PlanetMath: generatrices of one-sheeted hyperboloid

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generatrices of one-sheeted hyperboloid

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The one-sheeted hyperboloid is a ruled surface, which is seen from its equation written in the form

$\displaystyle \frac{y^2}{b^2}-\frac{z^2}{c^2} = 1-\frac{x^2}{a^2},$

(1)

$\displaystyle \left(\frac{y}{b}+\frac{z}{c}\right)\left(\frac{y}{b}-\frac{z}{c}\right) = \left(1+\frac{x}{a}\right)\left(1-\frac{x}{a}\right).$

(2)

In fact, (2) may be thought to be formed by multiplying the equations in the pair

$\begin{align*}\begin{cases}\displaystyle{\frac{y}{b}+\frac{z}{c} = h\left(1-\fra... ...b}-\frac{z}{c} = \frac{1}{h}\left(1+\frac{x}{a}\right),} \end{cases}\end{align*}$

(3)

which represents a line in the space;

is an arbitrary parameter. For any $h \neq 0$ , each point $(x,\,y,\,z)$ on the line (3) satisfies also (2). This means that the line (3) lies on the hyperboloid, i.e. it's a question of a generatrix (= ruling) of the one-sheeted hyperboloid.

Giving distinct real values to the parameter we get an infinite family of the generatrices (3). Further, one of these lines passes through every point of the hyperboloid. Actually, if the point $P_1 = (x_1,\,y_1,\,z_1)$ satisfies the equation (2) of the surface, we have the proportion equation

$\displaystyle \frac{\frac{y_1}{b}+\frac{z_1}{c}}{1-\frac{x_1}{a}} = \frac{1+\frac{x_1}{a}}{\frac{y_1}{b}-\frac{z_1}{c}},$

and if we assign in (3) to

the value of the left hand side of the proportion, then

satisfies also the equations (3).

But since the equation (2) may be splitted also as

$\begin{align*}\begin{cases}\displaystyle{\frac{y}{b}+\frac{z}{c} = k\left(1+\fra... ...b}-\frac{z}{c} = \frac{1}{k}\left(1-\frac{x}{a}\right),} \end{cases}\end{align*}$

(4)

the hyperboloid has as well the other family (4) of generatrices, containing similarly one generatrix through every point of the surface. The one-sheeted hyperboloid is doubly ruled -- having two distinct generatrices through every point. And the families (3) and (4) have really no common members, since otherwise we had an equation

$\displaystyle h\left(1-\frac{x}{a}\right) = k\left(1+\frac{x}{a}\right)$

for all

's; this would imply, by substituting

, that

and then the impossibility $\displaystyle{1-\frac{x}{a} \equiv 1+\frac{x}{a}}$ .

Note 1. One can solve from the equations (3) and (4) the coordinates for points of the one-sheeted hyperboloid:

$\displaystyle x = a\frac{h-k}{h+k},\quad y = b\frac{hk+1}{h+k},\quad z = c\frac{hk-1}{h+k}$

This is a parametric presentation of the surface.

Note 2. Furthermore one may prove, that two lines of the same family (3) or (4) cannot lie in a same plane, but two lines of distinct families (3) and (4) lie always in a same plane.

$\begin{pspicture}(-5,-6)(5,2) \psplot[linecolor=blue]{-4}{4}{1 x x mul 16 div su... ...6,-5.951) %324 \psline[linecolor=blue](3.804,-0.309)(-0,-6) %342 \end{pspicture}$

Bibliography

1: L. LINDELÖF: Analyyttisen geometrian oppikirja. Kolmas painos. Suomalaisen Kirjallisuuden Seura, Helsinki (1924).
2: LAURI PIMIÄ: Analyyttinen geometria. Werner Söderström Osakeyhtiö, Porvoo and Helsinki (1958).

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Other names:

rulings of one-sheeted hyperboloid

Also defines:

doubly ruled

Keywords:

ruled surface

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Cross-references: plane, parametric presentation, coordinates, imply, proportion equation, surface, infinite, real, ruling, generatrix, lies on, line, point, parameter, equation, ruled surface, one-sheeted hyperboloid
There are 3 references to this entry.

This is version 17 of generatrices of one-sheeted hyperboloid, born on 2007-09-08, modified 2008-01-15.
Object id is 9924, canonical name is GeneratricesOfOneSheetedHyperboloid.
Accessed 1132 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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