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Difference between revisions of "Galilean spiral"

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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.
 
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" />
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[[File:Galilean spiral.svg|center|300px|Galilean spiral with parameters (a,l)=(1,8)]]
 
 
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.
 
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====
 
====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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* {{Ref|1}} A.A. Savelov, "Planar curves", Moscow (1960) (In Russian)

Latest revision as of 19:26, 16 March 2023

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.

Galilean spiral with parameters (a,l)=(1,8)

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). Named after G. Galilei (1683) in connection with his studies on the free fall of solids.

References

  • [1] A.A. Savelov, "Planar curves", Moscow (1960) (In Russian)
How to Cite This Entry:
Galilean spiral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Galilean_spiral&oldid=52720
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article