Right circular cylinder - Revision history

Vipul: 8 revisions

2008-05-18T20:07:50Z

8 revisions

← Older revision	Revision as of 20:07, 18 May 2008
(No difference)

Vipul: /* Curvatures */

2007-08-01T03:36:05Z

Curvatures

← Older revision		Revision as of 03:36, 1 August 2007
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	The two principal directions at any point are the direction along the real line and along the circle. Along the real line, the curvature is 0, while along the circle, the curvature is the usual curvature associated with a circle. Thus, the [[Gaussian curvature]] at every point is zero.		The two principal directions at any point are the direction along the real line and along the circle. Along the real line, the curvature is 0, while along the circle, the curvature is the usual curvature associated with a circle. Thus, the [[Gaussian curvature]] at every point is zero.

			==Constructions==

			{{surface of revolution of\|real line}}

			The right circular cylinder can be obtained as the surface of revolution of a real line parallel to the axis of revolution. The perpendicular distance between the two lines in the radius of the cylinder.

			{{tube of curve of\|real line}}

			The right circular cylinder can also be described as the tube of the curve which is the real line, and the radius of the tube is the same as the radius of the cylinder.

	==Properties==		==Properties==

Vipul: /* Structure and symmetry */

2007-07-31T09:14:08Z

Structure and symmetry

← Older revision		Revision as of 09:14, 31 July 2007
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	* Rotation by an angle of <math>\pi</math> about the <math>z</math>-axis		* Rotation by an angle of <math>\pi</math> about the <math>z</math>-axis
	* Composite of the above two		* Composite of the above two

			Note that each of these isometries can be uniquely extended to isometrise fo <math>\R^3</math>, and that group of isometries is precisely the group of isometries of <math>\R^3</math> that send the axis to itself as a line.

	==Curvatures==		==Curvatures==

Vipul: /* Theorems exemplified */

2007-07-31T07:32:50Z

Theorems exemplified

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	This is on account of the Gaussian curvature (the two-dimensional particular case of [[sectional curvature]]) being zero at every point.		This is on account of the Gaussian curvature (the two-dimensional particular case of [[sectional curvature]]) being zero at every point.

	==~~Theorems exemplified~~==		==Related theorems==

	{{applicationinstance\|~~splitting~~ theorem}}		{{applicationinstance\|Splitting theorem}}

	The splitting theorem states that any Riemannian manifold with everywhere nonnegative Ricci curvature can be expressed isometrically as a direct product of a line-free Riemannian manifold, and a Euclidean space.		The splitting theorem states that any Riemannian manifold with everywhere nonnegative Ricci curvature can be expressed isometrically as a direct product of a line-free Riemannian manifold, and a Euclidean space.

Vipul at 04:58, 31 July 2007

2007-07-31T04:58:31Z

← Older revision		Revision as of 04:58, 31 July 2007
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	In the case of the right circular cylinder, the Ricci curvature is everywhere nonnegative (in fact, everywhere zero) and the manifold can indeed be expressed as a direct product of the circle (which is line-free) and the real line (which is one-dimensional Euclidean space).		In the case of the right circular cylinder, the Ricci curvature is everywhere nonnegative (in fact, everywhere zero) and the manifold can indeed be expressed as a direct product of the circle (which is line-free) and the real line (which is one-dimensional Euclidean space).

			{{applicationinstance\|Minding's theorem}}

			Minding's theorem states that two smooth surfaces with the same constant Gaussian curvature everywhere, must be locally isometric. Indeed, the cylinder and the [[plane]] both have everwhere zero Gaussian curvature, and the cylinder ''is'' locally isometric to the plane. This follows from the fact that the circle is locally isometric to a real line.

Vipul at 04:55, 31 July 2007

2007-07-31T04:55:12Z

← Older revision		Revision as of 04:55, 31 July 2007
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	This is on account of the Gaussian curvature (the two-dimensional particular case of [[sectional curvature]]) being zero at every point.		This is on account of the Gaussian curvature (the two-dimensional particular case of [[sectional curvature]]) being zero at every point.

			==Theorems exemplified==

			{{applicationinstance\|splitting theorem}}

			The splitting theorem states that any Riemannian manifold with everywhere nonnegative Ricci curvature can be expressed isometrically as a direct product of a line-free Riemannian manifold, and a Euclidean space.

			In the case of the right circular cylinder, the Ricci curvature is everywhere nonnegative (in fact, everywhere zero) and the manifold can indeed be expressed as a direct product of the circle (which is line-free) and the real line (which is one-dimensional Euclidean space).

Vipul: /* Structure and symmetry */

2007-07-31T04:51:00Z

Structure and symmetry

← Older revision		Revision as of 04:51, 31 July 2007
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	==Structure and symmetry==		==Structure and symmetry==

			{{direct product\|circle\|real line}}
	The infinite right circular cylinder is topologically, differentially, and geometrically the direct product of the [[circle]] <math>S^1</math> and the [[real line]] <math>\R</math>. Moreover, this is an [[isometry-invariant factorization]], in the sense that any isometry must take cross-sectional circles to cross-sectional circles, and lines parallel to the axis, to lines parallel to the axis.		The infinite right circular cylinder is topologically, differentially, and geometrically the direct product of the [[circle]] <math>S^1</math> and the [[real line]] <math>\R</math>. Moreover, this is an [[isometry-invariant factorization]], in the sense that any isometry must take cross-sectional circles to cross-sectional circles, and lines parallel to the axis, to lines parallel to the axis.

Vipul at 04:50, 31 July 2007

2007-07-31T04:50:06Z

← Older revision		Revision as of 04:50, 31 July 2007
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	==Structure and symmetry==		==Structure and symmetry==

	The infinite right circular cylinder is topologically, differentially, and geometrically the direct product of the [[circle]] <math>S^1</math> and the [[real line]] <math>\R</math>. Moreover, ~~both direct factors~~ are ~~metrically characteristic~~.		The infinite right circular cylinder is topologically, differentially, and geometrically the direct product of the [[circle]] <math>S^1</math> and the [[real line]] <math>\R</math>. Moreover, this is an [[isometry-invariant factorization]], in the sense that any isometry must take cross-sectional circles to cross-sectional circles, and lines parallel to the axis, to lines parallel to the axis.

			Using this, we can compute that the group of isometries of the right circular cylinder is the semidirect product of the group itself (acting via the regular representation) and the Klein-four group, comprising the following four operations:

			* The identity element
			* Reflections about the <math>xy</math>-plane
			* Rotation by an angle of <math>\pi</math> about the <math>z</math>-axis
			* Composite of the above two

			==Curvatures==

			The two principal directions at any point are the direction along the real line and along the circle. Along the real line, the curvature is 0, while along the circle, the curvature is the usual curvature associated with a circle. Thus, the [[Gaussian curvature]] at every point is zero.

			==Properties==

			{{homogeneous metric}}

			Since there is an underlying group structure compatible with the metric, there is an isometry taking any point to any other point. Thus, the right circular cylider is homogeneous.

			{{flat metric}}

			This is on account of the Gaussian curvature (the two-dimensional particular case of [[sectional curvature]]) being zero at every point.

Vipul at 04:23, 31 July 2007

2007-07-31T04:23:34Z

New page

==Definition==

A '''right circular cylinder''', or more precisely, a '''right circular cylindrical surface''', is a union of all circles centered at points on a given line, of fixed radius, and each circle in the plane perpendicular to the line at that point. The line is termed the '''axis''' of the cylinder.

The above is the ''infinite'' right circular cylindrical surface. The finite version is obtained by replacing the line with an open line segment.

The term '''solid cylinder''' is used to refer to the unino of discs, instead of circles. The cylindrical surface is then termed the ''lateral surface'' of the solid cylinder.

===Terminology===

* The line comprising centers is termed the ''axis'' of the cylinder.
* Planes perpendicular to the axis are termed ''cross-sectional planes''
* The equal radius of all circles is termed the ''radius'' or ''base radius'' of the cylinder and is denoted by <math>r</math>
* For a finite cylinder, the length of the open line segment is termed the ''height'' of the cylinder and is denoted by <math>h</math>

==Equational descriptions==

===Cartesian parametric description===

If the axis is the <math>z</math>-axis, the parametric description is in terms of <math>t \in \R</math> and <math>\theta</math> an angle.

<math>x = r \cos \theta, y = r \sin \theta, z = t</math>

For a finite cylinder, we have, instead of <math>t \in \R</math>, <math>t \in [t_0,t_1]</math> where <math>t_1 - t_0 = h</math>.

===Cartesian equational description===

If the axis is the <math>z</math>-axis, the equational description in Cartesian coordinates is:

<math>x^2 + y^2 = r^2</math>

In other words, the <math>z</math>-coordinate is arbitrary, and any section with the <math>xy</math>-plane looks like a circle of radius <math>r</math> centered at the origin.

===Polar parametric description===

===Polar equational description===

==Structure and symmetry==

The infinite right circular cylinder is topologically, differentially, and geometrically the direct product of the [[circle]] <math>S^1</math> and the [[real line]] <math>\R</math>. Moreover, both direct factors are metrically characteristic.