206 research outputs found

    On a property of plane curves

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    Let Ξ³:[0,1]β†’[0,1]2\gamma: [0,1] \to [0,1]^2 be a continuous curve such that Ξ³(0)=(0,0)\gamma(0)=(0,0), Ξ³(1)=(1,1)\gamma(1)=(1,1), and Ξ³(t)∈(0,1)2\gamma(t) \in (0,1)^2 for all t∈(0,1)t\in (0,1). We prove that, for each n∈Nn \in \mathbb{N}, there exists a sequence of points AiA_i, 0≀i≀n+10\leq i \leq n+1, on Ξ³\gamma such that A0=(0,0)A_0=(0,0), An+1=(1,1)A_{n+1}=(1,1), and the sequences Ο€1(AiAi+1β†’)\pi_1(\overrightarrow{A_iA_{i+1}}) and Ο€2(AiAi+1β†’)\pi_2(\overrightarrow{A_iA_{i+1}}), 0≀i≀n0\leq i \leq n, are positive and the same up to order, where Ο€1,Ο€2\pi_1,\pi_2 are projections on the axes.Comment: 8 page
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