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A plane transcendental curve whose equation in rectangular Cartesian coordinates has the form
 
A plane transcendental curve whose equation in rectangular Cartesian 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/t/t093/t093570/t0935701.png" /></td> </tr></table>
+
$$
 +
= \pm  a  \mathop{\rm ln} \
 +
 
 +
\frac{a + \sqrt {a  ^ {2} - y  ^ {2} } }{y }
 +
\mps
 +
\sqrt {a  ^ {2} - y  ^ {2} } .
 +
$$
  
The tractrix is symmetric about the origin (see Fig.), the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093570/t0935702.png" />-axis being an asymptote. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093570/t0935703.png" /> is a [[Cusp(2)|cusp]] with vertical tangent. The length of the arc measured from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093570/t0935704.png" /> is:
+
The tractrix is symmetric about the origin (see Fig.), the $  x $-
 +
axis being an asymptote. The point $  ( 0, a) $
 +
is a [[Cusp(2)|cusp]] with vertical tangent. The length of the arc measured from the point $  x = 0 $
 +
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/t/t093/t093570/t0935705.png" /></td> </tr></table>
+
$$
 +
= a  \mathop{\rm ln}  {
 +
\frac{a}{y}
 +
} .
 +
$$
  
 
The radius of curvature is:
 
The radius of curvature 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/t/t093/t093570/t0935706.png" /></td> </tr></table>
+
$$
 +
= a  \mathop{\rm cot}  {
 +
\frac{x}{y}
 +
} .
 +
$$
  
 
The area bounded by the tractrix and its asymptote is:
 
The area bounded by the tractrix and its asymptote 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/t/t093/t093570/t0935707.png" /></td> </tr></table>
+
$$
 +
= {
 +
\frac{\pi a  ^ {2} }{2}
 +
} .
 +
$$
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t093570a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t093570a.gif" />
Line 19: Line 52:
 
Figure: t093570a
 
Figure: t093570a
  
The rotation of the tractrix around the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093570/t0935708.png" />-axis generates a [[Pseudo-sphere|pseudo-sphere]]. The length of the tangent, that is, of the segment between the point of tangency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093570/t0935709.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093570/t09357010.png" />-axis, is constant. This property enables one to regard the tractrix as the trajectory of the end of a line segment of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093570/t09357011.png" />, when the other end moves along the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093570/t09357012.png" />-axis. The notion of a tractrix generalizes to the case when the end of the segment does not move along a straight line, but along some given curve; the curve obtained in this way is called the trajectory of the given curve.
+
The rotation of the tractrix around the $  x $-
 +
axis generates a [[Pseudo-sphere|pseudo-sphere]]. The length of the tangent, that is, of the segment between the point of tangency $  M $
 +
and the $  x $-
 +
axis, is constant. This property enables one to regard the tractrix as the trajectory of the end of a line segment of length $  a $,  
 +
when the other end moves along the $  x $-
 +
axis. The notion of a tractrix generalizes to the case when the end of the segment does not move along a straight line, but along some given curve; the curve obtained in this way is called the trajectory of the given curve.
  
 
====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>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Gomes Teixeira,  "Traité des courbes" , '''1–3''' , Chelsea, reprint  (1971)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Greenberg,  "Euclidean and non-Euclidean geometries" , Freeman  (1974)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  pp. 1–5</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  K. Fladt,  "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell.  (1962)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.D. Lawrence,  "A catalog of special plane curves" , Dover, reprint  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Gomes Teixeira,  "Traité des courbes" , '''1–3''' , Chelsea, reprint  (1971)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Greenberg,  "Euclidean and non-Euclidean geometries" , Freeman  (1974)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  pp. 1–5</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  K. Fladt,  "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell.  (1962)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.D. Lawrence,  "A catalog of special plane curves" , Dover, reprint  (1972)</TD></TR></table>

Revision as of 08:26, 6 June 2020


A plane transcendental curve whose equation in rectangular Cartesian coordinates has the form

$$ x = \pm a \mathop{\rm ln} \ \frac{a + \sqrt {a ^ {2} - y ^ {2} } }{y } \mps \sqrt {a ^ {2} - y ^ {2} } . $$

The tractrix is symmetric about the origin (see Fig.), the $ x $- axis being an asymptote. The point $ ( 0, a) $ is a cusp with vertical tangent. The length of the arc measured from the point $ x = 0 $ is:

$$ l = a \mathop{\rm ln} { \frac{a}{y} } . $$

The radius of curvature is:

$$ r = a \mathop{\rm cot} { \frac{x}{y} } . $$

The area bounded by the tractrix and its asymptote is:

$$ S = { \frac{\pi a ^ {2} }{2} } . $$

Figure: t093570a

The rotation of the tractrix around the $ x $- axis generates a pseudo-sphere. The length of the tangent, that is, of the segment between the point of tangency $ M $ and the $ x $- axis, is constant. This property enables one to regard the tractrix as the trajectory of the end of a line segment of length $ a $, when the other end moves along the $ x $- axis. The notion of a tractrix generalizes to the case when the end of the segment does not move along a straight line, but along some given curve; the curve obtained in this way is called the trajectory of the given curve.

References

[1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)

Comments

References

[a1] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a2] F. Gomes Teixeira, "Traité des courbes" , 1–3 , Chelsea, reprint (1971)
[a3] M. Greenberg, "Euclidean and non-Euclidean geometries" , Freeman (1974)
[a4] M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish pp. 1–5
[a5] K. Fladt, "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell. (1962)
[a6] J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)
How to Cite This Entry:
Tractrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tractrix&oldid=49006
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article