# Surface of revolution

This article defines a property that makes sense for a surface embedded in $\R^3$, viz three-dimensional Euclidean space. The property is invariant under orthogonal transformations and scaling, i.e., under all similarity transformations.
View other such properties

## Definition

A surface of revolution is a surface in $\R^3$ obtained by revolving, about the $x$-axis, a curve in the $xy$-plane.

## Examples

Some examples are given below:

Curve being revolved Surface of revolution
semicircle with endpoints for a circle whose endpoints lie on the $x$-axis 2-sphere in Euclidean space
ray terminating at the $x$ axis infinite cone
pair of rays terminating at the same point of the $x$-axis, and which have slopes of equal magnitude but opposite sign infinite double cone
line parallel to the $x$-axis infinite cylinder
circle not intersecting the $x$-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 $x$-axis

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

### Transverse planes and meridians

These are planes containing the $x$-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.

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### 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 $x$-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 $S^1$ is a subgroup of the group of diffeomorphisms of the surface of revolution, where each element of $S^1$ acts via the corresponding rotation about the $x$-axis. The other symmetries, if they exist, depend on the nature of the plane curve being rotated.