(Geometry: great circle distance) The great circle distance...

(Geometry: great circle distance) The great circle distance is the distance between two points on the surface of a sphere. Let $(x 1, y 1)$ and $(x 2, y 2)$ be the geographical latitude and longitude of two points. The great circle distance between the two points can be computed using the following formula: $$d=\text {radius } \times \arccos \left(\sin \left(x_{1}\right) \times \sin \left(x_{2}\right)+\cos \left(x_{1}\right) \times \cos \left(x_{2}\right) \times \cos \left(y_{1}-y_{2}\right)\right)$$ Write a program that prompts the user to enter the latitude and longitude of two points on the earth in degrees and displays its great circle distance. The average earth radius is $6,371.01 \mathrm{km} .$ Note that you need to convert the degrees into radians using the math. radians function since the Python trigonometric functions use radians. The latitude and longitude degrees in the formula are for north and west. Use negative to indicate south and east degrees. Here is a sample run:

(Geometry: great circle distance) The great circle distance is the distance between two points on the surface of a sphere. Let $(x 1, y 1)$ and $(x 2, y 2)$ be the geographical latitude and longitude of two points. The great circle distance between the two points can be computed using the following formula:
$$d=\text {radius } \times \arccos \left(\sin \left(x_{1}\right) \times \sin \left(x_{2}\right)+\cos \left(x_{1}\right) \times \cos \left(x_{2}\right) \times \cos \left(y_{1}-y_{2}\right)\right)$$
Write a program that prompts the user to enter the latitude and longitude of two points on the earth in degrees and displays its great circle distance. The average earth radius is $6,371.01 \mathrm{km} .$ Note that you need to convert the degrees into radians using the math. radians function since the Python trigonometric functions use radians. The latitude and longitude degrees in the formula are for north and west. Use negative to indicate south and east degrees. Here is a sample run:

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Key Concepts

Great Circle Distance

The great circle distance is the shortest path between two points on the surface of a sphere, lying along the sphere's surface. It is an important concept in spherical geometry and is used in navigation and geodesy to calculate distances on a curved surface like Earth.

Spherical Geometry

Spherical geometry deals with figures on the surface of a sphere, which differs from the flat, Euclidean geometry. It involves unique relationships and formulas—such as the spherical law of cosines—that are used to calculate angles and distances between points on a curved surface.

Conversion from Degrees to Radians

Most programming languages’ trigonometric functions assume angles are in radians rather than degrees. Therefore, when working with geographic coordinates expressed in degrees, it is essential to convert these values to radians to ensure accurate calculations in trigonometric formulas.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and arccosine are fundamental to calculating distances on a sphere. These functions help translate angular measurements into linear distances when used in formulas derived from spherical trigonometry, providing the mathematical framework to compute the great circle distance.

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