Paraboloid
In analytic geometry, a paraboloid is a quadric, a type of three-dimensional surface that is described by equations whose canonical form is of the type:
- (xa)2± ± (andb)2− − z=0{displaystyle left({frac {x}{a}}right)^{2}pm left({frac {y}{b}}{b}}right)^{2}-{z}=0}
Paraboloids can be elliptical or hyperbolic, depending on whether their quadratic terms (those containing variables raised squared, here indicated as x and y) have the same or different sign, respectively.
Hyperbolic Paraboloid
A paraboloid will be hyperbolic when the quadratic quantitative terms of its canonical equation are of opposite sign:
- (xa)2− − (andb)2− − z=0{displaystyle left({frac {x}{a}}}right)^{2}-left({frac {y}{b}{b}}right)^{2}-{z}=0}.
The hyperbolic paraboloid is a doubly ruled surface so it can be constructed from lines. Because of its appearance, it is also called a saddle surface.
The hyperbolic paraboloid is a surface generated by the displacement of a generating parabola that slides parallel to itself along another directrix parabola of opposite curvature located in its plane of symmetry.
Pringles snacks are characterized by having a hyperbolic paraboloid shape.
Elliptical Paraboloid
A paraboloid will be elliptic when the quadratic terms of its canonical equation have the same sign:
- (xa)2+(andb)2− − z=0{displaystyle left({frac {x}{a}}right)^{2}+left({frac {y}{b}}{b}}right)^{2}-z=0}
If it is also a = b, the elliptic paraboloid will be a paraboloid of revolution, which is the surface resulting from rotating a parabola around its axis of symmetry.
Satellite antennas are paraboloids of revolution, and have the property of reflecting incoming parallel rays towards their focus, the point where the receiver is located.