### Analytic Geometry

#### Wave Fronts

Consider a swimming pool, with children equally spaced along a straight edge of the pool, each of the children tapping the water with a rod in tune (at the same time) to the beat of a song that is played over loud speakers.

Each child generates a circular wave, and the circular waves combine away from the side of the pool to form parallel straight waves. The straight waves are called the “envelope” of the circular waves, or simply “wave fronts”.

Now, consider if the children are staggered in their timing, so that a given amount of time passes between each successive child tapping the water:

“the children are now independently addressed by listening to the music through earphones that are individually excited… The rhythm is the same for all the children, but the phase may be varied from one to the other. When all the children hit the surface ‘in phase,’ a rectilinear wave is generated parallel to the edge. If the phase repartition is varied linearly versus the distance from the child sitting in the middle, a rectilinear wave is still generated, but it propagates obliquely.”
—
Germain Chartier, Introduction to Optics Figure 1: Oblique wave front.

That is the idea behind phased array antennas: Figure 2: Phased array antenna transmitter (TX), depicted in slow motion. Antenna elements ‘A’ are each an antenna that transmits a signal when triggered by computer ‘C’ which lags the triggering to ‘steer’ the direction of the wave front.

This explains some of the properties of light, according to Huygens:

“Christiaan Huygens proposed that a light source emitted spherical waves of light, each point of which generated new spherical waves, thus propagating the wave forward. He derived all the previous optical laws from his wave theory: straight line motion, reflection, and refraction.”
—
Beeson & Mayer, Patterns of Light Figure 3: Dutch physicist Christiaan Huygens.

A beam of light can be considered to be a set of consecutive parallel wave fronts. Each wave front is a plane that is perpendicular to the light beam direction: Figure 4: Side view of a light beam from a spot light. Wave fronts with surface normals are shown in blue.

Wave front surface normals could be modeled, uniformly spaced across a wave front. The line of travel of each wave front surface normal is called a “ray”: Figure 5: Light rays shown as green lines.

The density of light rays (wave front surface normals) could represent the number of photons in the light beam, or more importantly the amount of energy of the light beam. We consider the latter.

The amount of energy in a light beam is referred to as “irradiance” (abbreviated I) and is measured as Watts per square meter (W/m²) on the wave front. Irradiance may also be referred to as “intensity”, and may be abbreviated E (for Energy).

Closer spacing of light rays (more surface normals per square meter) represents higher energy (more Watts per square meter).

Say that a flat surface is perpendicular to the direction of a light beam. In that case, a light beam wave front is parallel to the surface, and the beam irradiance (W/m² on the wavefront) equals the irradiance striking the surface (W/m² on the surface): Figure 6: Light rays perpendicular to a flat surface, b = a. In this case, the density of light rays on the wavefront equals the density of rays striking the surface.

On the other hand, if the surface and the light beam direction are not perpendicular, the unit vectors strike the flat surface further apart than on the wave front, resulting in less surface irradiance (W/m²): Figure 7: Light rays striking a flat surface obliquely, b > a. In this case, the light rays are more spread out on the surface than on a wavefront, illuminating a larger area on the surface but with less energy per unit area.

We refer to this geometric dissipation of energy as dihedral spreading, because the wave fronts are not parallel to the receiving surface, forming a dihedral angle between a wave front and the surface.

##### Dihedral Angle

A dihedral angle is the angle between two planes. Figure 8: Dihedral angle of two planes that intersect at common line ‘C’, with vector R on one plane, and vector R-prime on the other plane, both R and R-prime perpendicular to vector C that is on the line of interesection of the planes. In this example, the dihedral angle is denoted α. [Leyo]

Using the vectors of Figure 8 above, the dihedral angle is given by the following formula:

$\cos{\alpha} = \frac{ \mathbf{R} \cdot \mathbf{R'} } { \left|\mathbf{R}\right|\left|\mathbf{R'}\right| }$

[equation 1]

which is, in fact, the angle between any two vectors (in this case, the vectors R and R′). The vectors could be the surface normals of the two planes; that yields the same dihedral angle:

cos α = n · n

[equation 2]

This cosine (of the dihedral angle) is the “cosine factor” that is multiplied into the wavefront irradiance to give the (geometrically corrected) surface irradiance of the wavefront striking the surface:

[equation 3]

Abbreviating I for Irradiance, S for surface, and B for beam wavefront, the formula of eq. 3 becomes:

Is = (Ib) (cos α)

[equation 4]

This is the geometrically corrected surface irradiance of the wavefront striking the surface.

Dihedral spreading spreads the wavefront over a larger area of the receiving surface, the new area equal to the approaching beam (wavefront) width times the reciprocal of the cosine of the dihedral angle: Figure 9: Dihedral spreading of a wave front on a flat surface. In this example, the dihedral angle is denoted θ.

In this example, the distance  a  is always shorter than  a/cosθ  because the reciprocal of $\cos{\theta}$ will always be greater than one.

##### Exercise

Problem: Show that Figure 9 above is correct, using Figure 9 of the Angles lesson o.

Solution: Here is Figure 9 of the Angles lesson again: Figure 10: Trigonometry of a right triangle.

First, transpose the triangle (by reflection and rotation), maintaining all distances and angles (hence maintaining the trigonometric formulas): Figure 11: Right triangle of Figure 10 transposed.

Next, magnify the triangle, with constant magnification, to create a larger similar triangle o (with the same angles and with corresponding sides of the two triangles parallel), and collocate the similar triangles at the vertex of theta: Figure 12: Similar triangles. One of the similar triangles has a right angle side x and hypotenuse x′, and the other similar triangle has a right angle side c and hypotenuse c′.

Applying the formulas of Figure 11 to the formulas of Figure 12 gives:

\begin{align} \mathbf{x'} & = \frac{ \mathbf{x} } { \cos{\theta} } \\ & \\ \mathbf{c'} & = \frac{ \mathbf{c} } { \cos{\theta} } \end{align}

Subtracting x′ from c′ gives:

\begin{align} \mathbf{a'} & = \mathbf{c'} - \mathbf{x'} \\ & \\ & = \frac{ \mathbf{c} } { \cos{\theta} } - \frac{ \mathbf{x} } { \cos{\theta} } \\ & \\ & = \frac{ (\mathbf{c} - \mathbf{x}) } { \cos{\theta} } \\ & \\ & = \frac{ \mathbf{a} } { \cos{\theta} } \end{align}

which is the projection of  a  to  a/cosθ  of Figure 9 above (showing that projection is correct).

###### Analytic Geometry
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