The Workshop

Project Phoenix: Target Acquisition

Spherical to Rectilinear Conversion Challenge

Scenario: Phoenix 1 Last Known Coordinates

Your mission is to use the **3D-Compass** to visualize the drone's position and accurately convert the Spherical Coordinates ($\rho, \theta, \phi$) into the Rectilinear Coordinates ($x, y, z$) required by the ground recovery vehicle.

Radial Distance ($\rho$)

Value in kilometers (km)

Polar Angle ($\theta$)

Elevation Angle (Declination) in degrees

Azimuthal Angle ($\phi$)

Direction Angle (Azimuth) in degrees

Trigonometric Conversion Rules

Conversion Formulas (Astronomical Convention: $\theta$ is Elevation from $xy$-plane)

$$ \begin{align*} \mathbf{x} &= \rho \cos(\theta) \cos(\phi) \\ \mathbf{y} &= \rho \cos(\theta) \sin(\phi) \\ \mathbf{z} &= \rho \sin(\theta) \end{align*} $$

Deliverables: Calculated Rectilinear Coordinates

x Coordinate

-6.14 km

y Coordinate

10.65 km

z Coordinate

8.60 km

Positional Analysis

The Phoenix 1 drone is located **West** of the $yz$-plane (due to negative x), **North** of the $xz$-plane (due to positive y), and is **above** the command center (due to positive z). It is in the second spatial quadrant relative to the $xy$-plane.