Paraboloid

elliptic paraboloid, parabolic cylinder, hyperbolic paraboloid

The pencil of elliptic paraboloids

${\displaystyle z=x^{2}+{\frac {y^{2}}{b^{2}}},\ b>0,}$

and the pencil of hyperbolic paraboloids

${\displaystyle z=x^{2}-{\frac {y^{2}}{b^{2}}},\ b>0,}$

approach the same surface

${\displaystyle z=x^{2}}$

for ${\displaystyle b\rightarrow \infty }$, which is a parabolic cylinder (see image).

The elliptic paraboloid, parametrized simply as

${\displaystyle {\vec {\sigma }}(u,v)=\left(u,v,{\frac {u^{2}}{a^{2}}}+{\frac {v^{2}}{b^{2}}}\right)}$

has Gaussian curvature

${\displaystyle K(u,v)={\frac {4}{a^{2}b^{2}\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{2}}}}$

and mean curvature

${\displaystyle H(u,v)={\frac {a^{2}+b^{2}+{\frac {4u^{2}}{a^{2}}}+{\frac {4v^{2}}{b^{2}}}}{a^{2}b^{2}{\sqrt {\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{3}}}}}}$

which are both always positive, have their maximum at the origin, become smaller as a point on the surface moves further away from the origin, and tend asymptotically to zero as the said point moves infinitely away from the origin.

The hyperbolic paraboloid,[2] when parametrized as

${\displaystyle {\vec {\sigma }}(u,v)=\left(u,v,{\frac {u^{2}}{a^{2}}}-{\frac {v^{2}}{b^{2}}}\right)}$

has Gaussian curvature

${\displaystyle K(u,v)={\frac {-4}{a^{2}b^{2}\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{2}}}}$

and mean curvature

${\displaystyle H(u,v)={\frac {-a^{2}+b^{2}-{\frac {4u^{2}}{a^{2}}}+{\frac {4v^{2}}{b^{2}}}}{a^{2}b^{2}{\sqrt {\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{3}}}}}.}$

If the hyperbolic paraboloid

${\displaystyle z={\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}}$

is rotated by an angle of π/4 in the +z direction (according to the right hand rule), the result is the surface

${\displaystyle z=\left({\frac {x^{2}+y^{2}}{2}}\right)\left({\frac {1}{a^{2}}}-{\frac {1}{b^{2}}}\right)+xy\left({\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}\right)}$

and if a = b then this simplifies to

${\displaystyle z={\frac {2xy}{a^{2}}}}$.

Finally, letting a = √2, we see that the hyperbolic paraboloid

${\displaystyle z={\frac {x^{2}-y^{2}}{2}}.}$

is congruent to the surface

${\displaystyle z=xy}$

which can be thought of as the geometric representation (a three-dimensional nomograph, as it were) of a multiplication table.

The two paraboloidal ℝ² → ℝ functions

${\displaystyle z_{1}(x,y)={\frac {x^{2}-y^{2}}{2}}}$

and

${\displaystyle z_{2}(x,y)=xy}$

are harmonic conjugates, and together form the analytic function

${\displaystyle f(z)={\frac {z^{2}}{2}}=f(x+yi)=z_{1}(x,y)+iz_{2}(x,y)}$

which is the analytic continuation of the ℝ → ℝ parabolic function f(x) = x²/2.

The dimensions of a symmetrical paraboloidal dish are related by the equation

${\displaystyle 4FD=R^{2},}$

where F is the focal length, D is the depth of the dish (measured along the axis of symmetry from the vertex to the plane of the rim), and R is the radius of the rim. They must all be in the same unit of length. If two of these three lengths are known, this equation can be used to calculate the third.

A more complex calculation is needed to find the diameter of the dish measured along its surface. This is sometimes called the "linear diameter", and equals the diameter of a flat, circular sheet of material, usually metal, which is the right size to be cut and bent to make the dish. Two intermediate results are useful in the calculation: P = 2F (or the equivalent: P = R²/2D) and Q = √P² + R², where F, D, and R are defined as above. The diameter of the dish, measured along the surface, is then given by

${\displaystyle {\frac {RQ}{P}}+P\ln \left({\frac {R+Q}{P}}\right),}$

where ln x means the natural logarithm of x, i.e. its logarithm to base e.

The volume of the dish, the amount of liquid it could hold if the rim were horizontal and the vertex at the bottom (e.g. the capacity of a paraboloidal wok), is given by

${\displaystyle {\frac {\pi }{2}}R^{2}D,}$

where the symbols are defined as above. This can be compared with the formulae for the volumes of a cylinder (πR²D), a hemisphere (2π/3R²D, where D = R), and a cone (π/3R²D). πR² is the aperture area of the dish, the area enclosed by the rim, which is proportional to the amount of sunlight a reflector dish can intercept. The surface area of a parabolic dish can be found using the area formula for a surface of revolution which gives

${\displaystyle A={\frac {\pi R\left({\sqrt {(R^{2}+4D^{2})^{3}}}-R^{3}\right)}{6D^{2}}}.}$

• Ellipsoid – Quadric surface that looks like a deformed sphere
• Hyperboloid – Unbounded quadric surface
• Parabolic loudspeaker – Parabolic-shaped speaker producing coherent plane waves
• Parabolic reflector – Reflector that has the shape of a paraboloid

• Media related to Paraboloid at Wikimedia Commons