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Recently, I have come across a map showing the coverage of a missile given its radius on a Mercator projection. In a 3-D space, the boundary of the missile would be a circle of a sphere. However, I am curious as to how this is translated onto the 2-D Mercator projection. Note that the launch is from around North Korea, symbolised by the triangle, with a radius of 8000km.

Assuming that the Earth is perfectly spherical, and that it has a radius of $1$, if I have the latitude ($φ$) and longitude ($θ$) of a location, and the range $d$ of the missile, how would I find the formula giving me the 2-D graph upon the Mercator projection (in the form $y=f(x)$)? Currently, I am in the last year of secondary education (I would highly appreciate it if the method did not extend beyond this level too much).

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Some formulas exist that correspond to the curve of a buffer (circle) as seen on Mercator, in the special case of a spherical Earth. They are explained here. They are not quite in the form y = f(x), but they are the easiest that I know of. Note that 3 cases need to be handled, depending on the radius and the latitude of the center. However, I do not know how that relates to your current level of math, it requires a good understanding of trigonometric functions and identities to use them.

Another approach that is used a lot in software is to generate a series of points all around the center with either spherical trigonometry or the more accurate geodesic direct formulas, and then project (calculate X,Y from Lat, Lon) those points on Mercator with these formulas and use them to form and display a polygon.

As you may have noticed, a buffer around a point on Earth, when projected to Mercator, can result in all sorts of shapes that may seem couter-intuitive at first. That is because a map projection distorts reality and distances. Your map reminds me of the famous mapping error in the Independent that showed the range of missiles as simple circles on a Mercator map. As you can see, this mapping error vastly underestimated the distances (left), vs the corrected image to the right.

Error in the Independent

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