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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
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(1) |
or
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(2) |
In fact, (2) may be thought to be formed by multiplying the equations in the pair
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(3) |
which represents a line in the space; $h$ 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 $h$ 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 $$\frac{\frac{y_1}{b}+\frac{z_1}{c}}{1-\frac{x_1}{a}} = \frac{1+\frac{x_1}{a}}{\frac{y_1}{b}-\frac{z_1}{c}},$$ and if we assign in (3) to
$h$ the value of the left hand side of the proportion, then $P_1$ satisfies also the equations (3).
But since the equation (2) may be splitted also as
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(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 $$h\left(1-\frac{x}{a}\right) = k\left(1+\frac{x}{a}\right)$$ for all $x$ 's; this would imply, by substituting $x = 0$ , that $h = k$ 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: $$x = a\frac{h-k}{h+k},\quad y = b\frac{hk+1}{h+k},\quad z = c\frac{hk-1}{h+k}$$ 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.
- 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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"generatrices of one-sheeted hyperboloid" is owned by pahio. [ full author list (2) ]
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Cross-references: plane, parametric presentation, coordinates, imply, proportion equation, surface, generatrices, 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 2559 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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Pending Errata and Addenda
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