Paraboloid

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

  • Elliptic paraboloid, or
  • Hyperbolic paraboloid,

Described.

Obviously, both areas contain many parables as plane sections (see below). However, there are also significant differences:

  • P1 has ( z = const ) as vertical sections circles.
  • P2 has as vertical sections hyperbolas or straight line ( z = 0).
  • 3.1 tangent at P2
  • 3.2 level sections of P2

Properties of P1

Tangent at P1

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

For obtained for the equation of the tangent plane at the point

Plane sections of P1

The elliptic paraboloid P1 is a surface of revolution and rotation of the parabola formed by the z-axis. A plane section of P1 is:

  • A parabola if the plane perpendicular (parallel to z-axis).
  • An ellipse, or a dot or blank, if the plane is not vertical. A horizontal plane P1 intersects in a circle.
  • A point if the plane is a tangent plane.

Affine images of P1

Any elliptic paraboloid is an affine image of P1. The simplest affine transformations are scaling of the coordinate axes. They provide the paraboloids with equations

Still have the property that it is intersected by a vertical plane in a parabola. However, a horizontal plane intersects here in an ellipse, if.

Is

  • Symmetrically with respect to the xz or yz coordinate planes.
  • Symmetrical to the z- axis, i.e., leaves invariant.
  • Rotationally symmetric case is.

Comment:

Properties of P2

Tangent at P2

For the equation of the tangent plane (see above) is at the point

Plane sections of P2

P2 is (in contrast to P1) no rotation surface. But as with P1 at P2 are almost all vertical plane sections parables:

The intersection of a plane with P2 is

  • A parabola if the plane perpendicular (parallel to the z axis ) and has an equation.
  • A straight line, if the plane is perpendicular and an equation has.
  • A pair of lines to be cut if the plane is a tangent plane (see picture).
  • A hyperbola, if the plane is not perpendicular and no tangent plane is (see picture).

Comment:

Affine images of P2

Any hyperbolic paraboloid is an affine image of P2. The simplest affine transformations are scaling of the coordinate axes. They provide the hyperbolic paraboloids with equations

Is

  • Symmetrically with respect to the xz or yz coordinate planes.
  • Symmetrical to the z- axis, i.e., leaves invariant.

Comment:

Interface between crowds of ellipt. and hyperbolic. paraboloids

If we let in the equations

The parameters to run against, we obtain the equation of the common interface

  • .

This is the equation of a cylinder with a parabolic cross-section as ( parabolic cylinder), see Fig.

Formulas for a paraboloid of revolution

The formulas are valid for a paraboloid of revolution, which is from a perpendicular to the z- axis plane ( xy plane) cut in height. The cutting circle has the radius.

Ellipsoid
304656
de