# Difference between revisions of "Surface of revolution"

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==Definition== | ==Definition== | ||

− | A '''surface of revolution''' is a surface in <math | + | A '''surface of revolution''' is a surface in <math>\R^3</math> obtained by revolving, about the <math>x</math>-axis, a curve in the <math>xy</math>-plane. |

==Examples== | ==Examples== | ||

− | + | Some examples are given below: | |

− | + | {| class="sortable" border="1" | |

− | + | ! Curve being revolved !! Surface of revolution | |

− | + | |- | |

− | + | | semicircle with endpoints for a circle whose endpoints lie on the <math>x</math>-axis || [[2-sphere in Euclidean space]] | |

+ | |- | ||

+ | | ray terminating at the <math>x</math> axis || infinite cone | ||

+ | |- | ||

+ | | pair of rays terminating at the same point of the <math>x</math>-axis, and which have slopes of equal magnitude but opposite sign || infinite double cone | ||

+ | |- | ||

+ | | line parallel to the <math>x</math>-axis || infinite cylinder | ||

+ | |- | ||

+ | | circle not intersecting the <math>x</math>-axis || [[2-torus in Euclidean space]] | ||

+ | |} | ||

==Terminology associated with surfaces of revolution== | ==Terminology associated with surfaces of revolution== | ||

− | ===Sectional planes=== | + | ===Profile curve=== |

+ | |||

+ | The curve in the plane that is subjected to revolution is termed the '''profile curve'''. | ||

+ | |||

+ | ===Sectional planes and parallels=== | ||

These are planes perpendicular to the <math>x</math>-axis | These are planes perpendicular to the <math>x</math>-axis | ||

− | The intersection of the surface of revolution with any sectional plane is a union of concentric circles, centered at the <math>x</math>-axis | + | The intersection of the surface of revolution with any sectional plane is a circle, or a union of concentric circles, centered at the <math>x</math>-axis. These circles are termed '''parallels'''. |

− | ===Transverse planes=== | + | |

+ | ===Transverse planes and meridians=== | ||

These are planes containing the <math>x</math>-axis. | These are planes containing the <math>x</math>-axis. | ||

− | The intersection of the surface of revolution with any transverse plane | + | The intersection of the surface of revolution with any transverse plane gives a copy of the profile curve (the original curve which we revolved). Each such copy is termed a '''meridian'''. |

==Geometric constructions== | ==Geometric constructions== | ||

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{{fillin}} | {{fillin}} | ||

+ | |||

+ | ===Gaussian curvature=== | ||

+ | |||

+ | Since the curvature along the sectional direction is always positive, the sign of the Gaussian curvature at any point is the same as the sign of the curvature to the planar curve. Thus, any curve that opens ''upwards'' or away from the <math>x</math>-axis, gives rise to a surface of revolution with negative Gaussian curvature everywhere. | ||

+ | |||

+ | ==Automorphisms and symmetries== | ||

+ | |||

+ | ===Rotational symmetry=== | ||

+ | |||

+ | The surface of revolution enjoys many symmetries. In particular, the [[circle group]] <math>S^1</math> is a subgroup of the group of diffeomorphisms of the surface of revolution, where each element of <math>S^1</math> acts via the corresponding rotation about the <math>x</math>-axis. The other symmetries, if they exist, depend on the nature of the plane curve being rotated. | ||

==External links== | ==External links== |

## Latest revision as of 23:30, 29 July 2011

This article defines a property that makes sense for a surface embedded in , viz three-dimensional Euclidean space. The property is invariant under orthogonal transformations and scaling, i.e., under all similarity transformations.

View other such properties

## Contents

## Definition

A **surface of revolution** is a surface in obtained by revolving, about the -axis, a curve in the -plane.

## Examples

Some examples are given below:

Curve being revolved | Surface of revolution |
---|---|

semicircle with endpoints for a circle whose endpoints lie on the -axis | 2-sphere in Euclidean space |

ray terminating at the axis | infinite cone |

pair of rays terminating at the same point of the -axis, and which have slopes of equal magnitude but opposite sign | infinite double cone |

line parallel to the -axis | infinite cylinder |

circle not intersecting the -axis | 2-torus in Euclidean space |

## Terminology associated with surfaces of revolution

### Profile curve

The curve in the plane that is subjected to revolution is termed the **profile curve**.

### Sectional planes and parallels

These are planes perpendicular to the -axis

The intersection of the surface of revolution with any sectional plane is a circle, or a union of concentric circles, centered at the -axis. These circles are termed **parallels**.

### Transverse planes and meridians

These are planes containing the -axis.

The intersection of the surface of revolution with any transverse plane gives a copy of the profile curve (the original curve which we revolved). Each such copy is termed a **meridian**.

## Geometric constructions

### Tangent plane and principal directions

The tangent plane at each point has two directions: one, the tangent to the planar curve when taken in the transverse plane, and the other, the tangent to the circle when taken in the sectional plane. The two directions are mutually perpendicular. Furtherm these two directions are the principal directions. The principal curvature in the transversal plane equals the curvature of the planar curve, while the principal curvature in the sectional plane equals the curvature of the circle.

Both of these numbers can easily be described in terms of the equation of the planar curve.

*Fill this in later*

### Gaussian curvature

Since the curvature along the sectional direction is always positive, the sign of the Gaussian curvature at any point is the same as the sign of the curvature to the planar curve. Thus, any curve that opens *upwards* or away from the -axis, gives rise to a surface of revolution with negative Gaussian curvature everywhere.

## Automorphisms and symmetries

### Rotational symmetry

The surface of revolution enjoys many symmetries. In particular, the circle group is a subgroup of the group of diffeomorphisms of the surface of revolution, where each element of acts via the corresponding rotation about the -axis. The other symmetries, if they exist, depend on the nature of the plane curve being rotated.