Coordinate Systems#

RadarSimPy uses right-handed coordinate systems for all spatial representations. This page describes the global and local coordinate systems used throughout the library.

Global Coordinate System#

The global coordinate system defines the absolute reference frame for all simulations.

Cartesian Coordinates

  • axis (m): [x, y, z] - Position vector in meters

    • x: East-West axis

    • y: North-South axis

    • z: Vertical axis (up)

Spherical Angles

  • phi (φ, deg): Azimuthal angle in the x-y plane

    • Range: 0° to 360° (or -180° to 180°)

    • 0° corresponds to the positive x-axis

    • 90° corresponds to the positive y-axis

    • Measured counter-clockwise when viewed from above

  • theta (θ, deg): Polar angle from the z-axis

    • Range: 0° to 180°

    • 0° corresponds to the positive z-axis (zenith)

    • 90° corresponds to the x-y plane (horizon)

    • 180° corresponds to the negative z-axis (nadir)

Phi and Theta angle definitions in spherical coordinates

Local Coordinate System#

The local coordinate system defines object-specific reference frames using Euler angles and origin translations.

Euler Angle Rotations

Rotations in RadarSimPy are specified using the configuration array: [yaw, pitch, roll]. Rotations are applied in the order: yaw → pitch → roll (Z-Y-X convention).

  • yaw (deg): Rotation about the z-axis

    • Positive yaw rotates counter-clockwise from the positive x-axis toward the positive y-axis

    • Range: -180° to 180° (or 0° to 360°)

  • pitch (deg): Rotation about the y-axis

    • Positive pitch rotates the positive x-axis toward the positive z-axis

    • Range: -90° to 90°

  • roll (deg): Rotation about the x-axis

    • Positive roll rotates the positive y-axis toward the positive z-axis

    • Range: -180° to 180°

Example Configuration

# Define rotation: 45° yaw, 10° pitch, 5° roll
rotation = [45, 10, 5]

Origin

  • origin (m): [x, y, z] - The center point for rotation and translation operations

    • All rotations are performed about this point

    • The radar’s origin is always fixed at [0, 0, 0]

    • Target objects can have arbitrary origins

Yaw, pitch, and roll angle definitions

Radar-Centric Angles#

For radar applications, azimuth and elevation angles provide an intuitive alternative to phi and theta.

Angle Definitions

  • azimuth (deg): Horizontal angle in the local x-y plane

    • Range: -90° to 90°

    • Equivalent to φ in the range [-90°, 90°]

    • 0° is boresight (forward direction)

    • Positive values are to the left, negative values are to the right

  • elevation (deg): Vertical angle from the horizon

    • Range: -90° to 90°

    • Related to θ by: elevation = 90° - θ

    • 0° is the horizon (x-y plane)

    • Positive values are above the horizon, negative values are below

Relationship to Global Coordinates

\[\begin{split}\text{azimuth} &= \phi \quad \text{for } \phi \in [-90°, 90°] \\ \text{elevation} &= 90° - \theta\end{split}\]
Azimuth and elevation angle definitions

Notes#

  • All angles use degrees unless otherwise specified

  • All distances use meters as the base unit

  • Right-handed coordinate systems ensure consistency with standard conventions

  • The coordinate transformation order matters: always apply yaw, then pitch, then roll