Difference between revisions of "Curve of pursuit"
From Encyclopedia of Mathematics
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| − | A curve representing a solution of the "pursuit" problem, which is posed as follows: Let a point | + | {{TEX|done}} |
| + | 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 | ||
| − | + | $$\eta-y=\frac{dy}{dx}(\xi-x),\quad F(\xi,\eta)=0,$$ | |
| − | where | + | 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]]]. | ||
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
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