Question

As you may have guessed, there is a simple formula for expressing great-circle distance directly in terms of longitude and latitude. Let \(\left(\alpha_{1}, \beta_{1}\right)\) and \(\left(\alpha_{2}, \beta_{2}\right)\) be the longitude- latitude coordinates of two points on the surface of the earth, where we interpret \(\mathrm{N}\) and \(\mathrm{E}\) as positive and \(\mathrm{S}\) and \(\mathrm{W}\) as negative. Show that the great-circle distance between these points is \(3960 \gamma\) miles, where \(0 \leq \gamma \leq \pi\) and $$ \cos \gamma=\cos \left(\alpha_{1}-\alpha_{2}\right) \cos \beta_{1} \cos \beta_{2}+\sin \beta_{1} \sin \beta_{2} $$

Short answer

The great-circle distance is 3960\(\gamma\), using the cosine of \(\gamma\) calculated with given coordinates.

Step-by-step solution

Understanding the Formula

We are given a problem to find the great-circle distance between two points on Earth using latitude and longitude. The formula involves the angles \(\alpha_1, \beta_1\) and \(\alpha_2, \beta_2\) representing the coordinates of these points, where \(\alpha\) is longitude and \(\beta\) is latitude. We are tasked to show that the distance is \(3960 \gamma\), where \(\cos\gamma = \cos(\alpha_1 - \alpha_2) \cos\beta_1 \cos\beta_2 + \sin\beta_1 \sin\beta_2\).

Substituting Known Values

In this formula, apply the known values of the spherical trigonometric identities that relate to coordinate transformation. These identities link the angle \(\gamma\) (the central angle between two points) to the coordinates \((\alpha_1, \beta_1)\) and \((\alpha_2, \beta_2)\).

Cosine of the Central Angle

The equation given for \(\cos \gamma\) comes from the spherical law of cosines. This is a valid transformation of the longitudes and latitudes into a great-circle scenario where \(\cos \gamma = \cos(\alpha_1 - \alpha_2) \cos \beta_1 \cos \beta_2 + \sin \beta_1 \sin \beta_2\).

Applying Earth's Radius to Distance Formula

To convert the central angle into a physical distance, we multiply \(\gamma\) by the Earth's radius in miles. The average radius of the Earth is 3960 miles, so the distance becomes \(3960 \gamma\). This formula effectively provides the shortest distance over the Earth's surface between two points given their geographical coordinates.

Key concepts

Spherical Trigonometry

Spherical trigonometry is a branch of geometry that deals with triangles on the surface of a sphere, instead of the flat triangles dealt with in plane trigonometry. This field is crucial for navigation and astronomy.

Unlike flat triangles, which are constrained to 180 degrees, the angles of a spherical triangle add up to more than 180 degrees. This is due to the curvature of the sphere, which affects how distances and angles are measured.

When calculating distances on the Earth's surface, we use spherical trigonometry to account for its roughly spherical shape. The great-circle distance is the shortest path between two points on a sphere, resembling the straight line concept in plane geometry. Understanding this concept helps in computing routes for flights and mapping applications.

• Spherical triangles are formed by great circles—curves on the sphere's surface that represent the largest possible circles.
• Key elements involve spherical angles and sides, different from their plane counterparts.
• Spherical trigonometry also makes use of specific laws such as the spherical law of sines and cosines to solve problems.

Latitude and Longitude

Latitude and longitude are a pair of coordinates used to locate any position on Earth’s surface. Together, they create a grid that allows us to precisely pinpoint any location on the planet.

Latitude measures how far north or south a point is from the Equator, running parallel to it. It's expressed in degrees ranging from 0° at the Equator to 90° at the poles.

Longitude, on the other hand, measures how far east or west a point is from the Prime Meridian, which runs through Greenwich, England. It ranges from 0° at the Prime Meridian to 180° in both eastward and westward directions.

• Latitude lines are parallel to each other, hence they are sometimes called parallels.
• Longitude lines, or meridians, converge at the poles and are widest at the Equator.
• Both coordinates are vital for navigation, map-making, and even time-keeping on Earth.

Spherical Law of Cosines

The spherical law of cosines is a valuable tool in spherical trigonometry, especially when dealing with navigation or astronomy problems.

This law is used to find the angular distance between two points on a sphere given the latitudes and longitudes. The formula applies when the radius of the sphere is considered, making it ideal for solving problems about planets, like Earth.

The traditional form of the law for plane geometry is adapted for spheres to calculate the cosines of angles in spherical triangles.

• The formula: \[ \cos \gamma = \cos(\alpha_1 - \alpha_2) \cos \beta_1 \cos \beta_2 + \sin \beta_1 \sin \beta_2 \]connects the geographical coordinates directly to a central angle \( \gamma \).
• This angle \( \gamma \) represents the arc line distance over the sphere's surface, which helps determine the great-circle distance.
• The spherical law of cosines allows for accurate calculations necessitated by the Earth's curved surface.