মহা বৃত্ত ও গোলকীয় দূরত্ব

Great circles

Great circles are the circles on a sphere that have the same center as the sphere. They are the largest circles you can draw on a sphere.

A real-life geodesic would be the Earth's equator. In fact, great circles are also called equators.

Since there is no true "straight line" on a sphere, the great circles are considered the equivalent of straight lines, or geodesics, on a sphere.

Spherical distance

What's the flying distance between Los Angeles and New York City? At first, you might draw a straight line between them. However, remember that the Earth is (nearly) a sphere, and if you literally went with a line, you would fly through the Earth!

Instead, we must use great circles, the most "straight" path across a sphere.

Problem: Johnny rides an airplane from the North Pole to the South Pole of a spherical planet. If his airplane flies at 300 mph for 10 hours, what is the radius of the planet?

Draw a great circle connecting the north pole and south pole. Since the poles are on opposite sides of the sphere, Johnny travels exactly half of the great circle.

The length of half the great circle is $300 \text{ mph}\times 10\text{ h}=3000\text{ miles}$, so the length of the great circle is 6000 miles.

The great circle has the same center as the planet, so it has the same radius.

$$C=2\pi r\rightarrow 6000=2\pi r\rightarrow r=\frac{3000}{\pi}\text{ miles}$$