Tube of a curve - Revision history

Vipul: 4 revisions

2008-05-18T20:12:39Z

4 revisions

← Older revision	Revision as of 20:12, 18 May 2008
(No difference)

Vipul at 14:44, 30 August 2007

2007-08-30T14:44:30Z

← Older revision		Revision as of 14:44, 30 August 2007
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	==Symmetries of the tube and the curve==		==Symmetries of the tube and the curve==

	In general, any symmetry of the curve also gives a ~~symemtry~~ of the tube. There may be nontrivial symmetries of the tube which are trivial on the curve. For instance, for a planar curve, the reflection about its plane is a nontrivial symmetry of the tube that is trivial on the curve.		In general, any symmetry of the curve also gives a symmetry of the tube. There may be nontrivial symmetries of the tube which are trivial on the curve. For instance, for a planar curve, the reflection about its plane is a nontrivial symmetry of the tube that is trivial on the curve.

			==Curvatures of the tube==

			At any point on the tube, the two principal directions are:

			* The tangent to the foliation circle through that point. This circle lies in a cross-sectional plane, perpendicular to the tangent to the inner curve through the center.

			The principal curvature in this direction is <math>1/r</math>

			* The direction parallel to the tangent to the inner curve. This direction is perpendicular to the cross-sectional plane.

			The principal curvature in this direction is <math>\cos \theta</math> times the curvature of the inner curve at that point. Here <math>\theta</math> is the angle made by the radial to that point, with the normal vector at the center of the circle.

59.92.170.132: /* Examples */

2007-08-01T12:10:06Z

Examples

← Older revision		Revision as of 12:10, 1 August 2007
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	For the straight line, there is no distinguished normal direction, hence there is no distinguished inner rim, outer rim, top rim or bottom rim. All straight lines on the cylinder that are parallel to the axis, qualify as ''rims''.		For the straight line, there is no distinguished normal direction, hence there is no distinguished inner rim, outer rim, top rim or bottom rim. All straight lines on the cylinder that are parallel to the axis, qualify as ''rims''.

			Note that for any smooth open curve that is diffeomorphic to the real line, the tube about it is diffeomorphic to the right circular cylinder.

	===Tube of a circle===		===Tube of a circle===

	The tube of a circle is a [[torus]]. The circle is the ''base circle'' of the torus.		The tube of a circle is a [[torus]]. The circle is the ''base circle'' of the torus.

			Any smooth closed curve that is diffeomorphic to a circle, has a tube that is diffeomorphic to the torus.

	==Symmetries of the tube and the curve==		==Symmetries of the tube and the curve==

	In general, any symmetry of the curve also gives a symemtry of the tube. There may be nontrivial symmetries of the tube which are trivial on the curve. For instance, for a planar curve, the reflection about its plane is a nontrivial symmetry of the tube that is trivial on the curve.		In general, any symmetry of the curve also gives a symemtry of the tube. There may be nontrivial symmetries of the tube which are trivial on the curve. For instance, for a planar curve, the reflection about its plane is a nontrivial symmetry of the tube that is trivial on the curve.

Vipul at 03:31, 1 August 2007

2007-08-01T03:31:08Z

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 Note that when the curve is a [[planar curve]], viz it lies completely in a plane, the tube of that curve is symmetric about the plane.
 ==Examples==
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 The tube of a straight line is a [[right circular cylinder]]. The straight line is the axis of the cylinder.
 ===Tube of a circle===

Vipul at 09:40, 31 July 2007

2007-07-31T09:40:01Z

New page

{{curve-to-surface construct}}

==Definition==

Let <math>\gamma</math> be a smooth curve (viz, a curve having a tangent at every point) in <math>\R^3</math> and <math>r</math> a length. The ''tube'' of radius <math>r</math> about <math>\gamma</math> is the union of the following circles:

For each point on the curve, choose the circle of radius <math>r</math> about that point in the plane perpendicular to the tangent direction at that point.

<math>r</math> is termed the ''tube radius''.

For a closed curve that does not intersect itself, we can choose a tube radius such that no two of these circles described above intersect.

Note that when the curve is a [[planar curve]], viz it lies completely in a plane, the tube of that curve is symmetric about the plane.

==Examples==

===Tube of a straight line===

The tube of a straight line is a [[right circular cylinder]]. The straight line is the axis of the cylinder.

===Tube of a circle===

The tube of a circle is a [[torus]]. The circle is the ''base circle'' of the torus.

==Symmetries of the tube and the curve==

In general, any symmetry of the curve also gives a symemtry of the tube. There may be nontrivial symmetries of the tube which are trivial on the curve. For instance, for a planar curve, the reflection about its plane is a nontrivial symmetry of the tube that is trivial on the curve.