Exhaustive Principles of Navigation Study Notes

The Earth is not a perfect sphere; it is an oblate spheroid due to its equatorial diameter being greater than its polar diameter.
Equatorial diameter: $(7926.7)$ statute miles ( $(6378.16)$ km radius).
Polar diameter: $(7899.5)$ statute miles ( $(6356.77)$ km radius).
For most navigational purposes, the difference of approximately $(27)$ miles is small enough that the Earth can be treated as a true sphere.
Axis: The diameter about which the Earth rotates daily.
Poles: The two points (North and South) where the axis meets the Earth's surface.
Rotation: Occurs once each day towards the East. North is $(90^{\circ})$ to the left of East; South is $(90^{\circ})$ to the right; West is $(180^{\circ})$ from East.
Great Circle: A circle on the sphere's surface whose plane passes through the center of the sphere. It represents the shortest distance between two points.
Small Circle: A circle on the sphere's surface whose plane does not pass through the center.
Equator: A Great Circle perpendicular to the Earth's axis, dividing it into Northern and Southern hemispheres.
Parallels of Latitude: Small circles parallel to the Equator running East-West.
Meridians: Semi-great circles joining the poles, running North-South, and intersecting the Equator at $(90^{\circ})$ .
Prime Meridian: The meridian passing through Greenwich, from which longitude is measured East or West from $(0^{\circ})$ to $(180^{\circ})$ .

Geocentric Latitude: The angle at the center of the Earth between the Equator and the parallel of latitude passing through a specific place.
Geographical Latitude: The angle between the plane of the Equator and the vertical at a specific place. This is the latitude typically observed in navigation.
Difference in Latitude (d'lat): The arc of a meridian contained between the parallels of two places.
Longitude: The arc of the Equator or the angle at the poles between the Prime Meridian and the meridian of a place.
Difference in Longitude (d'long): The shorter arc of the Equator between the meridians of two places.
Mean Latitude: The arithmetic mean of two latitudes.

Nautical Mile: The length of one minute of arc of a meridian ( $(1')$ of d'lat). Because the Earth is not perfectly spherical, its length varies by latitude:
- At the poles: $(1861.7)$ m.
- At the Equator: $(1842.9)$ m.
- Standard (Mean) Nautical Mile: $(1852.3)$ m ( $(6080)$ ft).
Knot: A unit of speed equal to one nautical mile per hour.
Geographical Mile: The length of $(1')$ of arc of the Equator: $(1855.3)$ m.
Statute Mile: An arbitrary measure equal to $(5280)$ ft.
True Course: The angle between True North and the ship's fore-and-aft line.
Variation: The angle between the magnetic and geographic meridians caused by the magnetic poles not being at the geographic poles. It changes yearly (secular change).
Deviation: The angle the compass is deflected from Magnetic North due to the ship's own magnetism. It varies with the ship’s heading.
Compass Error: The algebraic sum of variation and deviation.

Rhumb Line (Loxodrome): A line crossing all meridians at the same angle. It is the most convenient track for navigation as the course remains constant.
Departure: The East-West distance between two places in nautical miles.
Parallel Sailing Formula: $\frac{\text{Departure}}{\text{d'long}} = \cos(\text{Lat})$ or $\text{d'long} = \text{Dep} \times \sec(\text{Lat})$ .
Plane Sailing: Simplified calculation treating d'lat, Departure, and Distance as sides of a right-angled plane triangle:
- $\text{Dep} = \text{Dist} \times \sin(\text{Course})$
- $\text{d'lat} = \text{Dist} \times \cos(\text{Course})$
- $\frac{\text{Dep}}{\text{d'lat}} = \tan(\text{Course})$
Mercator Sailing: Uses Meridional Parts to find rhumb line course and distance over long distances:
- $\tan(\text{Course}) = \frac{\text{d'long}}{\text{DMP}}$
- $\text{Distance} = \text{d'lat} \times \sec(\text{Course})$

Mercator Chart: A cylindrical orthomorphic projection. Meridians are parallel straight lines; parallels of latitude are also straight lines but their spacing increases poleward to maintain correctness of shape (orthomorphism).
Meridional Parts (MP): The length of a meridian from the Equator to a given latitude on a Mercator chart, expressed in units of the longitude scale.
Natural Scale: The ratio of a distance on the chart to the actual distance on Earth. On a Mercator chart, it varies with latitude.
Gnomonic Chart: Points on the sphere are projected from the center onto a tangent plane. All Great Circles appear as straight lines. It is used for Great Circle sailing to find the shortest route, which is then transferred to a Mercator chart as a series of rhumb lines.
Plan Chart: Used for very small areas like harbors. The Earth is treated as flat; latitude and longitude scales are constant.
Transverse Mercator: A Mercator projection rotated $(90^{\circ})$ , touching the Earth along a meridian instead of the Equator.

Celestial Sphere: An imaginary sphere of infinite radius with Earth at its center.
Declination: The angular distance of a body North or South of the Celestial Equator ( $(0^{\circ})$ to $(90^{\circ})$ ).
Hour Angle (GHA, LHA, SHA): Measurement of time/arc westward from Greenwich, the observer, or the First Point of Aries.
Sidereal Day: The period between two successive transits of the First Point of Aries ( $23\text{h } 56\text{m } 04.1\text{s}$ of mean solar time).
Apparent Solar Day: The interval between two transits of the True Sun; varies in length due to orbital eccentricity and the obliquity of the Ecliptic.
Equation of Time: The difference between Mean Time and Apparent Time ( $\text{LMT} - \text{LAT}$ ).
Solar System Motion: Planets follow Kepler’s Laws. Inferior planets (Mercury, Venus) are closer to the Sun than Earth; Superior planets (Mars, Jupiter, Saturn) are further away.
Twilight: Light received when the Sun is below the horizon ( $0^{\circ}$ to $18^{\circ}$ below). Classified as Civil ( $6^{\circ}$ ), Nautical ( $12^{\circ}$ ), and Astronomical ( $18^{\circ}$ ).