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You may just compute the (oriented) area of a triangle with vertices at $(0,0),(x_i,y_i),(x_{i+1},y_{i+1})$ and add the contributes to get: $$A = -\frac{1}{2}\sum_{i=1}^{n}\left(x_i y_{i+1} - x_{i+1} y_i\right)=\frac{1}{2}\sum_{i=1}^{n}(x_{i+1}-x_i)(y_{i+1}+y_i)$$ where the last equality follows from $\sum_{i=1}^{n}\left(x_{i+1}y_{i+1}-x_i y_i\right)=0$. ...

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Use Green's area formula $${\rm area\,}(P)={1\over2}\int_{\partial P}(x\>dy-y\>dx)$$ and parametrize the segments $\sigma_k$ making up $\partial P$ as $$\sigma_k:\quad t\mapsto \bigl((1-t)x_k+t x_{k+1}, \>(1-t)y_k+t y_{k+1}\bigr)\qquad(0\leq t\leq1)\ .$$

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The steps for such demostration are indicated in the book of Berg "Computational Geometry. Algorithms and aplications". Exercise 1.1. The convex hull of a set S is defined to be the intersection of all convex sets that contain S. For the convex hull of a set of points it was indicated that the convex hull is the convex set with smallest perimeter. We want to ...

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Such problems are tackled in group theory. My friend Ruby gave me this list of fundamental rotations and some of the aliases she knows: rots = [ k = 1: e=12345678 ["ee", "xxxx", "yyyy", "zzzz", "xexxx", "xxexx", "xxxex", "xxxxe", "yeyyy", "yyeyy", "yyyey", "yyyye", "zezzz", "zzezz", "zzzez", "zzzze"] k = 2: x=56218734 ["xe", "xee", "ex", "yxz", ...

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There may well be a slick dynamic programming solution, but I don't know it. Here is one approach that looks like it should work, but I don't quite see how to prove termination. The idea is to model threading a string along the road and pulling it tight. In more detail, find any polyline path contained in the road that connects the end points. Now look for a ...

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So your segment has endpoints $\mathbf p_1, \mathbf p_2$ and one of the planes is $\mathbf n \cdot \mathbf r = d$, where $\mathbf n$ points into the tetrahedron, then examine $\mathbf n \cdot \mathbf r - d$ for $\mathbf r = \mathbf p_1$ and $\mathbf r = \mathbf p_2$. If they are both negative for any of the 4 planes, then your segment does not intersect ...

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(I'm the author of that CGAL package) The problem is that this section is actually missing a definition, without which the case of (point) intersecting lines (i.e. non-collinears) is unclear. It says: The bisecting lines of two edges are the lines bisecting the supporting lines of the edges But there is no definition of "lines bisecting the ...

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