
The components of
= (
) and
= (
) are related by
with
and
in the ranges
and
. Hence a direction
can be represented in three equivalent ways: as (
) in Cartesian coordinates, and as (
,
) or (
,
) in polar coordinates.
The scalar (or dot) product between two direction vectors
and
can be written as
where
is the angle between directions
and
, and
denotes the (unit) length of vector
. The scalar product expressed in Cartesiancomponent form is
Equating these representations of
and recalling Eqs. (
) and (
) leads to
Equation (
) gives very useful connections between the various coordinate representations of
and
, and the included angle
. In particular, this equation allows us to compute the scattering angle
when light is scattered from an incident to a final direction.
The set of all directions
is called the unit sphere of directions, denoted by
. Referring to polar coordinates,
therefore represents all
values such that
and
. Two subsets of
frequently employed in optical oceanography are the downward (subscript d) and upward (subscript u) hemispheres of directions,
and
, defined by
Solid Angle
Closely related to the specification of directions in threedimensional space is the concept of solid angle, which is an extension of twodimensional angle measurement. As illustrated in panel (a) of Fig. , the plane angle
between two radii of a circle of radius r is
The angular measure of a full circle is therefore 2
rad. In panel (b) of Fig.
, a patch of area
A is shown on the surface of a sphere of radius
r. The boundary of
A is traced out by a set of directions
. The solid angle
of the set of directions defining the patch
A is by definition
Figure:
Geometry associated with the definition of plane angle (panel a) and solid angle (panel
b).

Since the area of a sphere is
, the solid angle measure of the set of all directions is
. Note that both plane angle and solid angle are independent of the radii of the respective circle and sphere. Both plane and solid angle are dimensionless numbers. However, they are given "units" of radians and steradians, respectively, to remind us that they are measures of angle.
Consider a simple application of the definition of solid angle and the observation that a full sphere has
. The area of Brazil is
and the area of the earth's surface is
. The solid angle subtended by Brazil as seen from the center of the earth is then
.
Figure:
Geometry used to obtain an element of solid angle in spherical coordinates.

The definition of solid angle as area on the surface of a sphere divided by radius of the sphere squared gives us a convenient form for a differential element of solid angle, as needed for computations. The blue patch shown in Fig. represents a differential element of area
on the surface of a sphere of radius
. Simple trigonometry shows that this area is
Thus the element of solid angle
about the direction
= (
) is given in polar coordinate form by
(The last equation is correct even though
=
=
. When the differential element is used in an integral and variables are changed from (
) to (
), the Jacobian of the transformation involves an absolute value.)
Example: Solid angle of a spherical cap
To illustrate the use of Eq. (), let us compute the solid angle of a ``polar cap'' of half angle
, i.e. all (
) such that
and
. Integrating the element of solid angle over this range of (
) gives

(7) 
or

(8) 
Note that
and
are special cases of a spherical cap (having
), and that
) =
.
Delta functions
It is sometimes convenient to specify directions using the Dirac delta function,
. This peculiar mathematical construction is defined (for our purposes) by

(9) 
and

(10) 
Here
is any function of direction. Note that
simply "picks out" the particular direction
from all direction in
. Note also in Eq. (
) that because the element of solid
has units of steradians, it follows that
has units of inverse steradians.
Equations () and () are a symbolic definition of
. The mathematical representation of
in spherical coordinates (
,
) is

(11) 
where
=
,
, and
Note that the
in the denominator of Eq. (
) is necessary to cancel the
factor in the element of solid angle when integrating in polar coordinates. Thus
Likewise, we can write

(12) 
where