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Difference between revisions of "Cochleoid"

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(Comment: inversion yields quadratrix of Hippias, ref Lawrence)
(gather refs)
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\rho = a\frac{\sin\varphi}{\varphi}.
 
\rho = a\frac{\sin\varphi}{\varphi}.
 
\end{equation}
 
\end{equation}
 
  
 
The cochleoid has infinitely many spirals, passing through its pole and touching the polar axis (see Fig.). The pole is a singular point of infinite multiplicity. Any straight line through the pole $O$ intersects the cochleoid; the tangents to the cochleoid at these intersection points pass through the same point.
 
The cochleoid has infinitely many spirals, passing through its pole and touching the polar axis (see Fig.). The pole is a singular point of infinite multiplicity. Any straight line through the pole $O$ intersects the cochleoid; the tangents to the cochleoid at these intersection points pass through the same point.
  
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Savelov,  "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR></table>
 
 
====Comments====
 
 
The [[Inversion|inverse]] of the cochleoid with respect to the origin is the [[quadratrix of Hippias]].
 
The [[Inversion|inverse]] of the cochleoid with respect to the origin is the [[quadratrix of Hippias]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.D. Lawrence,  "A catalog of special plane curves" , Dover  (1972) ISBN 0-486-60288-5  {{ZBL|0257.50002}}</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Savelov,  "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J.D. Lawrence,  "A catalog of special plane curves" , Dover  (1972) ISBN 0-486-60288-5  {{ZBL|0257.50002}}</TD></TR>
 +
</table>

Revision as of 11:16, 9 April 2023


Figure 1. The cochleoid (pdf)

A plane transcendental curve whose equation in polar coordinates is \begin{equation} \rho = a\frac{\sin\varphi}{\varphi}. \end{equation}

The cochleoid has infinitely many spirals, passing through its pole and touching the polar axis (see Fig.). The pole is a singular point of infinite multiplicity. Any straight line through the pole $O$ intersects the cochleoid; the tangents to the cochleoid at these intersection points pass through the same point.

The inverse of the cochleoid with respect to the origin is the quadratrix of Hippias.

References

[1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)
[a1] J.D. Lawrence, "A catalog of special plane curves" , Dover (1972) ISBN 0-486-60288-5 Zbl 0257.50002
How to Cite This Entry:
Cochleoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cochleoid&oldid=42500
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article