Page updated: June 2, 2020
Author: Curtis Mobley
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Geometry

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.

Directions

We have frequent need to specify directions. In order to do this in Euclidean three-dimensional space, let î1, î2, and î3 be three mutually perpendicular unit vectors that define a right-handed Cartesian coordinate system. We choose î1 to be in the direction that the wind is blowing over the ocean surface, (i.e. î1points downwind), and î3points downward, perpendicular to the mean position of the water surface; î2is then in the direction given by the cross (or vector) product î2= î3× î1. 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 î3 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 î1, î2, and î3, an arbitrary direction can be specified as follows. Let ξ̂denote a unit vector pointing in the desired direction. The vector ξ̂has components ξ1, ξ2, and ξ3in the î1, î2, and î3directions, respectively. We can therefore write ξ̂= ξ1î1+ ξ2î2+ ξ3î3, or just ξ̂= (ξ1,ξ2,ξ3) for notational convenience. Note that because ξ̂is of unit length, its components satisfy ξ12 + ξ22 + ξ32 = 1.

An alternative description of ξ̂is given by the polar coordinates 𝜃 and ϕ, defined as shown in Fig. 1. The nadir angle 𝜃 is measured from the nadir direction î3 and the azimuthal angle ϕ is measured positive counterclockwise from î1, when looking toward the origin along î3 (i.e. when looking in the î3 direction). The connection between ξ̂ = (ξ1,ξ2,ξ3) and ξ̂ = (𝜃,ϕ) is obtained by inspection of Fig. 1:

ξ1 = sin𝜃cosϕ ξ2 = sin𝜃sinϕ (1) ξ3 = cos𝜃 where 𝜃 and ϕ lie in the ranges 0 𝜃 π and 0 ϕ < 2π. The inverse transformation is

𝜃 = cos1(ξ 3) ϕ = tan1 ξ2 ξ1 (2) The polar coordinate form of ξ̂could be written as ξ̂= (r,𝜃,ϕ), but since the length of r is 1, we drop the radial coordinate for brevity.

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Figure 1: Definition of the polar coordinates (𝜃,ϕ) and of the upward (Ξu) and downward (Ξd) hemispheres of directions. ΔΩ(ξ̂) is an element of solid angle centered on ξ̂.

Another useful description of ξ̂is obtained using the cosine parameter

μ cos𝜃 = ξ3. (3)

The components of ξ̂= (ξ1,ξ2,ξ3) and ξ̂= (μ,ϕ) are related by

ξ1 = (1 μ2)12 cosϕ ξ2 = (1 μ2)12 sinϕ (4) ξ3 = μ, with μ and ϕ in the ranges 1 μ 1 and 0 ϕ < 2π. Hence a direction ξ̂ can be represented in three equivalent ways: as (ξ1,ξ2,ξ3) in Cartesian coordinates, and as (𝜃, ϕ) or (μ, ϕ) in polar coordinates.

The scalar (or dot) product between two direction vectors ξ̂ and ξ̂ can be written as

ξ̂ξ̂ ξ̂ξ̂cosψ = cosψ,

where ψ is the angle between directions ξ̂ and ξ̂, and |ξ̂| denotes the (unit) length of vector ξ̂. The scalar product expressed in Cartesian-component form is

ξ̂ ξ̂ = ξ 1ξ1 + ξ 2ξ2 + ξ 3ξ3.

Equating these representations of ξ̂ ξ̂ and recalling Eqs. (1) and (4) leads to

cosψ = ξ 1ξ1 + ξ 2ξ2 + ξ 3ξ3 = cos𝜃cos𝜃 + sin𝜃sin𝜃cos(ϕ ϕ) (5) = μμ + 1 μ 21 μ2 cos(ϕ ϕ)

Equation (5) gives very useful connections between the various coordinate representations of ξ̂ and ξ̂, and the included angle ψ. In particular, this equation allows us to compute the scattering angle ψ when light is scattered from an incident to a final direction.

The set of all directions ξ̂ is called the unit sphere of directions, denoted by Ξ. Referring to polar coordinates, Ξ therefore represents all (𝜃,ϕ) values such that 0 𝜃 π and 0 ϕ < 2π. Two subsets of Ξ frequently employed in optical oceanography are the downward (subscript d) and upward (subscript u) hemispheres of directions, Ξd and Ξu, defined by

Ξd  all (𝜃,ϕ) such that 0 𝜃 π2 and 0 ϕ < 2π, Ξu  all (𝜃,ϕ) such that π2 < 𝜃 π and 0 ϕ < 2π.

Solid Angle

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 𝜃 between two radii of a circle of radius r is

𝜃 arc length radius = r(rad).

The angular measure of a full circle is therefore 2π 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 ξ̂. The solid angle Ω of the set of directions defining the patch A is by definition

Ω area radius squared = A r2(sr).


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Figure 2: Geometry associated with the definition of plane angle (panel a) and solid angle (panel b).

Since the area of a sphere is 4πr2, the solid angle measure of the set of all directions is Ω(Ξ) = 4πrmsr. 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 4πsr. The area of Brazil is 8.5 106km2 and the area of the earth’s surface is 5.1 108km2. The solid angle subtended by Brazil as seen from the center of the earth is then 4π8.5 1065.1 108 = 0.21sr.


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Figure 3: 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. 3 represents a differential element of area dA on the surface of a sphere of radius r. Simple trigonometry shows that this area is dA = (rsin𝜃dϕ)(rd𝜃). Thus the element of solid angle dΩ(ξ̂) about the direction ξ̂= (𝜃,ϕ) is given in polar coordinate form by

dΩ(ξ̂) = dA r2 = (rsin𝜃dϕ)(rd𝜃) r2 = sin𝜃d𝜃dϕ = dμdϕ(sr). (6)

(The last equation is correct even though dμ = dcos𝜃 = sin𝜃d𝜃. When the differential element is used in an integral and variables are changed from (𝜃,ϕ) to (μ,ϕ), the 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 𝜃, i.e. all (𝜃,ϕ) such that 0 𝜃 𝜃 and 0 ϕ < 2π. Integrating the element of solid angle over this range of (𝜃,ϕ) gives

Ωcap = ϕ=02π𝜃=0𝜃 sin𝜃d𝜃dϕ = 2π(1 cos𝜃), (7)

or

Ωcap = ϕ=02πμ=μ1dμdϕ = 2π(1 μ). (8)

Note that Ξd and Ξu are special cases of a spherical cap (having 𝜃 = π2), and that Ω(Ξd) = Ω(Ξu) = 2πsr.

Delta functions

It is sometimes convenient to specify directions using the Dirac delta function, δ(ξ̂ ξ̂o). This peculiar mathematical construction is defined (for our purposes) by

δ(ξ̂ ξ̂o) 0ifξ̂ξ̂o, (9)

and

Ξf(ξ̂)δ(ξ̂ ξ̂o)dΩ(ξ̂) f(ξ̂o). (10)

Here f(ξ̂) is any function of direction. Note that δ(ξ̂ ξ̂o) simply ”picks out” the particular direction ξ̂o from all direction in Ξ. Note also in Eq. (10) that because the element of solid dΩ(ξ̂) has units of steradians, it follows that δ(ξ̂ ξ̂o) has units of inverse steradians.

Equations (9) and (10) are a symbolic definition of δ. The mathematical representation of δ(ξ̂ ξ̂o) in spherical coordinates (𝜃,ϕ) is

δ(ξ̂ ξ̂o) = δ(𝜃 𝜃o)δ(ϕ ϕo) sin𝜃 (sr1), (11)

where ξ̂ = (𝜃,ϕ), ξ̂o = (𝜃o,ϕo), and

0πf(𝜃)δ(𝜃 𝜃 o)d𝜃 f(𝜃o) 02πf(ϕ)δ(ϕ ϕ o)dϕ f(ϕo). Note that the sin𝜃 in the denominator of Eq. (11) is necessary to cancel the sin𝜃 factor in the element of solid angle when integrating in polar coordinates. Thus

Ξf(ξ̂)δ(ξ̂ ξ̂o)dΩ(ξ) = 02π0πf(𝜃,ϕ)δ(𝜃 𝜃o)δ(ϕ ϕo) sin𝜃 sin𝜃d𝜃dϕ = f(𝜃o,ϕo) = f(ξ̂o).

Likewise, we can write

δ(ξ̂ ξ̂o) = δ(μ μo)δ(ϕ ϕo)(sr1), (12)

where

11f(μ)δ(μ μ o)dμ f(μo).

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