Great-circle distance in the context of "Geodesic"

The contiguous United States, also known as the U.S. mainland, officially referred to as the conterminous United States, consists of the 48 adjoining U.S. states and the District of Columbia of the United States in central North America. The term excludes the only two non-contiguous states and the last two to be admitted to the Union, which are Alaska and Hawaii, and all other offshore insular areas, such as the U.S. territories of American Samoa, Guam, the Northern Mariana Islands, Puerto Rico, and the U.S. Virgin Islands. The colloquial term Lower 48 is also used, especially in relation to Alaska. The term The Mainland is used in Hawaii. The related but distinct term continental United States includes Alaska, which is also in North America, but separated from the 48 states by British Columbia in Canada, but excludes Hawaii and all the insular areas in the Caribbean and the Pacific.

The greatest distance on a great-circle route entirely within the contiguous U.S. is 2,802 miles (4,509 km), coast-to-coast between Florida and Washington state; the greatest north–south line is 1,650 miles (2,660 km). The contiguous United States occupies an area of 3,119,884.69 square miles (8,080,464.3 km). Of this area, 2,959,064.44 square miles (7,663,941.7 km) is actual land, composing 83.65 percent of the country's total land area, and is comparable in size to the area of Australia. Officially, 160,820.25 square miles (416,522.5 km) of the contiguous United States is water area, composing 62.66 percent of the nation's total water area.

In mathematics, two points of a sphere (or n-sphere, including a circle) are called antipodal or diametrically opposite if they are the endpoints of a diameter, a straight line segment between two points on a sphere and passing through its center.

Given any point on a sphere, its antipodal point is the unique point at greatest distance, whether measured intrinsically (great-circle distance on the surface of the sphere) or extrinsically (chordal distance through the sphere's interior). Every great circle on a sphere passing through a point also passes through its antipodal point, and there are infinitely many great circles passing through a pair of antipodal points (unlike the situation for any non-antipodal pair of points, which have a unique great circle passing through both). Many results in spherical geometry depend on choosing non-antipodal points, and degenerate if antipodal points are allowed; for example, a spherical triangle degenerates to an underspecified lune if two of the vertices are antipodal.

The Rocky Mountains, also known as the Rockies, are a major mountain range and the largest mountain system in North America. The Rocky Mountains stretch 3,000 mi (4,800 km) in a straight-line distance from the northernmost part of Western Canada, to New Mexico in the Southwestern United States. Depending on differing definitions between Canada and the U.S., its northern terminus is located either in northern British Columbia's Terminal Range south of the Liard River and east of the Trench, or in the northeastern foothills of the Brooks Range/British Mountains that face the Beaufort Sea coasts between the Canning River and the Firth River across the Alaska–Yukon border. Its southernmost point is near the Albuquerque metropolitan area, adjacent to the Rio Grande rift, and north of the Sandia–Manzano Mountain Range, also near Santa Fe, New Mexico. Being the easternmost portion of the North American Cordillera, the Rockies are distinct from the tectonically younger Cascade Range and Sierra Nevada, which both lie farther to its west.

The Rockies formed 55 million to 80 million years ago during the Laramide orogeny, in which a number of plates began sliding underneath the North American plate. The angle of subduction was shallow, resulting in a broad belt of mountains running down western North America. Since then, further tectonic activity and erosion by glaciers have sculpted the Rockies into dramatic peaks and valleys. At the end of the last ice age, humans began inhabiting the mountain range. After explorations of the range by Europeans, such as Sir Alexander Mackenzie, and Anglo-Americans, such as the Lewis and Clark Expedition, natural resources such as minerals and fur drove the initial economic exploitation of the mountains, although the range itself has never experienced a dense population.

In spherical geometry, a spherical circle (often shortened to circle) is the locus of points on a sphere at constant spherical distance (the spherical radius) from a given point on the sphere (the pole or spherical center). It is a curve of constant geodesic curvature relative to the sphere, analogous to a line or circle in the Euclidean plane; the curves analogous to straight lines are called great circles, and the curves analogous to planar circles are called small circles or lesser circles. If the sphere is embedded in three-dimensional Euclidean space, its circles are the intersections of the sphere with planes, and the great circles are intersections with planes passing through the center of the sphere.