Projection (cartography) Study Guide

📖 Core Concepts Map projection – a mathematical transformation that flattens the Earth’s curved surface onto a plane. Distortion – unavoidable when projecting a sphere/ellipsoid; any projection sacrifices at least one metric property (area, shape, distance, direction). Metric properties – area, shape (local angles), distance, direction/bearing. Developable surface – a surface (plane, cylinder, cone) that can be unfolded without stretching; used as the basis for most classic projections. Standard line / parallel – the line where the developable surface touches or cuts the globe; distortion is minimal there. Tissot’s Indicatrix – an ellipse drawn on a map that shows local scale factors: meridian scale h, parallel scale k, and angular deformation θ'. 📌 Must Remember Gauss’s Theorema Egregium – a sphere cannot be mapped to a plane without distortion. Conformal ≠ Equal‑area – no projection can be both; conformal preserves angles, equal‑area preserves area. East‑west stretch factor for a normal cylindrical projection: $\sec\phi$ (where $\phi$ = latitude). Cylindrical stretch summary Mercator (conformal): NS stretch = EW stretch = $\sec\phi$. Equirectangular (plate carrée): NS stretch = 1 (latitude spacing preserved). Equal‑area cylindrical: NS stretch = $\cos\phi$ (reciprocal of EW stretch). Azimuthal projection formulas (distance from centre $d$ on sphere of radius $R$) Gnomonic: $r = c\,\tan\frac{d}{R}$ Orthographic: $r = c\,\sin\frac{d}{R}$ Stereographic (conformal): $r = c\,\tan\frac{d}{2R}$ Azimuthal Equidistant: $r = c\,d$ Lambert Azimuthal Equal‑Area: $r = c\,\sin\frac{d}{2R}$ 🔄 Key Processes Select Earth model – sphere (simpler) vs. ellipsoid (more accurate shape). Choose projection surface – plane, cylinder, or cone (depends on region shape & intended use). Set aspect – normal (aligned with Earth's axis), transverse (perpendicular), or oblique (intermediate). Define standard lines/parallels – where the surface touches the globe; pick one (cylindrical) or two (conic) to limit distortion. Transform geographic coordinates – Convert $(\lambda,\phi)$ to Cartesian $(x,y)$ or polar $(r,\theta)$ using the chosen projection formulas. Apply scale factors – compute local scale using Tissot’s indicatrix if needed for error analysis. 🔍 Key Comparisons Cylindrical vs. Conic vs. Azimuthal Cylindrical: meridians → vertical lines, parallels → horizontal lines; best for equatorial‑wide regions. Conic: meridians → straight lines radiating from apex, parallels → arcs; ideal for mid‑latitude east‑west extents. Azimuthal: radial symmetry, preserves direction from centre; used for polar maps or great‑circle navigation. Mercator (conformal) vs. Gall‑Peters (equal‑area) Mercator: preserves angles & constant bearing → navigation; gross area distortion near poles. Gall‑Peters: preserves area → thematic maps; shape heavily distorted. Transverse Mercator vs. Normal Mercator Transverse: cylinder rotated 90°; minimal distortion along a chosen meridian → large‑scale national mapping. Normal: cylinder aligned with equator → world‑scale navigation maps. ⚠️ Common Misunderstandings “Mercator shows true size.” – It preserves angles, not area; high latitudes appear enlarged. “All cylindrical projections have the same distortion pattern.” – Stretch factors differ (see Must Remember). “Equal‑area means shapes are correct.” – Shapes are distorted; only area ratios stay true. “Azimuthal always preserves distances.” – Only the Azimuthal Equidistant projection does; others preserve direction or area instead. 🧠 Mental Models / Intuition “Stretch‑shrink analogy” – Imagine laying a rubber sheet over a globe; where the sheet touches (standard line) it fits perfectly, elsewhere it stretches (horizontal) or shrinks (vertical). “Cone‑cylinder‑plane hierarchy” – Think of fitting a flexible sheet: a cone matches mid‑latitude belts best, a cylinder fits equatorial belts, a plane fits a tiny region (local projection). Tissot’s ellipse – visual cue: a perfect circle → no distortion; elongated ellipse → more distortion in that direction. 🚩 Exceptions & Edge Cases Polar regions – cylindrical projections become extreme; azimuthal (e.g., stereographic) is preferred. Global thematic maps – compromise projections (Winkel Tripel, Robinson) are used when no single metric dominates. Transverse vs. Oblique – Oblique cylinders are rare; they are used when the area of interest is oriented diagonally relative to the meridian/equator. 📍 When to Use Which Navigation (constant bearing) → Mercator (normal or transverse). Large‑scale topographic mapping → Transverse Mercator or close variant. Continental east‑west extent → Lambert Conformal Conic (conformal) or Albers (equal‑area). Polar maps → Stereographic (conformal) or Lambert Azimuthal Equal‑Area. World‑wide visual maps → Winkel Tripel, Robinson, or Mollweide (compromise). Thematic data per unit area → Any equal‑area projection (e.g., Gall‑Peters, Sinusoidal, Albers). 👀 Patterns to Recognize “Two standard parallels = conic” – whenever a problem mentions two latitudes with minimal distortion, think conic (Albers, Lambert). “Secant vs. tangent” – secant surface cuts the globe (two standard lines), tangent touches at one (single line). “$\sec\phi$ factor” – appears in any normal cylindrical projection’s east‑west scaling. “Great‑circle = straight line” – only in gnomonic projection; spot when a map shows straight‑line routes across the globe. 🗂️ Exam Traps Choosing “Mercator” for area‑sensitive maps – distractor; Mercator is conformal, not equal‑area. Confusing transverse Mercator with normal Mercator – both are cylindrical but rotated; the exam may give a “central meridian” clue indicating transverse. Assuming all azimuthal maps preserve distances – only the azimuthal equidistant does; other azimuthal types preserve direction or area. Mixing up scale factors – remember east‑west stretch = $\sec\phi$; north‑south stretch varies by projection (1, $\sec\phi$, or $\cos\phi$). “Equal‑area” vs. “Equal‑scale” – equal‑area does not imply uniform scale; scale still varies with latitude. --- Quick Review Tip: Memorize the four classic families (cylindrical, conic, azimuthal, pseudocylindrical) and the single‑property each excels at (conformal, equal‑area, equidistant, azimuthal). Then match the map’s purpose to the appropriate family and property. Good luck!