Surface of revolution - Revision history

Vipul: /* Examples */

2011-07-29T23:30:46Z

Examples

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	==Examples==		==Examples==

	~~For a full list of surfaces of revolution, see [[:Category:Surfaces of revolution]]. Particular~~ examples are:		Some examples are given below:

	* [[~~Sphere~~]] ~~which is obtained from an upper semicircle~~		{\| class="sortable" border="1"
	* [[Cone]] which is obtained from a ray terminating at the <math>x</math>-~~axis~~		! Curve being revolved !! Surface of revolution
	* [[Double cone]] which is obtained from a pair of rays terminating at the same point of the <math>x</math>-axis, and which have slopes of equal magnitude but opposite sign		\|-
	* [[Cylinder]] which is obtained from a line parallel to the <math>x</math>-axis		\| 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==

Vipul: 5 revisions

2008-05-18T20:10:23Z

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Vipul: /* Terminology associated with surfaces of revolution */

2007-09-07T10:15:11Z

Terminology associated with surfaces of revolution

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	==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 is the original curve ~~whose revolution gave rise to the surface of revolution~~.		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==

Vipul: /* Geometric constructions */

2007-06-18T12:23:58Z

Geometric constructions

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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 <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==

Vipul: /* Definition */

2007-06-18T12:10:30Z

Definition

← Older revision		Revision as of 12:10, 18 June 2007
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	==Definition==		==Definition==

	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.		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==

Vipul: /* Geometric constructions */

2007-06-18T11:59:43Z

Geometric constructions

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			==External links==

			===Definition links===

			* {{mathworld\|SurfaceofRevolution}}
			* {{wp\|Surface_of_revolution}}

Vipul at 11:57, 18 June 2007

2007-06-18T11:57:51Z

New page

{{surface-in-3D property}}

==Definition==

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==

For a full list of surfaces of revolution, see [[:Category:Surfaces of revolution]]. Particular examples are:

* [[Sphere]] which is obtained from an upper semicircle
* [[Cone]] which is obtained from a ray terminating at the <math>x</math>-axis
* [[Double cone]] which is obtained from a pair of rays terminating at the same point of the <math>x</math>-axis, and which have slopes of equal magnitude but opposite sign
* [[Cylinder]] which is obtained from a line parallel to the <math>x</math>-axis

==Terminology associated with surfaces of revolution==

===Sectional planes===

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
===Transverse planes===

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

The intersection of the surface of revolution with any transverse plane is the original curve whose revolution gave rise to the surface of revolution.

==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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