View source for Parabolic spiral

A transcendental plane curve whose equation in polar coordinates has the form

<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/p/p071/p071250/p0712501.png" /></td> </tr></table>

To each value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071250/p0712502.png" /> correspond two values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071250/p0712503.png" />, one positive and one negative.

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

Figure: p071250a

The curve has infinitely many double points and one point of inflection (see Fig.). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071250/p0712504.png" />, 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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Savelov,   "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Gomez Teixeira,   "Traité des courbes" , '''1–3''' , Chelsea, reprint  (1971)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.D. Lawrence,   "A catalog of special plane curves" , Dover, reprint  (1972)</TD></TR></table>