We now develop the mathematical tools needed to specify directions and angles in three-dimensional space. These mathematical concepts are fundamental to the speciﬁcation of how much light there is and what direction it is traveling.
We have frequent need to specify directions. In order to do this in Euclidean three-dimensional space, let , , and be three mutually perpendicular unit vectors that deﬁne a right-handed 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 ”wind-based” coordinate system simpliﬁes the mathematical speciﬁcation of sea-surface wave spectra, as must be done when we discuss radiative transfer across a wind-blown 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. 1. 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. 1:
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.
Another useful description of is obtained using the cosine parameter
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 Cartesian-component form is
Equation (5) 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
Closely related to the speciﬁcation of directions in three-dimensional space is the concept of solid angle, which is an extension of two-dimensional angle measurement. As illustrated in panel (a) of Fig. 2, 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. 2, 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
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 .
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. 3 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
Jacobian of the transformation involves an absolute value.)
Example: Solid angle of a spherical cap
To illustrate the use of Eq. (6), 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
Note that and are special cases of a spherical cap (having ), and that ) = .
It is sometimes convenient to specify directions using the Dirac delta function, . This peculiar mathematical construction is deﬁned (for our purposes) by
Here is any function of direction. Note that simply ”picks out” the particular direction from all direction in . Note also in Eq. (10) that because the element of solid has units of steradians, it follows that has units of inverse steradians.
where = , , and
Note that the in the denominator of Eq. (11) is necessary to cancel the factor in the element of solid angle when integrating in polar coordinates. Thus
Likewise, we can write