GEN 07: Projections

📄 Detailed Notes⚡ Cheat Sheet
 

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📄 Notes ⚡ Cheat Sheet
📄 Notes ⚡ Cheat Sheet

Q1. A non-perspective chart:

is produced by mathematically adjusting a light projection of the Reduced Earth –

  • Perspective projections are geometric (light source)- is produced directly from a light projection of a Reduced Earth.

    Non-perspective charts (like Mercator) are derived mathematically to preserve specific properties like conformality.

Q2. Parallels of latitude, except the equator, are:

Rhumb lines –

  • Parallels cut all meridians at 90°, making them Rhumb Lines. Only the Equator is a Great Circle.

    • Q3. On a direct Mercator, great circles can be represented as:

    Straight lines and curves –

  • Generally, Great Circles are curves concave to the equator. However, Meridians and the Equator are Great Circles and appear as straight lines. Thus, “Straight lines and curves” is the precise answer.

    • Q4. A normal Mercator chart is a … projection? (i) Cylindrical (ii) Perspective (iii) Non-perspective (iv) Conformal (v) Conical (vi) Azimuthal

    (i), (iii) and (iv) –

  • Mercator is Cylindrical (wrapped around equator), Non-perspective (mathematically modified stretch), and Conformal (angles preserved).

    • Q5. At 60S on a Mercator chart, chart convergence is:

    less than Earth convergency –

  • On a Mercator chart, meridians are parallel, so Chart Convergence is Zero. Earth Convergency at 60S is significant ($dLong times sin(60)$). Therefore, Chart Conv (0) is less than Earth Conv.

    • Q6. On a normal Mercator chart, rhumb lines are represented as:

    Straight lines –

  • The defining property of a Mercator chart is that straight lines represent Rhumb Lines (lines of constant bearing).

    • Q7. On a Mercator chart, the rhumb line track from Durban (30S 032E) to Perth (30S 116E) is 090°(T). What is the great circle track from Perth to Durban?

    249°(T) –

  • RL Track (Perth to Durban): Reciprocal of 090 is 270°.
  • Conversion Angle (CA): $0.5 times dLong times sin(Lat)$. $dLong = 116 – 32 = 84^{circ}$. $CA = 0.5 times 84 times 0.5 = 21^{circ}$.
  • Direction: In Southern Hemisphere, GC is nearer the pole (South) than RL. From Perth (East) to Durban (West), track is West. GC curves South (Left) of RL.
  • Calc: $270^{circ} – 21^{circ} = 249^{circ}$.

    • Q8. On a Direct Mercator chart, meridians are:

    parallel, equally spaced, vertical straight lines –

  • Meridians are projected as parallel vertical lines with equal spacing (unlike parallels of latitude which expand).

    • Q9. The angle between a straight line on a Mercator chart and the corresponding great circle is:

    Conversion angle –

  • A straight line on Mercator is a Rhumb Line. The angle between the Rhumb Line and the Great Circle is the Conversion Angle.

    • Q10. If the rhumb line track from Turin (45N 008E) to Khartoum (15N 032E) is 145°(T), what is the direction of the great circle track from Khartoum to Turin?

    331°(T) –

  • Mean Lat: $(45+15)/2 = 30^{circ}$.
  • dLong: $32-8 = 24^{circ}$.
  • CA: $0.5 times 24 times sin(30) = 12 times 0.5 = 6^{circ}$.
  • RL Reciprocal: $145+180 = 325^{circ}$.
  • Direction: Northern Hemisphere. GC is North (Poleward) of RL. Track is NW. GC is to the Right of RL.
  • Calc: $325^{circ} + 6^{circ} = 331^{circ}$.

    • Q11. Mercator charts use … projections.

    cylindrical –

  • The projection surface is a cylinder tangent at the equator.

    • Q12. How does scale change on a normal Mercator chart?

    expands directly with the secant of the latitude –

  • Scale factor at latitude $phi = sec(phi) = 1/cos(phi)$. Scale increases moving away from the Equator.

    • Q13. On a Direct Mercator chart a great circle will be represented by a:

    curve concave to the equator –

  • Great Circles bulge towards the pole to find the shortest distance, appearing concave to the Equator on the chart.

    • Q14. If the rhumb line track from Turin (45N 008E) to Khartoum (15N 032E) is 145°(T) what is the direction of the great circle track measured at Turin?

    139°(T) –

  • dLong: 24°. Mean Lat: 30°. CA: $0.5 times 24 times 0.5 = 6^{circ}$.
  • Direction: NH. Track SE. GC is North (Left) of RL.
  • Calc: $145^{circ} – 6^{circ} = 139^{circ}$.

    • Q15. On a direct Mercator, with the exception of the meridians and the Equator, great circles are represented as:

    Curves concave to the Equator –

  • Same concept as previous questions; they bulge poleward.

    • Q16. Which one of the following, concerning great circles on a Direct Mercator chart, is correct?

    With the exception of meridians and the equator, they are curves concave to the equator –

  • This option includes the necessary exception for Meridians and Equator which are straight lines.

    • Q17. Parallels of latitude on a Direct Mercator chart are:

    parallel straight lines unequally spaced –

  • They are parallel horizontal lines, but the spacing increases as latitude increases (due to the $1/cos(text{lat})$ expansion).

    • Q18. A direct Mercator graticule is:

    Rectangular –

  • Meridians and Parallels cross at 90° and are straight lines, forming rectangles.

    • Q19. On a Mercator chart, the scale:

    varies as 1/cosine of latitude (1/cosine= secant) –

  • Mathematical definition of the Mercator expansion factor.

    • Q20. A Lambert conformal conic projection, with two standard parallels:

    the scale is only correct along the standard parallels –

  • The cone cuts the earth at two parallels; scale is exact (1.0) at these intersections.

    • Q21. On a Lambert conformal conic chart, with two standard parallels, the quoted scale is correct:

    along the two standard parallels –

  • See previous explanation.

    • Q22. The convergence factor of a Lambert conformal conic chart is quoted as 0.78535. At what latitude on the chart is earth convergency correctly represented?

    51°45′ –

  • Formula: Constant of Cone (n) = $sin(text{Parallel of Origin})$.
  • Calc: $text{Lat} = arcsin(0.78535) approx 51.75^{circ} = 51^{circ}45’$. Earth convergency equals chart convergence exactly at the Parallel of Origin.

    • Q23. The scale on a Lambert conformal conic chart:

    is constant along a parallel of latitude –

  • Since the projection is symmetrical around the rotation axis, scale depends only on latitude.

    • Q24. The two standard parallels of a conical Lambert projection are at N10°40’N and N41°20′. The cone constant of this chart is approximately

    0.44 –

  • Formula: $n = sin(text{Parallel of Origin})$. Parallel of Origin is approx the average of Standard Parallels: $(10.66 + 41.33)/2 = 26^{circ}$. $sin(26) = 0.438$.
  • Strict Formula: $n = frac{log(cos phi_1) – log(cos phi_2)}{log(tan(45+phi_2/2)) – log(tan(45+phi_1/2))}$. The average sine method is sufficient for estimation.

    • Q25. On a Lambert Conformal Conic chart earth convergency is most accurately represented at the:

    parallel of origin –

  • Chart Convergence = $dLong times n$. Earth Convergence = $dLong times sin(text{Lat})$. These are equal where $n = sin(text{Lat})$, which is the Parallel of Origin.

    • Q26. On a Lambert conformal conic chart, the distance between parallels of latitude spaced the same number of degrees apart:

    reduces between, and expands outside, the standard parallels –

  • Scale is 1.0 outside (expanded).

    • Q27. A straight line on a Lambert Conformal Projection chart for normal flight planning purposes:

    is approximately a Great Circle –

  • This is the primary advantage of LCC charts for aviation; straight lines approximate Great Circles.

    • Q28. The parallels on a Lambert Conformal Conic chart are represented by:

    arcs of concentric circles –

  • The cone is unrolled, resulting in circular arcs for parallels.

    • Q29. On a Lambert chart (standard parallels 37°N and 65°N), with respect to the straight line drawn on the map between A(N49° W030°) and B (N48° W040°), the:

    great circle and rhumb line are to the south

    Q30. The constant of the cone, on a Lambert chart where the convergence angle between longitudes 010°E and 030°W is 30°, is:

    0.75 –

  • Formula: $n = text{Chart Conv} / dLong$.
  • dLong: $10E$ to $30W = 40^{circ}$.
  • Calc: $30 / 40 = 0.75$.

    • Q31. On a Lambert Conformal Conic chart great circles that are not meridians are:

    curves concave to the parallel of origin –

  • Because the projection is “flattened” slightly compared to a sphere, Great Circles curve slightly towards the Parallel of Origin (center of the map projection).

    • Q32. The nominal scale of a Lambert conformal conic chart is the:

    scale at the standard parallels –

  • This is where the scale factor is exactly 1.0.

    • Q33. On a Lambert conformal conic chart the convergence of the meridians:

    is the same as earth convergency at the parallel of origin –

  • Chart Convergence is constant ($dLong times n$). Earth Conv varies. They match only at the single latitude where $n = sin(text{lat})$.

    • Q34. Which one of the following statements is correct concerning the appearance of great circles, with the exception of meridians, on a Polar Stereographic chart whose tangency is at the pole ?

    The higher the latitude the closer they approximate to a straight line –

  • A straight line through the Pole (meridian) is a GC. As you move away from the pole, GCs curve slightly. Near the pole (high lat), they are nearly straight.

    • Q35. What is the value of the convergence factor on a Polar Stereographic chart?

    1.0 –

  • The projection surface is a plane tangent at the Pole ($sin(90) = 1$). Meridians radiate as straight lines from the pole, so Chart Conv = dLong. Factor = 1.0.

    • Q36. The chart that is generally used for navigation in polar areas is based on a:

    Stereographical projection –

  • Preserves angles (conformal) and handles the pole without distortion (unlike Mercator).

    • Q37. Which one of the following describes the appearance of rhumb lines, except meridians, on a Polar Stereographic chart?

    Curves concave to the Pole –

  • Rhumb lines spiral towards the pole, appearing as curves concave to the pole.

    • Q38. Transverse Mercator projections are used for:

    maps of large north/south extent –

    • Distortion is low along the meridian of tangency (N/S), making it ideal for N/S areas (e.g., Chile, UK).

    Q39. An Oblique Mercator projection is used specifically to produce:

    charts of the great circle route between two points –

  • The cylinder is tangent along a specific Great Circle path.

    • Q40. On a Transverse Mercator chart, scale is exactly correct along the:

    meridian of tangency –

  • Scale is 1.0 along the central meridian where the cylinder touches the earth.

    • Q41. On a transverse Mercator chart, with the exception of the Equator, parallels of latitude appear as:

    ellipses –

    • The projection geometry results in parallels appearing as ellipses concave to the nearest pole.
    • Rhumb Line – Complex curve.
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