Definition

Noun: A spherical polygon is a geometric figure on the surface of a sphere. It is bounded by three or more arcs of great circles. These arcs are the sides of the polygon, and the points where they intersect are its vertices.

Usage

The term is used in spherical geometry, a branch of mathematics that deals with figures on a sphere's surface, contrasting with plane geometry. * In navigation and astronomy, the concept of a spherical polygon is fundamental for calculating distances and angles on the celestial sphere or the Earth. * The simplest spherical polygon is a spherical triangle, which has three sides.

Examples

• "To calculate the area, the navigator treated the region as a spherical polygon."
• "In this model, each face of the polyhedron corresponds to a spherical polygon on the circumscribed sphere."
• "The sum of the interior angles of a spherical polygon is greater than that of a plane polygon with the same number of sides."

Advanced Usage

• Spherical Excess: A key property of a spherical polygon is its spherical excess (E), which is the amount by which the sum of its interior angles exceeds the sum for a plane polygon. For a polygon with sides, the area is often proportional to this excess (A = E * R², where R is the sphere's radius).
• Geodesic Polygon: A spherical polygon is a specific type of geodesic polygon, where all the sides are geodesic segments (the shortest paths between points on the sphere).

Variants and Related Words

• Spherical Triangle (n): A spherical polygon with exactly three sides and three vertices. It is the most commonly studied case.
• Spherical Geometry (n): The non-Euclidean geometry of the two-dimensional surface of a sphere, where figures like the spherical polygon are defined.
• Lune (n): A spherical polygon bounded by two halves of great circles. It is a digon, which is a degenerate case with two sides.

Synonyms

• Spherical figure: A general term for any shape on a sphere's surface.
• Geodesic polygon on a sphere: A more technical synonym emphasizing the sides are great circle arcs.

Related Concepts (Not Phrasal Verbs or Idioms)

• Great Circle: A circle on the surface of a sphere whose center is the sphere's center. The arcs of great circles form the sides of a spherical polygon.
• Plane Polygon: The analogous figure in flat, Euclidean geometry, which differs in its properties (e.g., angle sums).
• Solid Angle: The three-dimensional angle subtended by a spherical polygon at the sphere's center can be measured in steradians.

Noun

1. a figure on the surface of a sphere bounded by arcs of 3 or more great circles