Namespaces
Variants
Actions

Difference between revisions of "Galilean spiral"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
Line 1: Line 1:
 +
{{TEX|done}}
 
A plane curve whose equation in polar coordinates is
 
A plane curve whose equation in polar coordinates is
  
<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/g/g043/g043080/g0430801.png" /></td> </tr></table>
+
$$\rho=a\phi^2-l,\quad l\geq0.$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043080/g0430802.png" /> 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" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043080a.gif" />
Line 9: Line 10:
 
Figure: g043080a
 
Figure: g043080a
  
The polar axis of a Galilean spiral contains infinitely many double points, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043080/g0430803.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043080/g0430804.png" />. 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>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Savelov,  "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR></table>

Revision as of 11:12, 26 July 2014

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.

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). 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=32536
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article