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A plane curve whose equation in polar coordinates is

$$\rho=a\phi^2-l,\quad l\geq0.$$

The spiral is symmetric with respect to the polar axis (see Fig.) and has a double point at the pole with tangents forming angles equal to $\pm\sqrt{l/a}$ with the polar axis.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043080a.gif" />

Figure: g043080a

The polar axis of a Galilean spiral contains infinitely many double points, for which $\rho=ak^2\pi^2-l$, where $k=1,2,\ldots$. The Galilean spiral is a so-called algebraic spiral (cf. [[Spirals|Spirals]]). Named after G. Galilei (1683) in connection with his studies on the free fall of solids.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Savelov,   "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR></table>