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Difference between revisions of "Conoid"

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A [[Catalan surface|Catalan surface]] all straight line generators of which intersect a fixed straight line, called an axis of the conoid. For example, a hyperbolic paraboloid is a conoid with two axes.
 
A [[Catalan surface|Catalan surface]] all straight line generators of which intersect a fixed straight line, called an axis of the conoid. For example, a hyperbolic paraboloid is a conoid with two axes.
  
 
The position vector of a conoid is given by
 
The position vector of a conoid is given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025210/c0252101.png" /></td> </tr></table>
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$$r=\{u\cos v+\alpha f(v),u\sin v+\beta f(v),\gamma f(v)\},$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025210/c0252102.png" /> is a unit vector having the same direction as an axis of the conoid and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025210/c0252103.png" /> is some function. For a right conoid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025210/c0252104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025210/c0252105.png" />, and then its axis is a line of striction. A right conoid with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025210/c0252106.png" /> is a [[Helicoid|helicoid]].
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where $\{\alpha,\beta,\gamma\}$ is a unit vector having the same direction as an axis of the conoid and $f(v)$ is some function. For a right conoid $\alpha=\beta=0$, $\gamma=1$, and then its axis is a line of striction. A right conoid with $f(v)=av$ is a [[Helicoid|helicoid]].
  
  

Revision as of 17:30, 1 August 2014

A Catalan surface all straight line generators of which intersect a fixed straight line, called an axis of the conoid. For example, a hyperbolic paraboloid is a conoid with two axes.

The position vector of a conoid is given by

$$r=\{u\cos v+\alpha f(v),u\sin v+\beta f(v),\gamma f(v)\},$$

where $\{\alpha,\beta,\gamma\}$ is a unit vector having the same direction as an axis of the conoid and $f(v)$ is some function. For a right conoid $\alpha=\beta=0$, $\gamma=1$, and then its axis is a line of striction. A right conoid with $f(v)=av$ is a helicoid.


Comments

References

[a1] M. Berger, B. Gostiaux, "Géométrie différentielle: variétés, courbes et surfaces" , Presses Univ. France (1987)
[a2] M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976)
How to Cite This Entry:
Conoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conoid&oldid=18035
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article