# Calculate ellipse semi-axises from center point on Earth given distance

How would one calculate the vertices of an ellipse on the surface of the Earth given a center coordinate and a distance (in KM)? Since the distance between two points in the Y (Latitude) direction get closer as one approaches the poles, when generating a "buffer" around a point, the buffer becomes an ellipse and not a perfect circle. As such I need to calculate the semi-major and semi-minor axis values to determine the vertices.

Not sure of the formulas for such a task

• If I read the question correctly the figure you want is a circle: the set of points a fixed (great circle) distance from a fixed point. Distance on the surface of the Earth is independent of latitude, although finding an equation for points on that circle isn't. – Ethan Bolker Dec 20 '17 at 14:27
• Correct. However, how does one determine the terminal points in the X/Y directions from a fixed point? I know great circle to be used in the calculation of the distance between two fixed points on Earth (Haversine). – pstatix Dec 20 '17 at 14:29
• @pstatix Can you add a sketch of what you want? – cgiovanardi Dec 21 '17 at 0:50

That said, if you are willing to accept some inaccuracies, much simpler and less time-consuming formulae exist. For example, assuming your distances are fairly short, and not too close to a pole, you could simply use $\frac{s}{r{\text{ cos }}\phi}$ ($\phi =$ latitude, $s =$ buffer distance and $r =$ 6371 km) as a semi major-axis, and $s/r$ as a semi minor-axis, multiply these 2 results by $180/\pi$ and the resulting "ellipse" (analog to Tissot's indicatrix) on the lat-lon square grid (called plate carree equirectangular projection) would give you an approximation of the actual buffer, which could then be used to generate lat/lon coordinates for other vertices. It works fairly well for shorts distances (ex. a 5 km buffer typically gives an max error around 20 meters), however, for a distance of, say, 1000 km, the max error can be as large as several tens of km, and more if the buffer is close to the poles. The haversine and spherical trigonometry method is another way to solve the problem, with errors no larger than 0.5% for any length.