Hyperboloid

In geometry, a hyperboloid of revolution, sometimes called circular hyperboloid, is a surface that may be generated by rotating a hyperbola around one of its principal axes. A hyperboloid is a surface that may be obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.The Adziogol Lighthouse, Ukraine, 1911.Kobe Port Tower, Japan, 1963.Saint Louis Science Center's James S. McDonnell Planetarium, St. Louis, Missouri, 1963.Newcastle International Airport control tower, Newcastle upon Tyne, England, 1967.Ještěd Transmission Tower, Czech Republic, 1968.Cathedral of Brasília, Brazil, 1970.Hyperboloid water tower with toroidal tank, Ciechanów, Poland, 1972.Roy Thomson Hall, Toronto, Canada, 1982.The THTR-300 cooling tower for the now decommissioned thorium nuclear reactor in Hamm-Uentrop, Germany, 1983.The Corporation Street Bridge, Manchester, England, 1999.The Killesberg observation tower, Stuttgart, Germany, 2001.BMW Welt, (BMW World), museum and event venue, Munich, Germany, 2007.The Tornado Tower skyscraper in Doha, Qatar, 2008.The Canton Tower, China, 2010.The Essarts-le-Roi water tower, France. In geometry, a hyperboloid of revolution, sometimes called circular hyperboloid, is a surface that may be generated by rotating a hyperbola around one of its principal axes. A hyperboloid is a surface that may be obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation. A hyperboloid is a quadric surface, that is a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, a hyperboloid is characterized by not being a cone or a cylinder, having a center of symmetry, and intersecting many planes into hyperbolas. A hyperboloid has also three pairwise perpendicular axes of symmetry, and three pairwise perpendicular planes of symmetry. Given a hyperboloid, if one chooses a Cartesian coordinate system whose axes are axes of symmetry of the hyperboloid, and origin is the center of symmetry of the hyperboloid, then the hyperboloid may be defined by one of the two following equations: