The last module priced the two lines an aircraft can fly; this lesson prices the paper they are drawn on. A projection is a controlled distortion, and aviation's rules for it are short: angles must survive, and the line a pilot draws with a ruler should be worth flying. Everything in this lesson follows from those two demands.
What aviation demands of a chart
ICAO Annex 4, the standard for aeronautical charts, asks for two properties above all:
conformality — angles and bearings on the Earth appear unchanged on the chart, so a track measured with a protractor is the track to fly, and shapes stay locally true;
a straight line drawn on the chart should approximate a great circle, so the ruler's line is also the shortest route.
The second demand is really a statement about convergence. A chart draws its meridians converging at some rate of its own, the chart convergence (the syllabus sometimes writes convergency), and on conical charts that rate is set by the constant of the cone, n, which equals the sine of the parallel of origin (the parallel of origin is the parallel the cone is built around: the one it touches when it simply rests on the globe): chart convergence = change of longitude × n. A chart whose parallel of origin is 40°N has n = sin 40°, which is 0.643, and two meridians 20° apart converge on it by 20 × 0.643, about 13°. A straight line approximates a great circle where the chart convergence matches the Earth convergence of the last module, Δlong × sin(mean latitude): choose the parallel of origin near the mean latitude of the area, and sin(parallel of origin) ≈ sin(mean latitude), so the ruler and the geometry agree.
Scale
A chart declares how much it has shrunk the world in three interchangeable ways:
a representative fraction such as 1:500,000, one chart unit to 500,000 Earth units in any unit you like;
a statement in words, such as one inch to ten nautical miles;
a graduated scale line printed on the sheet, to be walked with dividers.
The arithmetic is one proportion: Earth distance = chart length × the scale denominator. On a 1:500,000 chart, 27 cm is 27 × 500,000 cm, which is 135 km, and dividing by 1.852 makes it about 73 NM. Run backwards it identifies a chart: if 5 cm represents 20 NM, then 20 × 1.852 km is 3,704,000 cm of Earth, and 3,704,000 over 5 gives a scale of about 1:740,800.
The three families
Every classical projection presses the globe onto one of three developable surfaces, and the families are named for them:
azimuthal — from a flat plane;
cylindrical — from a wrapped cylinder;
conical — from a resting cone.
Where the surface touches the globe the chart is honest; away from the touch the distortion grows, which is why each family owns the region its surface hugs: planes the poles, cylinders the equator, cones the middle latitudes. The plane's chart and the cylinder's fill the rest of this lesson; the cone's, the Lambert chart that carries most en-route flying, is the next lesson's whole subject.