Difference between revisions of "Conoid"
From Encyclopedia of Mathematics
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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 | ||
| − | + | $$r=\{u\cos v+\alpha f(v),u\sin v+\beta f(v),\gamma f(v)\},$$ | |
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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]]. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Berger, B. Gostiaux, "Géométrie différentielle: variétés, courbes et surfaces" , Presses Univ. France (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Berger, B. Gostiaux, "Géométrie différentielle: variétés, courbes et surfaces" , Presses Univ. France (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976)</TD></TR></table> | ||
Latest revision as of 10:48, 16 March 2023
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.
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
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