Editing Paraboloid (section)

{{Short description|Quadric surface with one axis of symmetry and no center of symmetry}}
{{more citations needed|date=June 2020}}
[[Image:Paraboloid of Revolution.svg|thumb|right|Paraboloid of revolution]]
In [[geometry]], a '''paraboloid''' is a [[quadric surface]] that has exactly one [[axial symmetry|axis of symmetry]] and no [[central symmetry|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 (geometry)#A line and a plane|parallel]] to the axis of symmetry is a parabola. The paraboloid is '''hyperbolic''' if every other plane section is either a [[hyperbola]], or two crossing lines (in the case of a section by a tangent plane). The paraboloid is '''elliptic''' if every other nonempty plane section is either an [[ellipse]], or a single point (in the case of a section by a tangent plane). A paraboloid is either elliptic or hyperbolic.

Equivalently, a paraboloid may be defined as a quadric surface that is not a [[cylinder]], and has an [[implicit surface|implicit equation]] whose part of degree two may be factored over the [[complex number]]s into two different linear factors. The paraboloid is hyperbolic if the factors are real; elliptic if the factors are [[complex conjugate]].

An elliptic paraboloid is shaped like an oval cup and has a [[maximum]] or minimum point when its axis is vertical. In a suitable [[coordinate system]] with three axes {{math|''x''}}, {{math|''y''}}, and {{math|''z''}}, it can be represented by the equation<ref>{{cite book |title=Thomas' Calculus 11th ed. |last=Thomas |first=George B. |author2=Maurice D. Weir |author3=Joel Hass |author3-link=Joel Hass |author4=Frank R. Giordiano |year=2005 |publisher= Pearson Education, Inc |isbn=0-321-18558-7 |page=892}}</ref>
<math display="block">z = \frac{x^2}{a^2} + \frac{y^2}{b^2}.</math>
where {{math|''a''}} and {{math|''b''}} are constants that dictate the level of curvature in the {{math|''xz''}} and {{math|''yz''}} planes respectively. In this position, the elliptic paraboloid opens upward.

[[Image:HyperbolicParaboloid.svg|thumb|right|Hyperbolic paraboloid]]
A hyperbolic paraboloid (not to be confused with a [[hyperboloid]]) is a [[doubly ruled surface]] shaped like a [[saddle]]. In a suitable coordinate system, a hyperbolic paraboloid can be represented by the equation<ref name="Weisstein">Weisstein, Eric W. "Hyperbolic Paraboloid." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HyperbolicParaboloid.html</ref><ref>{{cite book |title=Thomas' Calculus 11th ed. |last=Thomas |first=George B.|author2=Maurice D. Weir |author3=Joel Hass |author4=Frank R. Giordiano |year=2005 |publisher= Pearson Education, Inc |isbn=0-321-18558-7 |page=896}}</ref>
<math display="block">z = \frac{y^2}{b^2} - \frac{x^2}{a^2}.</math>
In this position, the hyperbolic paraboloid opens downward along the {{math|''x''}}-axis and upward along the {{math|''y''}}-axis (that is, the parabola in the plane {{math|''x'' {{=}} 0}} opens upward and the parabola in the plane {{math|''y'' {{=}} 0}} opens downward).

Any paraboloid (elliptic or hyperbolic) is a [[Translation surface (differential geometry)|translation surface]], as it can be generated by a moving parabola directed by a second parabola.