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Great circles in many ways play the same logical role in spherical geometry as lines in Euclidean geometry, e.g., as the sides of (spherical) triangles. This is more than an analogy; spherical and plane geometry and others can all be unified under the umbrella of geometry built from distance measurement , where "lines" are defined to mean ...
Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle ) is called the reference plane (sometimes fundamental plane ).
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Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are great circles. Spherical trigonometry is of great importance for calculations in astronomy, geodesy, and ...
It also differs from the approach taken in Menelaus' Spherics, a treatise of the same title written 3 centuries later, which treats the geometry of the sphere intrinsically, analyzing the inherent structure of the spherical surface and circles drawn on it rather than primarily treating it as a surface embedded in three-dimensional space.
A sphere (from Greek σφαῖρα, sphaîra) [1] is a geometrical object that is a three-dimensional analogue to a two-dimensional circle.Formally, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. [2]
Spherics (sometimes spelled sphaerics or sphaerica) is a term used in the history of mathematics for historical works on spherical geometry, [1] [2] exemplified by the Spherics (Ancient Greek: τὰ σφαιρικά tá sphairiká), a treatise by the Hellenistic mathematician Theodosius (2nd or early 1st century BC), [3] and another treatise of the same title by Menelaus of Alexandria (c. 100 AD).
where C is the center of the sphere, A is the center of the small circle, and B is a point in the boundary of the small circle. Therefore, knowing the radius of the sphere, and the distance from the plane of the small circle to C, the radius of the small circle can be determined using the Pythagorean theorem.