Navigation Systems Lecture 3: Spherical Trigonometry and Coordinate Systems

Vocabulary and key concepts from Lecture 3 regarding navigation calculations, Earth modeling, and coordinate frames.

Effective Sky Temperature

Also referred to as antenna temperature, it describes the noise environment for GNSS signals, which is much hotter than the cosmic microwave background radiation (CMBR).

Great Circle

The intersection of the surface of a sphere with a plane which passes through the origin $O$; it is the spherical equivalent of a line. All lines of longitude are great circles, but only the equator is a great circle among lines of latitude.

Spherical Triangle

A shape on a sphere's surface where the summed interior angles are not equal to $180^\text{o}$; for example, a triangle covering $1/8$th of the globe has a sum of $270^\text{o}$.

Geodesic

The term for the shortest path between two points on a spherical surface.

Rhumb Line

A course of constant heading; on a globe, this path forms a spiral shape called a loxodrome.

Gnomonic Projection

A map projection that represents all great circles as straight lines, making it useful for plotting shortest-path headings.

Stellar Positioning Timing Error

In celestial navigation, a $1$ minute error in time translates to approximately $281,km$ of position error at the equator.

ECEF

Earth Centered, Earth Fixed; a non-inertial cartesian coordinate frame where the origin is the center of the planet, the $Z$ axis points through the North Pole, and the $X$ axis aligns with the prime meridian. It rotates at $15^\text{o}$ per hour.

Local Level / Tangent Frames

Coordinate systems such as East North Up (ENU) or North East Down (NED) that are useful for local representations but rotate relative to the earth as the user moves.

UTM

Universal Transverse Mercator; a grid-based coordinate system using Northing and Easting in metres. It is noted for limitations such as not covering the poles and having non-uniform grid distribution, specifically in Norway.

Oblate Spheroid

The mathematical model of the Earth as a vertically compressed or flattened sphere.

Semi-major axis ($a$)

The radius of the Earth at the equator, defined in the WGS84 model as $6378137,m$.

Semi-minor axis ($b$)

The polar radius of the Earth, calculated as $b = (1 - f) \times a$, where $f$ is the flattening term. In WGS84, this is approximately $6356752.314,m$.

Flattening ($f$)

The ratio that defines the compression of the Earth's shape, specifically $1/298.257223563$ for the WGS84 ellipsoid.

First Numerical Eccentricity ($e^2$)

A parameter for the ellipsoidal Earth calculated as $e^2 = \frac{a^2 - b^2}{a^2}$. In the WGS84 model, it is approximately $0.00669437999014$.

Ellipsoidal Height

The height coordinate used in the LLH (Latitude, Longitude, Height) system based on a smooth mathematical ellipsoid model of the Earth.

Geoid Height / Mean Sea Level

A height system that accounts for the gravitational potential variations over the planet, currently modeled by the Earth Gravitational Model 2008 (EGM2008).

Orthometric Height

The vertical distance between the terrain surface and the equipotential surface of the Geoid at a given point.

Barometric Height

Also known as pressure altitude, this height varies with local atmospheric pressure and temperature conditions.

Radar Altimeter

A sensor that directly measures height above local terrain or obstacles, independent of satellite-based reports.

Deflection of the Vertical

The phenomenon where local gravity does not point straight down relative to the ellipsoidal model due to topographical features like mountains; it can shift the gravity direction by $> 1^\text{o}$.