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A sphere can be constructed as the surface formed by rotating a circle one half revolution about any of its diameters; this is very similar to the traditional definition of a sphere as given in Euclid's Elements. Since a circle is a special type of ellipse, a sphere is a special type of ellipsoid of revolution.
For example, one sphere that is described in Cartesian coordinates with the equation x 2 + y 2 + z 2 = c 2 can be described in spherical coordinates by the simple equation r = c. (In this system—shown here in the mathematics convention—the sphere is adapted as a unit sphere, where the radius is set to unity and then can generally be ignored ...
Taking the distance of the Sun as 1,266 Earth radii, he was forced to place the sphere of Venus above the sphere of the Sun; as a further refinement, he added the planet's diameters to the thickness of their spheres. As a consequence, his version of the nesting spheres model had the sphere of the stars at a distance of 140,177 Earth radii. [34]
Visualization of a celestial sphere. In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth.All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphere, which may be centered on Earth or the observer.
There are different notations for expressing these uniform solutions, Wythoff symbol, Coxeter diagram, and Coxeter's t-notation. Simple tiles are generated by Möbius triangles with whole numbers p,q,r, while Schwarz triangles allow rational numbers p,q,r and allow star polygon faces, and have overlapping elements.
The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. A sphere with a spherical triangle on it.
Diagram of several terms in positional astronomy. Spherical astronomy, or positional astronomy, is a branch of observational astronomy used to locate astronomical objects on the celestial sphere, as seen at a particular date, time, and location on Earth. It relies on the mathematical methods of spherical trigonometry and the measurements of ...
Diagram showing a section through the centre of a cone (1) subtending a solid angle of 1 steradian in a sphere of radius r, along with the spherical "cap" (2). The external surface area A of the cap equals r2 only if solid angle of the cone is exactly 1 steradian. Hence, in this figure θ = A/2 and r = 1.