Ivan Shishkin, Rye (1878)

Problems/TopologyExerciseUnreviewed

A simple path homotopy

by Ancient Tree·
30
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Find a path homotopy between the red and green half-circles below.

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Solution by Ancient Tree

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The two half-circles can be parametrized as follows:
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))(note that we use the letter ss because the letter tt will be used for the homotopy parameter).

So a natural path homotopy is:
H(t,s)=(cos⁡(s),(1−2t)sin⁡(s)).H(t,s)=\left(\cos(s),(1-2t)\sin(s)\right).

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