Paraboloid

Elliptical paraboloid

Hyperbolic paraboloid

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

• ${\displaystyle P1\colon \ z=x^{2}+y^{2},}$ elliptical paraboloid , or
• ${\displaystyle P2\colon \ z=x^{2}-y^{2},}$ 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:

${\displaystyle P1}$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. ${\displaystyle z=x^{2}}$
${\displaystyle P2}$ ${\displaystyle P2}$ ${\displaystyle x=0}$ ${\displaystyle z=-y^{2}}$

However, there are also significant differences:

• ${\displaystyle P1}$has circles as height sections ( ) . (In the general case there are ellipses (see below), which is reflected in the name affix), ${\displaystyle z={\text{const}}}$

• ${\displaystyle P2}$has hyperbolas or straight lines (for ) as height cuts , which justifies the addition hyperbolic . ${\displaystyle z=0}$

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 ${\displaystyle (x_{0},y_{0},f(x_{0},y_{0}))}$ ${\displaystyle f}$

${\displaystyle z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})}$.

For the equation of the tangential plane at the point results ${\displaystyle f(x,y)=x^{2}+y^{2}}$ ${\displaystyle (x_{0},y_{0},x_{0}^{2}+y_{0}^{2})}$

${\displaystyle z=2x_{0}x+2y_{0}y-(x_{0}^{2}+y_{0}^{2})}$.

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: ${\displaystyle P1}$ ${\displaystyle z=x^{2}}$ ${\displaystyle z}$ ${\displaystyle P1}$

• a parabola if the plane is perpendicular (parallel to the -axis). ${\displaystyle z}$
• an ellipse or a point or empty if the plane is not perpendicular . A horizontal plane cuts in a circle . ${\displaystyle P1}$
• a point if the plane is a tangent plane .

Affine images of P 1

Rotating parabolic antenna for satellite communication

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 ${\displaystyle P1}$

${\displaystyle P1_{ab}\colon z={\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}},\ a,b>0}$.

${\displaystyle P1_{ab}}$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. ${\displaystyle a\neq b}$

${\displaystyle P1_{ab}}$ is

• symmetrical to the or coordinate planes. ${\displaystyle xz}$ ${\displaystyle yz}$
• symmetrical to the -axis, d. H. leaves invariant. ${\displaystyle z}$ ${\displaystyle (x,y,z)\rightarrow (-x,-y,z)}$ ${\displaystyle P1_{ab}}$
• rotationally symmetric, if is. ${\displaystyle a=b}$

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. ${\displaystyle a=b}$
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 ${\displaystyle f(x,y)=x^{2}-y^{2}}$ ${\displaystyle (x_{0},y_{0},x_{0}^{2}-y_{0}^{2})}$

${\displaystyle z=2x_{0}x-2y_{0}y-x_{0}^{2}+y_{0}^{2}}$.

Plane sections from P 2

${\displaystyle P2}$is (in contrast to ) not a surface of revolution. But like are in almost all vertical plane sections parables: ${\displaystyle P1}$ ${\displaystyle P1}$ ${\displaystyle P2}$

The intersection of a plane with is ${\displaystyle P2}$

• a parabola if the plane is perpendicular (parallel to the -axis) and has an equation . ${\displaystyle z}$ ${\displaystyle ax+by+c=0,a\neq \pm b}$
• a straight line if the plane is perpendicular and has an equation . ${\displaystyle y=\pm x+c}$
• 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 . ${\displaystyle xz}$ ${\displaystyle yz}$ ${\displaystyle z=x^{2}}$
2. ${\displaystyle P2}$is a sliding surface . is created by moving the parabola with its vertex along the parabola . ${\displaystyle P2}$ ${\displaystyle z=x^{2},y=0}$ ${\displaystyle z=-y^{2},x=0}$
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 . ${\displaystyle P2}$
5. ${\displaystyle P2}$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 . ${\displaystyle 0}$ ${\displaystyle <0}$ ${\displaystyle >0}$
7. By rotating the coordinate system around the -axis by 45 degrees, the equation changes into the simpler equation . ${\displaystyle z}$ ${\displaystyle z=x^{2}-y^{2}}$ ${\displaystyle z=2xy}$

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 ${\displaystyle P2}$

${\displaystyle P2_{ab}:\ z={\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}},\ a,b>0}$.

${\displaystyle P2_{ab}}$ is

• symmetrical to the or coordinate planes. ${\displaystyle xz}$ ${\displaystyle yz}$
• symmetrical to the -axis, d. H. leaves invariant. ${\displaystyle z}$ ${\displaystyle (x,y,z)\rightarrow (-x,-y,z)}$ ${\displaystyle P2_{ab}}$

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 : ${\displaystyle \ \mathbf {a} _{1},\;\mathbf {a} _{2},\;\mathbf {b} _{1},\;\mathbf {b} _{2}\ }$

• ${\displaystyle {\mathbf {x} }(u,v)={\begin{pmatrix}1-u&u\end{pmatrix}}{\begin{pmatrix}{\bf {a_{1}}}&{\bf {b_{1}}}\\{\bf {a_{2}}}&{\bf {b_{2}}}\end{pmatrix}}{\begin{pmatrix}1-v\\v\end{pmatrix}}\ }$

${\displaystyle =(1-v){\big (}(1-u)\mathbf {a} _{1}+u\mathbf {a} _{2}{\big )}\ +\ v{\big (}(1-u)\mathbf {b} _{1}+u\mathbf {b} _{2}{\big )}\ }$.

The network of parameter lines consists of straight lines.

For the image shown in the example . The hyperbolic paraboloid thus described has the equation . ${\displaystyle \ \mathbf {a} _{1}=(0,0,0)^{T},\;\mathbf {a} _{2}=(1,0,0)^{T},\;\mathbf {b} _{1}=(0,1,0)^{T},\;\mathbf {b} _{2}=(1,1,1)^{T}\ }$ ${\displaystyle z=xy}$

Interface between families of elliptical and hyperbolic paraboloids

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

Leaves in the equations

${\displaystyle z=x^{2}+{\frac {y^{2}}{b^{2}}}}$ (Family of elliptical paraboloids)

and

${\displaystyle z=x^{2}-{\frac {y^{2}}{b^{2}}}}$ (Family of hyperbolic paraboloids)

the parameter run against , one obtains the equation of the common interface ${\displaystyle b}$ ${\displaystyle \infty }$

${\displaystyle z=x^{2}}$.

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

• Hyperbolic paraboloid shell
• Paraboloid of revolution
• Ellipsoid
• Hyperboloid
• Rotational hyperboloid
• cone
• Conoid
• cylinder

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

• Animated paraboloid