Rotational hyperboloid

Mae West by tram

The single-shell hyperboloid of revolution is a second order surface , which can be by rotating a straight line about a her crooked straight (axis) can imagine emerged. It is a special case of the single-shell hyperboloid . Its Gaussian curvature is negative at every point; it is therefore an anticlastically curved surface.

Note: There is also a two-shell rotational hyperboloid (see hyperboloid ).

application

The form of the hyperboloid of revolution is used, among other things, in the construction industry for hyperboloid structures. Vladimir Shukhov built the first tower in the world in this form for the All-Russian Industrial and Crafts Exhibition in 1896 .

The architect Antoni Gaudí used the shape as a design principle. The work of art Mae West in Munich is also a 52 meter high rotation hyperboloid made of CFRP .

equation

The equation for the single-shell rotational hyperboloid with a circular cross-section results from the equation

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

of a single-shell hyperboloid with a generally elliptical cross-section by setting : ${\ displaystyle \ a = b = r \}$

{\ displaystyle {\ frac {x ^ {2} + y ^ {2}} {r ^ {2}}} - {\ frac {z ^ {2}} {c ^ {2}}} = 1 \. }

A section with a horizontal plane is always a circle. The smallest circle results for . He has the radius . This hyperboloid can be generated by rotating the hyperbola in the xz-plane with the equation around the z-axis. ${\ displaystyle z = z_ {0}}$ ${\ displaystyle z = 0}$ ${\ displaystyle r}$ ${\ displaystyle {\ tfrac {x ^ {2}} {r ^ {2}}} - {\ tfrac {z ^ {2}} {c ^ {2}}} = 1 \}$

Generation of a single-shell rotational hyperboloid by rotating a straight line (red)

A generation that is more suitable for the application allows a straight line (rod) that is skewed to the z-axis to rotate around the z-axis:

The straight line with the parametric equation

{\ displaystyle {\ vec {x}} (u) = {\ begin {pmatrix} r \\ 0 \\ 0 \ end {pmatrix}} + u {\ begin {pmatrix} 0 \\\ cos (\ gamma) \\\ sin (\ gamma) \ end {pmatrix}}}

is parallel to the yz-plane, has the distance to the z-axis and the angle of inclination compared to the xy-plane (see picture). ${\ displaystyle r}$ ${\ displaystyle \ gamma}$

If you let this straight line rotate around the z-axis , you get a surface with the parametric equation

${\ displaystyle {\ vec {x}} (u, v) = {\ begin {pmatrix} r \ cos (v) \\ r \ sin (v) \\ 0 \ end {pmatrix}} + u {\ begin {pmatrix} - \ cos (\ gamma) \ sin (v) \\\ cos (\ gamma) \ cos (v) \\\ sin (\ gamma) \ end {pmatrix}}}$ .

One calculates that in the case of the coordinates of the surface points the above equation of a rotational hyperboloid also fulfills. You can also see: the straight line with the angle of inclination creates the same hyperboloid (see picture). So two straight lines (bars) go through every point of the hyperboloid, which increases the stability of a model considerably. (In the case , the straight line lies in the xy plane and sweeps the outside of the circle with the equation . If is, a cylinder with a radius is created .) ${\ displaystyle \; 0 <\ gamma <\ pi / 2 \;}$ ${\ displaystyle \; c = r \ tan \ gamma \;}$ ${\ displaystyle - \ gamma}$
${\ displaystyle \ gamma = 0}$ ${\ displaystyle x ^ {2} + y ^ {2} = r ^ {2}}$ ${\ displaystyle \ gamma = \ pi / 2}$ ${\ displaystyle r}$

literature

Rotational hyperboloid. In: Klaus-Jürgen Schneider, Rüdiger Wormuth (Ed.): Building dictionary. Explanation of important construction terms. 2nd, expanded edition. Bauwerk ua, Berlin 2009, ISBN 978-3-89932-159-3 .