Hyperboloid of one sheet Lines on the surface • A hyperboloid of one sheet contains two pencils of lines. It is a
doubly ruled surface. If the hyperboloid has the equation {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2}= 1 then the lines g^{\pm}_{\alpha}: \mathbf{x}(t) = \begin{pmatrix} a\cos\alpha \\ b\sin\alpha \\ 0\end{pmatrix} + t\cdot \begin{pmatrix} -a\sin\alpha\\ b\cos\alpha\\ \pm c\end{pmatrix}\ ,\quad t\in \R,\ 0\le \alpha\le 2\pi\ are contained in the surface. In case a = b the hyperboloid is a surface of revolution and can be generated by rotating one of the two lines g^{+}_{0} or g^{-}_{0}, which are skew to the rotation axis (see picture). This property is called ''
Wren's theorem''. The more common generation of a one-sheet hyperboloid of revolution is rotating a
hyperbola around its
semi-minor axis (see picture; rotating the hyperbola around its other axis gives a two-sheet hyperbola of revolution). A hyperboloid of one sheet is
projectively equivalent to a
hyperbolic paraboloid.
Plane sections For simplicity the plane sections of the
unit hyperboloid with equation \ H_1: x^2+y^2-z^2=1 are considered. Because a hyperboloid in general position is an affine image of the unit hyperboloid, the result applies to the general case, too. • A plane with a slope less than 1 (1 is the slope of the lines on the hyperboloid) intersects H_1 in an
ellipse, • A plane with a slope equal to 1 containing the origin intersects H_1 in a
pair of parallel lines, • A plane with a slope equal 1 not containing the origin intersects H_1 in a
parabola, • A tangential plane intersects H_1 in a
pair of intersecting lines, • A non-tangential plane with a slope greater than 1 intersects H_1 in a
hyperbola. Obviously, any one-sheet hyperboloid of revolution contains circles. This is also true, but less obvious, in the general case (see
circular section).
Hyperboloid of two sheets The hyperboloid of two sheets does
not contain lines. The discussion of plane sections can be performed for the
unit hyperboloid of two sheets with equation H_2: \ x^2+y^2-z^2 = -1. which can be generated by a rotating
hyperbola around one of its axes (the one that cuts the hyperbola) • A plane with slope less than 1 (1 is the slope of the asymptotes of the generating hyperbola) intersects H_2 either in an
ellipse or in a
point or not at all, • A plane with slope equal to 1 containing the origin (midpoint of the hyperboloid) does
not intersect H_2, • A plane with slope equal to 1 not containing the origin intersects H_2 in a
parabola, • A plane with slope greater than 1 intersects H_2 in a
hyperbola. Obviously, any two-sheet hyperboloid of revolution contains circles. This is also true, but less obvious, in the general case (see
circular section).
Remark: A hyperboloid of two sheets is
projectively equivalent to a sphere.
Other properties Symmetries The hyperboloids with equations \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 , \quad \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1 are •
pointsymmetric to the origin, •
symmetric to the coordinate planes and •
rotational symmetric to the z-axis and symmetric to any plane containing the z-axis, in case of a=b (hyperboloid of revolution).
Curvature Whereas the
Gaussian curvature of a hyperboloid of one sheet is negative, that of a two-sheet hyperboloid is positive. In spite of its positive curvature, the hyperboloid of two sheets with another suitably chosen metric can also be used as a
model for hyperbolic geometry. == In more than three dimensions ==