https://diffgeom.subwiki.org/w/index.php?action=history&feed=atom&title=Helix_of_a_curveHelix of a curve - Revision history2026-10-03T22:06:56ZRevision history for this page on the wikiMediaWiki 1.41.2https://diffgeom.subwiki.org/w/index.php?title=Helix_of_a_curve&diff=678&oldid=prevVipul: 1 revision2008-05-18T19:46:52Z<p>1 revision</p>
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</td></tr></table>Vipulhttps://diffgeom.subwiki.org/w/index.php?title=Helix_of_a_curve&diff=677&oldid=prevVipul at 12:40, 1 August 20072007-08-01T12:40:02Z<p></p>
<p><b>New page</b></p><div>==Definition==<br />
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Let <math>\gamma</math> be a parametrized curve on a manifold <math>M</math>. The ''helix'' or ''spiral'' of <math>\gamma</math> is a parametrized curve in <math>\R \times M</math> that sends <math>t</math> in the parameter domain to <math>(t,f(t))</math>. In other words, the helix of a curve is the Cartesian product of the curve with the identity-parametrized curve.<br />
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Note that the above definition requires a parametrized curve. However, for a [[Riemannian manifold]], any smooth curve in the manifold acquires a natural parametrization by arc-length, and we can thus give a natural choice of helix, or spiral, associated with the curve.<br />
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For a [[closed curve]], viz a map <math>S^1 \to M</math>, we construct the helix using the map <math>\R \to M</math> that composes the ''modulo'' map with the curve parametrization.</div>Vipul