Ivan Shishkin, Rye (1878)

Problems/TopologyExerciseUnreviewedEdited since review

Un exemple simple d’homotopie de chemins

by Ancient Tree·translated by Catalpa·
30
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Unreviewed. This problem changed after its last review and should be reviewed again.

Trouvez une homotopie du plan entre les demi-cercles vert et rouge ci-dessous.

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Solution by Ancient Tree · translated by Catalpa

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Les deux demi-cercles peuvent être paramétrés comme suit :
γ0(s)=(cos⁡(s),sin⁡(s)),  γ1(s)=(cos⁡(−s),sin⁡(−s))=(cos⁡(s),−sin⁡(s))\gamma_{0}(s)=(\cos(s),\sin(s)),\;\gamma_{1}(s)=(\cos(-s),\sin(-s))=(\cos(s),-\sin(s))(remarquez que nous utilisons la lettre ss parce que la lettre tt sera utilisée pour le paramètre de l’homotopie).

Donc une homotopie naturelle est :
H(t,s)=(1−t)γ0(s)+tγ1(s)=(cos⁡(s),(1−2t)sin⁡(s))).H(t,s)=(1-t)\gamma_0(s)+t\gamma_1(s)=\left(\cos(s),(1-2t)\sin(s))\right).

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