My Math Forum Geometry piecewise line segments

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 December 2nd, 2011, 06:56 PM #1 Member   Joined: Dec 2011 Posts: 41 Thanks: 0 Geometry piecewise line segments Let $\overline{AB}$ be the line segment connecting $A= ( \alpha_{1},\alpha_{ 2}) and B = ( \beta_{1},\beta_{ 2})$ A does not equal B. Since AB is a subset of a straight line $g_{\sigma,\tau}\ := x\in R^2\mid\ x\ =\sigma\t\ +\tau, t\in R}\ with\ \sigma\ = (\sigma_{1},\sigma{2}),\ \tau\ =(\tau_{1},\tau_{2}),$ we can consider$\overline{AB}$ as a trace of a parameter curve, namely $\gamma(t)\=\gamma(t)\ +\tau$ for t $\in\ [a,b]$ with some interval [a,b]. Further, given $[a,b]\subset\ R$ there exists$\gamma: [a,b]\rightarrow\R^2$ such that $\gamma([a,b])=\overline{AB}$ for this we need to find $\gamma,\tau\in\ R^2$ such that $\alpha_{1}\=\ \sigma_{1}a\ +\tau_{1}$ $\beta_{1}\=\sigma_{1}b\ +\tau_{1}$ and $\alpha_{2}\=\sigma_{2}b\ +\tau_{2}$ $\beta_{2}\=\sigma_{2}b\ +\tau_{2}$ and take $\gamma(t)\=\sigma(t)\ +\tau$ 1. Find for $A=(\alpha_{1},\alpha_{2})\in\ R^2,\ B=(\beta_{1},\beta_{2}\in\ R^2\ the\ explicit\ values\ of\ \sigma,\tau\in R^2$ 2. also let $\gamma: [a,b]\rightarrow R^n$ be a continuous mapping, gamma is a piecewise curve. Prove that the polygon $P(A_{1},A_{2},A_{3},A_{4}) where A_{1}=(1,2),A_{2}=(5,2),A_{3}=(5,4),A_{4}=(1,4)$ is a piece wise infinite class curve.

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