# 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

### 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

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 ${\ displaystyle P1}$ ${\ displaystyle P1_ {ab} \ colon z = {\ frac {x ^ {2}} {a ^ {2}}} + {\ frac {y ^ {2}} {b ^ {2}}}, \ a, b> 0}$ .

${\ displaystyle P1_ {from}}$ 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. ${\ displaystyle a \ neq b}$ ${\ displaystyle P1_ {from}}$ is

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

### 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 .${\ displaystyle z}$ ${\ displaystyle z = x ^ {2} -y ^ {2}}$ ${\ displaystyle z = 2xy}$ ## 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 ${\ displaystyle P2}$ ${\ displaystyle P2_ {ab}: \ z = {\ frac {x ^ {2}} {a ^ {2}}} - {\ frac {y ^ {2}} {b ^ {2}}}, \ a, b> 0}$ .

${\ displaystyle P2_ {from}}$ is

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

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

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.