surface of revolution
If a curve in rotates about a line, it generates a surface of revolution. The line is called the axis of revolution. Every point of the curve generates a circle of latitude. If the surface is intersected by a half-plane beginning from the axis of revolution, the intersection curve is a meridian curve. One can always think that the surface of revolution is generated by the rotation of a certain meridian, which may be called the 0-meridian.
Let be a curve of the -plane rotating about the -axis. Then any point of this 0-meridian draws a circle of latitude, parallel to the -plane, with centre on the -axis and with the radius . So the - and -coordinates of each point on this circle satisfy the equation
This equation is thus satisfied by all points of the surface of revolution and therefore it is the equation of the whole surface of revolution.
More generally, if the equation of the meridian curve in the -plane is given in the implicit form , then the equation of the surface of revolution may be written
The catenoid is the only surface of revolution being also a minimal surface.
The quadratic surfaces of revolution:
- 1 Lauri Pimiä: Analyyttinen geometria. Werner Söderström Osakeyhtiö, Porvoo and Helsinki (1958).
|Title||surface of revolution|
|Date of creation||2013-03-22 17:17:08|
|Last modified on||2013-03-22 17:17:08|
|Last modified by||pahio (2872)|
|Defines||surface of revolution|
|Defines||axis of revolution|
|Defines||circle of latitude|
|Defines||cone of revolution|