The latitude and longitude of a point P in the Northern...

The latitude and longitude of a point $P$ in the Northern Hemisphere are related to spherical coordinates $\rho, \theta, \phi$ as follows. We take the origin to be the center of the earth and the positive $z$ -axis to pass through the North Pole. The positive $x$ -axis passes through the point where the prime meridian (the meridian through Greenwich, England) intersects the equator. Then the latitude of $P$ is $\alpha=90^{\circ}-\phi^{\circ}$ and the longitude is $\beta=360^{\circ}-\theta^{\circ} .$ Find the great-circle distance from Los Angeles (lat. $34.06^{\circ} \mathrm{N}$, long. $\left.118.25^{\circ} \mathrm{W}\right)$ to Montréal (lat. $45.50^{\circ} \mathrm{N}$, long. $\left.73.60^{\circ} \mathrm{W}\right)$. Take the radius of the earth to be $6370 \mathrm{~km} .$ (A great circle is the circle of intersection of a sphere and a plane through the center of the sphere.)

Key Concepts

Spherical Coordinate System

The spherical coordinate system represents points in three?dimensional space using a radial distance and two angles. This framework is especially useful for problems involving spheres, as it directly relates to measurements of distance from the center of the sphere and angles that determine a point's position on the surface.

Conversion Between Geographic and Spherical Coordinates

Geographic coordinates (latitude and longitude) are naturally related to spherical coordinates through specific transformations. In this context, the latitude relates to the polar angle and the longitude to the azimuthal angle. This conversion is essential to situate locations on the sphere when using mathematical models of the Earth.

Great Circle Distance

A great circle is a circle on the sphere whose center coincides with the center of the sphere, representing the largest possible circle on the sphere. The shortest path between two points on a spherical surface is along this circle. Computing the great circle distance involves finding the central angle between two points and then multiplying it by the sphere’s radius.

Spherical Law of Cosines

The spherical law of cosines is a fundamental relation in spherical trigonometry, which is used to calculate the central angle between two points on a sphere based on their spherical coordinates. This formula is pivotal for computing the great circle distance, as it directly relates the angular separation to the positions (latitude and longitude) of the points.

Earth as a Sphere

In many mathematical models, the Earth is approximated as a perfect sphere. This simplification allows for the use of spherical geometry and trigonometry to compute distances and angles on the Earth’s surface, with the radius of the Earth serving as the key scaling factor in these calculations.

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