|
|
| Line 1: |
Line 1: |
| [[File:Helixtgtdev.png|thumb|right|300px|The tangent developable of a helix]]
| | Claude is her name and she totally digs that name. The thing I adore most bottle tops gathering and now I have time to consider on new things. Her family lives in Idaho. His day occupation is a monetary officer but he ideas on changing it.<br><br>my page [http://Letuschaton.com/blogs/post/27011 extended car warranty] |
| The '''tangent developable''' of a [[space curve]] <math>\gamma(t)</math> is a [[developable surface]] formed by the union of the [[tangent line]]s to the curve.
| |
| | |
| ==Parameterization==
| |
| Let <math>\gamma(t)</math> be a parameterization of a smooth space curve. That is, <math>\gamma</math> is a [[differentiable function|twice-differentiable function]] with nowhere-vanishing derivative that maps its argument <math>t</math> (a [[real number]]) to a point in space; the curve is the image of <math>\gamma</math>. Then a two-dimensional surface, the tangent developable of <math>\gamma</math>, may be parameterized by the map
| |
| :<math>(s,t)\mapsto \gamma(t) + s\gamma'(t).</math><ref>{{citation
| |
| | last = Pressley | first = Andrew
| |
| | isbn = 1-84882-890-X
| |
| | page = 129
| |
| | publisher = Springer
| |
| | title = Elementary Differential Geometry
| |
| | year = 2010}}.</ref>
| |
| The original curve forms a boundary of the tangent developable, and is called its directrix. | |
| | |
| ==Properties==
| |
| The tangent developable is a [[developable surface]]; that is, it is a surface with zero [[Gaussian curvature]]. It is one of three fundamental types of developable surface; the other two are the generalized cones (the surface traced out by a one-dimensional family of lines through a fixed point), and the cylinders (surfaces traced out by a one-dimensional family of [[parallel lines]]). (The [[plane (geometry)|plane]] is sometimes given as a fourth type, or may be seen as a special case of either of these two types.) Every developable surface in three-dimensional space may be formed by gluing together pieces of these three types; it follows from this that every developable surface is a [[ruled surface]], a union of a one-dimensional family of lines.<ref name="lawrence"/> However, not every ruled surface is developable; the [[helicoid]] provides a counterexample.
| |
| | |
| ==History==
| |
| Tangent developables were first studied by [[Leonhard Euler]] in 1772.<ref>{{citation
| |
| | last = Euler | first = L. | author-link = Leonhard Euler
| |
| | journal = Novi Commentarii academiae scientiarum Petropolitanae
| |
| | language = Latin
| |
| | pages = 3–34
| |
| | title = De solidis quorum superficiem in planum explicare licet
| |
| | url = http://www.math.dartmouth.edu/~euler/pages/E419.html
| |
| | volume = 16
| |
| | year = 1772}}.</ref> Until that time, the only known developable surfaces were the generalized cones and the cylinders. Euler showed that tangent developables are developable and that every developable surface is of one of these types.<ref name="lawrence">{{citation
| |
| | last = Lawrence | first = Snežana
| |
| | doi = 10.1007/s00004-011-0087-z
| |
| | issue = 3
| |
| | journal = Nexus Network Journal
| |
| | pages = 701–714
| |
| | title = Developable surfaces: their history and application
| |
| | volume = 13
| |
| | year = 2011}}.</ref>
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| ==External links==
| |
| *{{MathWorld|title=Tangent Developable|urlname=TangentDevelopable}}
| |
| | |
| {{DEFAULTSORT:Tangent Developable}}
| |
| [[Category:Differential geometry of surfaces]]
| |
| [[Category:Surfaces]]
| |
Claude is her name and she totally digs that name. The thing I adore most bottle tops gathering and now I have time to consider on new things. Her family lives in Idaho. His day occupation is a monetary officer but he ideas on changing it.
my page extended car warranty