Geodesic

A geodesic is, in its most basic terms, the shortest path between two points along a surface. The exact nature of a geodesic varies depending on whether the concept is being considered in terms of Euclidean or non-Euclidean geometry. In Euclidean geometry, a geodesic is simply a straight line between two points on a surface. In non-Euclidean geometry, a geodesic is typically described as a segment of a great circle. In most cases, this sort of geodesic line is actually curved to some degree. Outside of Euclidean geometry, geodesics are most commonly associated with a specific form of non-Euclidean geometry known as Riemannian geometry. While geodesics have many everyday applications, they are perhaps most recognized for the key role they play in the design and construction of special structures called geodesic domes. Such domes are spherical space-frame structures that are made of a self-bracing framework of straight geodesic lines that ultimately form a spherical surface.

Background

In order to fully understand and appreciate the concept of geodesics, it helps to first have an adequate understanding of the difference between Euclidean and non-Euclidean geometry. As a whole, geometry is the field of mathematics that deals with ideas like points, lines, angles, shapes, and the properties and relationships these things share. Euclidean geometry is the common form of geometry that most students study in the classroom. This form of geometry takes its name from the ancient Greek mathematician Euclid. More than 2,000 years ago, in a book called The Elements, Euclid described the geometric properties of objects that exist in a flat two-dimensional plane. For this reason, Euclidean geometry is also known as plane geometry. In addition to laying the groundwork for traditional geometry, Euclid's descriptions in The Elements highlight the main difference between Euclidean and non-Euclidean geometry. Where Euclidean geometry applies only to objects on a flat two-dimensional plane, non-Euclidean forms of geometry apply to objects on planes that are not flat and may involve more than two dimensions. Put more simply, non-Euclidean geometry encompasses any form of geometry that does not exist in a flat world.

One example of non-Euclidean geometry is spherical geometry. Spherical geometry is a variant of plane geometry that is warped such that it can be applied to the surface of a sphere. The geometric properties of triangles provide a simple example of how plane and spherical geometry differ. In plane geometry, the interior angles of a triangle always add to 180°. In spherical geometry, the same angles add up to more than 180°. One form of spherical geometry is Riemannian geometry. Riemannian geometry is the study of curved surfaces and higher dimensional spaces. It was first developed in the nineteenth century by a German mathematician named Bernhard Riemann. Riemann unified and generalized several types of geometry into a new single form of geometry that helped applied plane geometry to curved surfaces. His work changed the way people looked at the world and led the way to higher dimensional geometry. One of the key elements of Riemannian geometry was the geodesic, which demonstrated how the shortest path between two points was different on spherical planes than it was on flat planes.

Overview

At the most basic level, a geodesic is simply defined as the shortest path between two points. On a flat plane as seen in Euclidean geometry, a geodesic is a segment of straight line that joins two points. On a spherical plane as seen in Riemannian geometry and other forms of non-Euclidean geometry, a geodesic is the shorter arc of a great circle that joins two points. Geodesics can also be viewed in relation to the theory of general relativity. In that sense, a geodesic is the path followed by a particle that is not acted upon by electromagnetic forces.

While they certainly have a role in Euclidean geometry, geodesic lines are of particular interest in the study of non-Euclidean forms of geometry. One of the easiest ways to visualize the nature of geodesic lines on a spherical plane is to apply the concept to a typical globe. A line that runs between the north and south poles is an example of a geodesic line in that it directly connects the two poles and necessarily curves as it extends from one pole to the other. This example also helps to illustrate another important detail about geodesics: the geodesic lines on a spherical plane are not strictly parallel to one another as they are on a flat plane. Instead, all the geodesic lines that extend from pole to pole on a globe ultimately converge at either pole.

Geodesic lines have many practical uses and can be seen in all sorts of places. Among the most common places where one can find examples of geodesic lines is on maps that show the shortest distance between two cities for an airplane's flight path. Because these particular geodesic lines connect points on the surface of a spherical body, they are curved rather than straight. Geodesic lines can also play into political boundaries. The best example of this can be seen in the disputed Beaufort Sea International Border where American and Canadian territory intersects in and off the coast of northern Alaska. Each country has a different interpretation of how the borderline they share in the Beaufort Sea should be plotted. Canada recognizes the so-called A-B line that follows the land border between Alaska and the Yukon Territory and follows a geodesic curve to the west, while the United States recognizes a straighter border that skews east.

By far, the most well-known example of geodesic lines in a practical application is the geodesic dome. A geodesic dome is a type of geodesic structure, or a spherical structure made of interconnecting geodesic lines instead of curved surfaces. The most basic example of such a structure is the geodesic playdome, or jungle gym. American architect R. Buckminster Fuller is traditionally credited as the inventor of the geodesic dome. He led the construction of several of the first such domes in the 1940s. Many notable geodesic domes have been built since that time. Some of these include the Tacoma Dome in Tacoma, Washington; Mitchell Park Conservatory in Milwaukee, Wisconsin; and the St. Louis Climatron in St. Louis, Missouri. However, the world's best-known geodesic dome is Spaceship Earth in Walt Disney World's EPCOT theme park in Orlando, Florida.

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