### Reading Review

- SECS, Chapter 7: Applying the Angles to Shadows and Tracking

This is a brief recap to set the stage for orthographic projections and polar projections used in the shading analysis project to come.

In a spherical coordinate system, the **angles are the coordinates**. So, if I were standing in a field in North Dakota, looking at something tall like an enormous wind turbine, I could define the position of the top of the nacelle relative to me by stating the general **azimuth angle** ($\gamma $, the rotation across the horizon from due South) and the general **altitude angle** ($\alpha $, the rotation up from the horizon). Effectively, this is my *x* ($\gamma $) and *y* ($\alpha $) coordinates on an **orthographic projection** of the **sky dome** on a flat surface.

Many of you will be familiar with **Cartesian coordinates** in space (*x, y, z*). However, when dealing with spatial relations of spherical objects like the Earth and the Celestial Sphere, we find that working with basic spherical coordinate systems makes **trigonometry** available to us to solve for space and time equations. For **spherical coordinates,** we need information of **radial distance, zenith angle, and azimuth**. However, in solar positioning studies, a radius of one (unit radius) is all we need to establish a **unit vector**, and we are left with equations for only the zenith angle and azimuth (and the complement of the zenith angle, the **altitude angle**). Note how the zenith angle in Figure 2.12, above (the generic $\theta $ angle), is congruent with the **solar zenith** (${\theta}_{z}$) of the Sun, and the generic azimuth angle ($\phi $) is congruent with the **solar azimuth** (${\gamma}_{s}$).

The following equations describe the Cartesian coordinates (*x, y, z*) for **unit vectors**, followed by the equivalent functionals using the complementary altitude angle.

$$\begin{array}{c}\begin{array}{ccccc}z& =& \mathrm{cos}\theta & =& \mathrm{sin}\alpha \\ x& =& \mathrm{sin}\theta \mathrm{cos}\phi & =& \mathrm{cos}\alpha \mathrm{cos}\phi \\ y& =& \mathrm{sin}\theta \mathrm{sin}\phi & =& \mathrm{cos}\alpha \mathrm{sin}\phi \end{array}\end{array}$$

As seen in Figure 2.13, we need to pull back to our old trigonometry mneumonics! **Recall "Soh Cah Toa" when looking at the figure:**

**Sine(A): Opposite over Hypotenuse (a/h)****Cosine(A): Adjacent over Hypotenuse (b/h)****Tangent(A): Opposite over Adjacent (a/b)**