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Paraboloid of revolution. In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plane section of a paraboloid made by a plane parallel to
The red paraboloid corresponds to τ=2, the blue paraboloid corresponds to σ=1, and the yellow half-plane corresponds to φ=-60°. The three surfaces intersect at the point P (shown as a black sphere) with Cartesian coordinates roughly (1.0, -1.732, 1.5).
Paraboloidal coordinates are three-dimensional orthogonal coordinates (,,) that generalize two-dimensional parabolic coordinates.They possess elliptic paraboloids as one-coordinate surfaces.
In the theory of quadratic forms, the parabola is the graph of the quadratic form x 2 (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form x 2 + y 2 (or scalings), and the hyperbolic paraboloid is the graph of the indefinite quadratic form x 2 − y 2. Generalizations to more variables yield ...
The hyperbolic paraboloid and the hyperboloid of one sheet are doubly ruled surfaces. The plane is the only surface which contains at least three distinct lines through each of its points ( Fuchs & Tabachnikov 2007 ).
A saddle point (in red) on the graph of z = x 2 − y 2 (hyperbolic paraboloid). In mathematics, a saddle point or minimax point [1] is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. [2]
A hyperbolic paraboloid (the graph of the function z = xy) is a differentiable surface and an algebraic surface. It is also a ruled surface, and, for this reason, is often used in architecture . A two-sheet hyperboloid is an algebraic surface and the union of two non-intersecting differentiable surfaces.
Hyperbolic paraboloid drawn using PSTricks package pst-3dplot. multido provides basic loop functionality for programming graphs with repeating elements: Plotting a graph while varying parameters with multido. pst-eucl is a beta extension for easy creation of geometrical drawings. Circumcircle of a triangle, illustrated with pst-eucl