Difference between revisions of "Cochleoid"
From Encyclopedia of Mathematics
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| + | <span id="Fig1"> | ||
| + | [[File:Cochleoid-1.png| right| frame| Figure 1. The cochleoid ([[Media:Cochleoid-1.pdf|pdf]]) ]] | ||
| + | </span> | ||
A plane transcendental curve whose equation in polar coordinates is | 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. | |
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| − | 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 | ||
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| + | 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"> | + | <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> | ||
Latest revision as of 20:46, 5 December 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=18847
Cochleoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cochleoid&oldid=18847
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article