PlanetMath: generatrices of hyperbolic paraboloid

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generatrices of hyperbolic paraboloid

(Topic)

Since the equation

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

of hyperbolic paraboloid can be gotten by multiplying the equations in the pair

$\begin{align*}\begin{cases}\displaystyle{\frac{x}{a}+\frac{y}{b} = \frac{2z}{h}}... ...ace{15pt} \\ \displaystyle{\frac{x}{a}-\frac{y}{b} = h}, \end{cases}\end{align*}$

(1)

of equations of planes, the intersection line of these planes is contained in the surface of the hyperbolic paraboloid, for each value of the parameter

. So the surface has the family of generatrices (= rulings) given by all real values of

. The same concerns the other family

$\begin{align*}\begin{cases}\displaystyle{\frac{x}{a}-\frac{y}{b} = \frac{2z}{k}}... ...pace{15pt}\\ \displaystyle{\frac{x}{a}+\frac{y}{b} = k,} \end{cases}\end{align*}$

(2)

of lines. It is easily seen that any point of the hyperbolic paraboloid is passed through by exactly two generatrices, one from the family (1) and the other from family (2). Thus the surface is a doubly ruled surface.

The latter of the equations (1) tells that all generatrices the first family are parallel to the vertical plane

$\displaystyle \frac{x}{a}-\frac{y}{b} = 0,$

the so-called director plane of the hyperbolic paraboloid; this plane is also a plane of symmetry of the surface. According to the latter equation (2), one may say the corresponding things of the alternative director plane

$\displaystyle \frac{x}{a}+\frac{y}{b} = 0.$

Note 1. We can solve from (1) and (2) the coordinates of a point lying on the surface:

$\displaystyle x = a\!\cdot\!\frac{h\!+\!k}{2},\quad y = b\!\cdot\!\frac{h\!-\!k}{2},\quad z = \frac{hk}{2}$

This is a parametric presentation of the hyperbolic paraboloid.

Note 2. One can check that two distinct lines of one family (1) resp. (2) are never in a same plane, but on the contrary, any line of one family intersects always all lines of the other family (in finity or in infinity).

Bibliography

1: LAURI PIMIÄ: Analyyttinen geometria. Werner Söderström Osakeyhtiö, Porvoo and Helsinki (1958).

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Other names:

rulings of hyperbolic paraboloid

Also defines:

director plane

Keywords:

ruled surface

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Cross-references: infinity, parametric presentation, lying on, coordinates, symmetry, parallel, doubly ruled, point, real, rulings, generatrices, parameter, surface, contained, planes, line, intersection, equations of plane, hyperbolic paraboloid, equation
There is 1 reference to this entry.

This is version 7 of generatrices of hyperbolic paraboloid, born on 2007-09-11, modified 2007-09-15.
Object id is 9929, canonical name is GeneratricesOfHyperbolicParaboloid.
Accessed 908 times total.

Classification:

AMS MSC:	51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries)
	51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)

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