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Surfaces of rotation

Let some curve located in the plane have an equation . If rotate this curve around the axis then its every point will describe a circumference.

The set of points of which the coordinates satisfy the equation is called a surface of rotation.

Example. The following surfaces are surfaces of rotation:

an ellipsoid of rotation (the axis of rotation is ).

an one-sheeted hyperboloid of rotation (the axis of rotation is ).

Example. Find an equation of the surface obtained at rotating the line around the axis .

Solution: The surface of rotation is the cone with the vertex in the point . Let an arbitrary point of the required surface have the coordinates . The point on the line is corresponded to . The points and lie on one plane that is perpendicular to the axis of rotation . Then . Substituting these expressions in the equation of the line, we obtain the equation of the required surface: or

, i.e. .

Let an orthonormal system of coordinates be given in the space. We say a surface is called an algebraic surface of the second order if its equation in a given system of coordinates has the form: ,

where the numbers are not equal to zero simultaneously, and and are the coordinates of radius-vector of a point lying on a given surface .

Theorem. For every surface of the second order there is an orthonormal system of coordinates in which an equation of this surface has one of the following 17 canonic forms:

Empty sets Points, lines and planes Cylinders and cones
        Isolated point Line Pair of intersecting planes Pair of parallel (or coinciding) planes Elliptic cylinder Hyperbolic cylinder Parabolic cylinder Cone

 

Non-degenerate surfaces
Ellipsoids Paraboloids Hyperboloids
Elliptic paraboloid Hyperbolic paraboloid One-sheeted hyperboloid Two-sheeted hyperboloid

 

where

 

Ellipsoid

A surface given in some orthonormal system of coordinates by canonic equation of the form

: is called an ellipsoid.

Properties of an ellipsoid:

1. An ellipsoid is a restricted surface since from its canonic equation follows that , , .

2. An ellipsoid has central symmetry regarding to the origin of coordinates; axial symmetry regarding the coordinate axes; planar symmetry regarding to the coordinate planes.

3. At cutting an ellipsoid by a plane that is orthogonal to any of the coordinate axes an ellipse is obtained. For example, considering the cutting plane , where , we obtain the following equation of line of cutting being an ellipse.

 


Date: 2015-12-18; view: 464


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