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Difference between revisions of "Curve of pursuit"

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A curve representing a solution of the  "pursuit"  problem, which is posed as follows: Let a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027400/c0274001.png" /> be moved uniformly along a given curve. The trajectory has to be found of the uniform movement of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027400/c0274002.png" />, such that the tangent drawn towards the trajectory at any moment of the movement would pass through the position of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027400/c0274003.png" /> corresponding to that moment in time.
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A curve representing a solution of the  "pursuit"  problem, which is posed as follows: Let a point $M$ be moved uniformly along a given curve. The trajectory has to be found of the uniform movement of a point $N$, such that the tangent drawn towards the trajectory at any moment of the movement would pass through the position of $M$ corresponding to that moment in time.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c027400a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c027400a.gif" />
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In a plane, the system of equations which the curve of pursuit must satisfy takes the form
 
In a plane, the system of equations which the curve of pursuit must satisfy takes 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/c/c027/c027400/c0274004.png" /></td> </tr></table>
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$$\eta-y=\frac{dy}{dx}(\xi-x),\quad F(\xi,\eta)=0,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027400/c0274005.png" /> is the slope of the curve of pursuit, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027400/c0274006.png" /> is the equation of the given curve.
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where $dy/dx$ is the slope of the curve of pursuit, and $F(\xi,\eta)=0$ is the equation of the given curve.
  
 
The  "pursuit"  problem was posed by Leonardo da Vinci and solved by P. Bouguer (1732). For a generalization of it see the last Chapter of [[#References|[2]]].
 
The  "pursuit"  problem was posed by Leonardo da Vinci and solved by P. Bouguer (1732). For a generalization of it see the last Chapter of [[#References|[2]]].

Latest revision as of 07:43, 21 June 2014

A curve representing a solution of the "pursuit" problem, which is posed as follows: Let a point $M$ be moved uniformly along a given curve. The trajectory has to be found of the uniform movement of a point $N$, such that the tangent drawn towards the trajectory at any moment of the movement would pass through the position of $M$ corresponding to that moment in time.

Figure: c027400a

In a plane, the system of equations which the curve of pursuit must satisfy takes the form

$$\eta-y=\frac{dy}{dx}(\xi-x),\quad F(\xi,\eta)=0,$$

where $dy/dx$ is the slope of the curve of pursuit, and $F(\xi,\eta)=0$ is the equation of the given curve.

The "pursuit" problem was posed by Leonardo da Vinci and solved by P. Bouguer (1732). For a generalization of it see the last Chapter of [2].

References

[1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)
[2] J.E. Littlewood, "A mathematician's miscellany" , Methuen (1953)
How to Cite This Entry:
Curve of pursuit. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Curve_of_pursuit&oldid=15061
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article