Paraboloid

from Wikipedia, the free encyclopedia
Elliptical paraboloid
Hyperbolic paraboloid

A paraboloid is a surface of the second order (quadric) and is in the simplest cases either by an equation

  • elliptical paraboloid , or
  • hyperbolic paraboloid ,

described.

One encounters elliptical paraboloids as surfaces of satellite dishes .
Hyperbolic paraboloids are saddle surfaces . They contain straight lines and are therefore used by architects as easily modelable roof shapes ( hyperbolic parabolic shells ).

The equations show that both surfaces contain many parabolas , which contributed to the naming:

can be imagined as having arisen by rotating the parabola in the xz-plane with the equation around the z-axis. is not a surface of revolution. But with two exceptions, every section with a plane through the z-axis is also a parabola. For example, the intersection with the plane (yz plane) is the parabola . Both surfaces (elliptical or hyperbolic) can be understood as sliding surfaces and can be generated by moving a parabola along a second parabola.

However, there are also significant differences:

  • has circles as height sections ( ) . (In the general case there are ellipses (see below), which is reflected in the name affix),
  • has hyperbolas or straight lines (for ) as height cuts , which justifies the addition hyperbolic .

One should not confuse a hyperbolic paraboloid with a hyperboloid .

Properties of P 1

Paraboloid of revolution with parabolas and circles of height

Tangential planes at P 1

The tangent plane in a surface point on the graph of a differentiable function has the equation

.

For the equation of the tangential plane at the point results

.

Plane sections from P 1

The elliptical paraboloid is a surface of revolution and is created by rotating the parabola around the axis. A flat section of is:

  • a parabola if the plane is perpendicular (parallel to the -axis).
  • an ellipse or a point or empty if the plane is not perpendicular . A horizontal plane cuts in a circle .
  • a point if the plane is a tangent plane .

Affine images of P 1

Any elliptical paraboloid is an affine image of . The simplest affine mappings are scaling of the coordinate axes. You provide the paraboloids with the equations

.

still has the property that it is intersected by a perpendicular plane in a parabola. However, a horizontal plane cuts in an ellipse here, if the following applies. That any elliptical paraboloid always contains circles is shown in the circular section plane.

is

  • symmetrical to the or coordinate planes.
  • symmetrical to the -axis, d. H. leaves invariant.
  • rotationally symmetric, if is.
Rotating water glass

Comment:

  1. A paraboloid of revolution (i.e. ) is of great technical importance as a parabolic mirror because all parabolas with the axis of rotation as an axis have the same focal point.
  2. If a glass filled with water is allowed to rotate around its axis of symmetry at a constant speed, the water rotates with the glass after a while. Its surface then forms a paraboloid of revolution.
  3. An elliptical paraboloid is often called a paraboloid for short .
  4. An elliptical paraboloid is projectively equivalent to the unit sphere (see projective quadric ).

Properties of P 2

hyperbolic paraboloid: parabolas, straight lines
hyperbolic paraboloid: straight line

Tangential planes at P 2

For is the equation of the tangential plane (see above) in the point

.

Plane sections from P 2

is (in contrast to ) not a surface of revolution. But like are in almost all vertical plane sections parables:

The intersection of a plane with is

  • a parabola if the plane is perpendicular (parallel to the -axis) and has an equation .
  • a straight line if the plane is perpendicular and has an equation .
  • an intersecting pair of lines if the plane is a tangential plane (see picture).
  • a hyperbola , if the plane is not perpendicular and not a tangent plane (see picture).

Other properties

  1. The section parabolas with planes parallel to the or plane are all congruent to the norm parabola .
  2. is a sliding surface . is created by moving the parabola with its vertex along the parabola .
  3. A non-perpendicular plane that contains a straight line always contains a second straight line and is a tangential plane.
  4. Since the surface contains straight lines, it is a ruled surface .
  5. is a conoid .
  6. A hyperbolic paraboloid contains straight lines (like cylinders and cones), but cannot be developed (like cylinders and cones), since the Gaussian curvature is not present at every point . The Gaussian curvature is everywhere . (With a sphere the Gaussian curvature is everywhere .) Thus a hyperbolic paraboloid is a saddle surface .
  7. By rotating the coordinate system around the -axis by 45 degrees, the equation changes into the simpler equation .
hyperbolic paraboloid with hyperbolas as vertical sections

Affine images of P 2

Any hyperbolic paraboloid is an affine image of . The simplest affine mappings are scaling of the coordinate axes. They provide the hyperbolic paraboloids with the equations

.

is

  • symmetrical to the or coordinate planes.
  • symmetrical to the -axis, d. H. leaves invariant.

Comment:

  1. Hyperbolic paraboloids are used by architects for the construction of roofs (see picture) because they can easily be modeled with straight lines (bars).
  2. A hyperbolic paraboloid is projectively equivalent to a single-shell hyperboloid .

Hyperbolic paraboloid as an interpolation surface of 4 points

hyperbolic paraboloid as an interpolation surface of 4 points

A hyperbolic paraboloid can also be understood as a bilinear interpolation surface of four points that are not in one plane :

.

The network of parameter lines consists of straight lines.

For the image shown in the example . The hyperbolic paraboloid thus described has the equation .

Interface between families of elliptical and hyperbolic paraboloids

ellipt. Paraboloid, parabolic. Cylinder (interface), hyperbolic. Paraboloid

Leaves in the equations

(Family of elliptical paraboloids)

and

(Family of hyperbolic paraboloids)

the parameter run against , one obtains the equation of the common interface

.

This is the equation of a cylinder with a parabola as a cross section (parabolic cylinder) , s. Image.

Stacked chips resemble a hyperbolic paraboloid in shape to increase stability.
Warszawa Ochota station, example of a hyperbolic paraboloid as a roof

See also

literature

  1. ^ G. Farin: Curves and Surfaces for Computer Aided Geometric Design , Academic Press, 1990, ISBN 0-12-249051-7 , p. 250

Web links

Commons : Paraboloid  - collection of pictures, videos and audio files