# Difference between revisions of "Parabolic spiral"

From Encyclopedia of Mathematics

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− | To each value of | + | A transcendental plane curve whose equation in [[Polar_coordinates | polar coordinates]] has the form |

+ | \begin{equation} | ||

+ | \rho = a\sqrt{\phi} + l,\quad l>0. | ||

+ | \end{equation} | ||

+ | To each value of $\phi$ correspond two values of $\sqrt{\phi}$, one positive and one negative. | ||

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p071250a.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p071250a.gif" /> | ||

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Figure: p071250a | Figure: p071250a | ||

− | The curve has infinitely many double points and one point of inflection (see Fig.). If | + | The curve has infinitely many double points and one [[Point_of_inflection | point of inflection]] (see Fig.). If $l=0$, then the curve is called the [[Fermat spiral|Fermat spiral]]. The parabolic spiral is related to the so-called algebraic spirals (see [[Spirals|Spirals]]). |

====References==== | ====References==== |

## Latest revision as of 09:32, 30 November 2012

A transcendental plane curve whose equation in polar coordinates has the form
\begin{equation}
\rho = a\sqrt{\phi} + l,\quad l>0.
\end{equation}
To each value of $\phi$ correspond two values of $\sqrt{\phi}$, one positive and one negative.

Figure: p071250a

The curve has infinitely many double points and one point of inflection (see Fig.). If $l=0$, then the curve is called the Fermat spiral. The parabolic spiral is related to the so-called algebraic spirals (see Spirals).

#### References

[1] | A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian) |

#### Comments

#### References

[a1] | F. Gomez Teixeira, "Traité des courbes" , 1–3 , Chelsea, reprint (1971) |

[a2] | J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972) |

**How to Cite This Entry:**

Parabolic spiral.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Parabolic_spiral&oldid=28966

This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article