Charts

061 General NavigationChapter 3

Navigation charts must be orthomorphic (conformal) — correct shape locally, meridians perpendicular to parallels, and scale equal in all directions at a point.

Scale & orthomorphism

Definition — Orthomorphic

Chart is conformal — bearings correct, shape preserved locally. Requires meridians ⊥ parallels and equal scale in all directions at any point.

Formula — Reduced earth scale

RE Scale = Distance on RE / Distance on Earth
Chart Scale = Chart Distance (CD) / Earth Distance (ED)
Scale Factor (SF) = Chart Scale / RE Scale

SF = 1 → correct scale. Constant if 0.99 ≤ SF < 1.01.

Key fact

Large scale = small area, more detail (e.g. 1:500 000). Small scale = large area, less detail.

Direct Mercator

Schematic

Direct Mercator — straight rhumb lines; scale expands with secant(latitude).

Key fact

Cylindrical, mathematically constructed (non-perspective). Meridians parallel straight lines, equally spaced. Parallels straight, unequally spaced (expand toward poles). Poles cannot be shown.

Key fact

Straight line = rhumb line. Great circles curve concave to the equator. Chart convergency = 0 (equals earth convergency at equator only).

Formula — Mercator scale expansion

Scale at lat λ = Scale at equator × sec λ
Distance at λ = Distance at equator × cos λ

Scale denominator decreases (scale increases) away from the equator.

Key fact

Used for nautical and met charts. Aviation use in tropics 8°N–8°S where scale change can be treated as constant (≈1/100 to 1/99).

Exam trap

Rhumb line track 30°S 100°E → 40°S 110°E — track is more than 135°T (curves toward pole, not a straight 135° rhumb on the globe).

Key fact

Mercator types: Normal/Direct, Transverse (scale expands E/W — UK, Chile), Oblique (neither equator nor meridian as great circle).

Lambert conformal conic

Schematic

Lambert — straight line ≈ great circle; scale minimum at parallel of origin.

Key fact

Simple conic — scale correct only at parallel of origin (PO). Lambert modification uses two standard parallels (SP) with PO midway between them.

Key fact

Rule of sixths — near-constant scale within ±12° of latitude spread from SPs. Scale contracts toward PO; minimum at PO. Scale correct at both SPs.

Formula — Constant of the cone (n)

n = sin(parallel of origin)
Chart convergency = Change in longitude × sin(PO)

Also called convergence factor. Example: n = 0.75 → PO ≈ 48°35′.

Key fact

Straight line ≈ great circle (slightly concave to PO). Meridian = exact straight GC. Rhumb line concave to pole of projection. Chart convergency is constant (equals earth convergency at PO).

Polar stereographic

Schematic

Polar stereographic — minimum scale at the pole; used ~70°–90° latitude.

Key fact

Plane projection — paper tangential to pole, light source at opposite pole. Parallels = concentric circles, concave to nearest pole.

Key fact

Minimum scale at the pole. Scale expands as sec²(½ co-latitude) — ~1% at 80°, ~3% at 70°. Used 70°N/S to 90°N/S.

Key fact

Above ~78°, great circles appear as straight lines. Chart convergency (C)CV = change in longitude (constant, equals earth convergency at pole).

Earth vs chart convergency

Key fact

Earth convergency (E)CV — angle between meridians = change in longitude × sin(mean lat). Zero at equator, max at poles.

Key fact

Conversion angle = ½ (E)CV — difference in direction between GC and rhumb line at the mean meridian: RL track = GC track ± CA.

Key fact

GC paths concave to equator. RL paths concave to nearer pole.

Chart comparison summary

Projection	Straight line	Scale	Convergency
Direct Mercator	Rhumb line	Expands (sec λ)	0 (at equator = EC)
Lambert	≈ Great circle	Min at PO	Constant = EC at PO
Polar stereo	≈ GC above ~78°	Min at pole	= Δ longitude

Grid charts

Key fact

Near poles, grid north replaces true/magnetic for stable direction reference. Grivation = convergence + variation.

Formula — Grid heading

Grid Heading = True Heading + West longitude (− East longitude)