We are given the lengths of all sides of a polygon. We need to determine if the given polygon is convex or concave. How can this be done? What is the propery applied to determine this?
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3$\begingroup$ It can't be done. $\endgroup$– LucianDec 6, 2015 at 19:02
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$\begingroup$ ohkk.. thanx a lot . In case if we are given some straight lines (with length of each of them) , how can we determine if convex polygon formation is possible or not using each of those lines? $\endgroup$– user249117Dec 6, 2015 at 19:05
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1$\begingroup$ @user249117 If it's possible to form a polygon from given line segments, then it's possible to form a convex one from these. $\endgroup$– WojowuDec 6, 2015 at 19:15
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$\begingroup$ @Wojowu thanks a lot :) $\endgroup$– user249117Dec 6, 2015 at 19:18
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$\begingroup$ What is a "concave polygon", by the way? $\endgroup$– user147263Dec 6, 2015 at 20:14
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1 Answer
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Summary of comments: Lucian gave an example of two polygons with the same sidelengths, only one of which is convex.
Wojowu added that "If it's possible to form a polygon from given line segments, then it's possible to form a convex one from these."
Additional remark: if you know the coordinates $V_j$ of vertices, then the signs of scalar cross-products of vectors $V_jV_{j+1}$ can be used to determine convexity. I.e., the determinants such as $$ \begin{vmatrix} V_2^x-V_1^x & V_3^x-V_2^x \\ V_2^y-V_1^y & V_3^y-V_2^y \end{vmatrix} $$ must be all $\ge 0$, or all $\le 0$. (Including one with $V_n$ and $V_1$ to close the loop.)