If I know the center and radius of a massive circle C, how can I construct a smaller ellipse E to approximate the arc I'm interested in within a range of confidence R?
Approximate an Arc with an Ellipse
Basically, this is a Navigation problem relating to the Circles of Equal Altitude. If we know the angular altitude of the Sun, we can determine a circle around the Globe of our possible location. Since we can only be within a certain range of where we were when we measured Latitude and Longitude, that circle is reduced to just an arc. And since we can't locate the centre of the circle in a map to draw that arc, an ellipse would suffice.
I know there's the curvature of the Earth to take into account, but for now, could such 2D approximation be constructed?
The radius of curvature at the end of the minor axis of an ellipse is $a^2/b$, where $a$ is the semi-major axis and $b$ is the semi-minor axis. Therefore choose any $b$ (perhaps the biggest possible value that will fit on your page); then set $a = \sqrt{br}$, where $r$ is the radius of the circle.