The gnomonic map projection maps into straight lines all Great Circles, even those not passing through the central point, but can present even less than one hemisphere (unless the map were of infinite size with corresponding distortions, which is obviously not possible).
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In plane geometry it is a straight line. If you want to know the shortest line between two points on a globe, it will be the intervening section or arc of the great circle route that connects the points. The great circle will be a circle that cuts the globe into exactly equal parts, like the equator.
Let's be very careful here: The "great circle" of a sphere is a circle that lies on the surface ofthe sphere, so there's no way the great circle can "pass through" the sphere's center.However, in order for the circle to be a "great circle", its center must be the center of the sphere.
Yes the great circle formula is the same formula for any other circle.
A small circle is a simple circle on a two dimensional plane. The great circle is the circle around a sphere that is on the plane that intersects with the center of the sphere. It is the reason that on a flat map the paths of planes seem to curve to go to and from Europe and the North American continent.
In geometry it is called the "Great Circle".