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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク radiodrome ： ウィキペディア英語版 radiodrome In geometry, a radiodrome is the pursuit curve followed by a point that is pursuing another linearly-moving point. The term is derived from the Latin word ''radius'' (beam) and the Greek word ''dromos'' (running). The classical (and best-known) form of a radiodrome is known as the "dog curve"; this is the path a dog follows when it swims across a stream with a current after food it has spotted on the other side. Because the dog drifts downwards with the current, it will have to change its heading; it will also have to swim further than if it had computed the optimal heading. This case was described by Pierre Bouguer in 1732. A radiodrome may alternatively be described as the path a dog follows when chasing a hare, assuming that the hare runs in a straight line at a constant velocity. It is illustrated by the following figure: ==Mathematical analysis== Introduce a coordinate system with origin at the position of the dog at time zero and with y-axis in the direction the hare is running with the constant speed $V\_t$. The position of the hare at time zero is $(A\_x\backslash \; ,\backslash \; A\_y)$ and at time $t$ it is # The dog runs with the constant speed $V\_d$ towards the momentary position of the hare. The differential equation corresponding to the movement of the dog, $(x(t)\backslash \; ,\backslash \; y(t))$, is consequently # |}} # |}} It is possible to obtain a closed form analytical expression $y=f(x)$ for the motion of the dog From () and () follows that # Multiplying both sides with $T\_x-x$ and taking the derivative with respect to $x$ using that # one gets # |}} or # =\frac |}} From this relation follows that # where $B$ is the constant of integration that is determined by the initial value of $y\text{'}$ at time zero, i.e. # From () and () follows after some computations that # } - \frac)^}}} If now $V\_t\; \backslash neq\; V\_d$ this relation is integrated as # - \frac) ^} } = \frac and the chase starts with the hare at position $(A\_x\backslash \; ,\backslash \; -0.6\backslash \; A\_x)$ what means that $y\text{'}(0)\; =\; -0.6$. From () one therefore gets that the hare is caught at position $(A\_x\backslash \; ,\backslash \; 1.21688\backslash \; A\_x)$ and consequently that the hare will run the total distance $(1.21688\backslash \; +\backslash \; 0.6)\backslash \; A\_x$ before being caught. If $V\_t\; \backslash geq\; V\_d$ one gets from () and () that $\backslash lim\_y(x)\; =\; \backslash infty$ what means that the hare never will be caught whenever the chase starts. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「radiodrome」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.