[11Bravo]

Celestial navigation depends on the mathematical principles of spherical trigonometry. All the inputs and outputs are angles. Latitude is an angle. Longitude is an angle. The altitudes of the heavenly bodies are angles. Only the final result is converted to a linear distance using the scale factor of that spherical world.

We have been working in km using the SH3 scale factor of 1°=120km, but we were always really working with angles...minutes of arc. And we plotted our error triangle using the arcminute difference between Ho and Hc.

Our error triangle is located a certain distance from a longitude line. That means it is located a certain angle from that line. And that angle was calculated with the assumption the world was a sphere. So we need to distort that angle onto the surface of a cylinder.

The mathematical function that causes earth's longitude lines to converge at the poles is the cosine of the latitude.

The cosine of 0° is 1
The cosine of 50° is 0.6428
And the cosine of 80° is 0.1736

Recall the 1° distances between longitudes on earth.

111 km at 0° = 111 cosine 0°
71.3 km at 50° = 111 cosine 50°
19.3 km at 80° = 111 cosine 80°

So when spherical trigonometry gives you an angle between longitudes, the distances get shrunk according to the cosine of the latitude.

So it stands to reason we need to stretch them back so we can chart them on a cylinder. Since spherical trigonometry has already multiplied the angle by the cosine of the latitude, we are going to undo that by dividing by the cosine of the latitude.

Our scale factor will depend on latitude and the function will be 1/cos(latitude).

The 1/cosine of 0° is 1
The 1/cosine of 50° is 1.556
And the 1/cosine of 80° is 5.759