Surfaces of rotationLet 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 onesheeted 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 radiusvector 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

Nondegenerate surfaces
 Ellipsoids
 Paraboloids
 Hyperboloids

 Elliptic paraboloid
Hyperbolic paraboloid
 Onesheeted hyperboloid
Twosheeted 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: 20151218; view: 1055
