Hyperboloid

A hyperboloid in the simplest case a surface which is created by rotation of a hyperbola about one of its axes.

• When rotating a hyperbola about its minor axis produces a single-shell hyperboloid. It consists of a continuous Fächenstück.
• When rotating a hyperbola around its main axis produces a two-shell hyperboloid. It consists of two separate Fächenstücken.

Both surfaces can be described by a quadratic equation (analogous to the equations of ellipses and hyperbola ). They are therefore special cases of quadrics ( sphere, cone, paraboloid, ...) and are typically cut by planes in conics. A major difference between a one - or - two-sheeted hyperboloid is:

• The hyperboloid of one sheet containing straight lines, the bivalve not.

This property makes the hyperboloid for architects and civil engineers interesting, as can be single-shell hyperboloid model of straight lines easily: eg Cooling towers. Single- hyperboloids also play a role in synthetic geometry: A Minkowski plane, the geometry of plane sections of a single- hyperboloid. While the hyperboloid of tangent planes is cut into two intersecting straight lines ( see below), has therefore used a two-shell hyperboloid with tangential always only one point in common and geometrically more with a ball.

• 2.1 The bivalve Einheitshyperboloid H2
• 2.2 tangent planes of H2
• 2.3 level sections of H2
• 2.4 Affine images of H2

Single-leaf hyperboloid

Single skin Einheitshyperboloid H1

Leaving aside the hyperbole in the xz -plane about its minor axis (ie, z -axis) rotate (see picture), we obtain the single- unit hyperboloid with the equation

• .

(If the rotation is replaced by. ) Obviously any height section with a plane is a circle with radius. The plane yields the two intersection lines. By rotating this straight line is obtained parametric equations of all lines on the hyperboloid:

The hyperboloid So just by the rotation of the line, or ( skew to the axis of rotation ) produce (see image ).

Tangent planes of H1

The equation of the tangent plane of an implicitly given surface in a point.

For H1 results

Plane sections of H1

• Levels with a slope less than 1 (1 is the slope of the line on the hyperboloid ) intersect H1 in an ellipse,
• Levels with a slope equal to 1 cut through the zero point H1 in a parallel pair of lines,
• Levels with a slope equal to 1 do not cut through the zero point H1 in a parabola,
• Tangent planes intersect H1 in an intersecting pair of lines,
• Planes with an inclination greater than 1 that are not tangential planes intersect H1 in a hyperbola.

Note: A plane containing a hyperboloid straight, is either a tangent plane and thus contains a second intersecting hyperboloid Straight or contains a parallel to hyperboloid straight and is thus " tangent plane at a remote point."

Affine images of H1

Analogously as any ellipse can be thought of as affine image of the unit circle, any single-shell hyperboloid is the affine image of Einheitshyperboloids. The simplest affine images is obtained by scaling the coordinate axes:

Only in the case of the vertical sections circuits are still (otherwise ellipses). Such a hyperboloid is called single-shell hyperboloid of revolution.

Since any single-shell hyperboloid (like the Einheitshyperboloid ) line contains, it is a ruled surface. As each tangent plane ( a single- hyperboloid ) near its contact point intersects the surface, it has a negative Gaussian curvature, and is therefore non- developable, in contrast to the control surfaces of a cone or cylinder ( Gaussian curvature 0). From the usual Parametrdarstellung a hyperbola with hyperbolic functions we obtain the following Parametrdarstellung of the hyperboloid

Note: The hyperbolic paraboloid hyperboloid and are projectively equivalent.

Two-shell hyperboloid

The bivalve Einheitshyperboloid H2

Leaving aside the hyperbole in the xz -plane around its main axis (ie, z -axis) rotate (see picture), we obtain the two-shell unit hyperboloid with the equation or in the usual form

• .

The plane is a circle with (in case) or a period ( if applicable) or empty (in case). consists of two parts, corresponding to the two parts of the hyperbola.

Tangent planes of H2

The tangent plane of H2 in a point, the equation (see above)

Plane sections of H2

• Levels with a slope less than 1 ( slope of the asymptotes of the hyperbola -producing ) intersect H2 in either an ellipse or a point or not,
• Levels with a slope equal to 1 and passes through the origin do not intersect H2,
• Levels with a slope equal to 1 and not through the origin intersect H2 in a parabola,
• Levels with a slope greater than 1 cut H2 into a hyperbola.

Affine images of H2

An arbitrary two-shell hyperboloid is the affine image of Einheitshyperboloids. The simplest affine images is obtained by scaling the coordinate axes:

Only in the case of the non-trivial amount cuts circles are still (otherwise ellipses). Such a hyperboloid is called the two-shell hyperboloid of revolution.

For a two-shell hyperboloid we have the following parametric representation:

Note: The two -sheeted hyperboloid is projective to the unit sphere equivalent.

Symmetry properties of hyperboloids

How ellipses and hyperbola have also hyperboloids apex and side parting and symmetries. The hyperboloids are obviously

• Point symmetric to the origin of coordinates,
• Symmetrical with respect to the coordinate planes and
• Rotationally symmetrical to the z-axis and symmetrically with respect to each plane by the z- axis, is appropriate.

Double Cone

The double cone may be considered as the interface between the multitudes of one-or two -shell hyperboloid respectively. It is formed by rotation of the common asymptotes of the hyperbolas producer.

Common parametric representation

There are several ways to parameterize hyperboloids. A simple way to parameterize the single- and two -sheeted hyperboloid and the cone is:

For results in a single skin, for a two-shell hyperboloid and for a double cone.