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Difference between revisions of "Si-ci-spiral"

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A plane curve whose equation in rectangular Cartesian coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084980/s0849801.png" /> has the form
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A plane curve whose equation in rectangular Cartesian coordinates $(x,y)$ 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/s/s084/s084980/s0849802.png" /></td> </tr></table>
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$$x=\operatorname{ci}(t),\quad y=\operatorname{si}(t),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084980/s0849803.png" /> is the [[Integral cosine|integral cosine]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084980/s0849804.png" /> is the [[Integral sine|integral sine]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084980/s0849805.png" /> is a real parameter (see Fig.).
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where $\operatorname{ci}$ is the [[Integral cosine|integral cosine]], $\operatorname{si}$ is the [[Integral sine|integral sine]] and $t$ is a real parameter (see Fig.).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s084980a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s084980a.gif" />
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Figure: s084980a
 
Figure: s084980a
  
The arc length from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084980/s0849806.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084980/s0849807.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084980/s0849808.png" />, and the curvature is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084980/s0849809.png" />.
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The arc length from $t=0$ to $t=t_0$ is equal to $\log t_0$, and the curvature is equal to $\kappa=t_0$.
  
 
====References====
 
====References====

Revision as of 14:06, 3 August 2014

A plane curve whose equation in rectangular Cartesian coordinates $(x,y)$ has the form

$$x=\operatorname{ci}(t),\quad y=\operatorname{si}(t),$$

where $\operatorname{ci}$ is the integral cosine, $\operatorname{si}$ is the integral sine and $t$ is a real parameter (see Fig.).

Figure: s084980a

The arc length from $t=0$ to $t=t_0$ is equal to $\log t_0$, and the curvature is equal to $\kappa=t_0$.

References

[1] E. Jahnke, F. Emde, F. Lösch, "Tafeln höheren Funktionen" , Teubner (1966)


Comments

References

[a1] M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1972)
How to Cite This Entry:
Si-ci-spiral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Si-ci-spiral&oldid=32696
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article