Plot of Curves of equal Altitude on the Flat Earth

From zenith angles to stars and using a Nautical Almanac we can get the ground positions (GP) of each star and the distances of the observer from this GPs, see McToon's Challenge.

On the Globe we then draw circles with centers at the GPs and radii equal to the distances from the GPs, measured along the surface. Everywhere on such a circle we would see the corresponding star at the same altitude (elevation or zenith angle). That's why this circles are called Circles of equal Altitude. The observers location on the Globe is where all 3 Circles of equal Altitude intersect.

If however we draw the same Circles of equal Altitude on a Flat Earth, we can not get an intersection of all 3 circles at the same location and therefore it is impossible to get the correct location of the observer.

Only if we calculate the Circles of equal Altitude on a Globe model and then project them onto a Flat Earth Map using the right projection for the type of map used, can we get the correct location of the observer on a Flat Earth. We need a Globe for that.

The projected circles are curves as shown in the image below for the AE map. See Globe Arc Projection to Flat Earth for the math of this projection, an App to create this image and the JavaScript of the App.

The red crosses are the GPs of the 3 sighted stars of McToon's Challenge. The intersection of the 3 curves is the correct location in Minnesota from his Challenge:

McToon's Challenge

McToon created the following Challenge to find a position on earth from the zenith angles to 3 stars, measured using a sextant or any other device that gives the zenith angles directly or indirectly:

Date: 2022-03-28, Times as Central Time or GMT -5

He gave the local times and the names of 3 sighted stars with their elevation angle, measured using a theodolite App on the smartphone.

The GHA of Aries (Greenwich Hour Angle), the SHA of the Star (Sidereal Hour Angle) and the Declination of the Star (Dec) can be found for the actual date and times in the Nautical Almanac 2022.

Then the latitude and longitude of the Ground Positions (GP) of each star and the distance of the observer from the GP can be calculated as follows:

• GP Lat = Declination Star
• GP Long = GHA Aries + SHA Star, normalized to a value between +/-180°
• GP Distance = Zenith Angle * 60 nmi, because the earth is a Globe with a circumference of 360·60 nmi (radius = 3438 nmi).
• Zenith Angle = 90° - Elevation Angle, if we have an Elevation Angle instead of the Zenith Angle.

The GP Distance is the radius of the Circle of equal Altitude, measured along the curved surface of the Globe.