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POMathematica: Non-intersecting line segments
 

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  1. POMathematica: Non-intersecting line segments
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    <p>How can we tell <em>Mathematica</em> to gives us a set of non-intersecting lines? In this case two lines intersect if they have a point (not an endpoint) in common. Consider this simple case:</p> <pre><code>l1 = {{-1, 0}, {1, 0}}; l2 = {{0, -1}, {0, 1}}; lines = {l1, l2}; </code></pre> <p>The idea is to create a function which, given a set a lines, returns a set of non-intersecting lines. If such function exists say <code>split</code> then the output of</p> <pre><code>split[lines] </code></pre> <p>would be</p> <pre><code>{ {{-1, 0}, {0,0}}, {{ 0, 0}, {1,0}}, {{ 0,-1}, {0,0}}, {{ 0, 0}, {0,1}} } </code></pre> <p>The function detected that <code>{0,0}</code> is the intersection between the two lines in the set and in order to have non-intersecting lines it broke the line segments at the intersections thus generating 2 more lines. This process gets more complicated if the original input contains more lines. Does anyone know how to do this efficiently in <em>Mathematica</em> without using loops? It might help to know an algorithm to find if <a href="https://stackoverflow.com/questions/563198/how-do-you-detect-where-two-line-segments-intersect/563275#563275">two lines are intersecting</a>.</p> <p><strong>Note</strong></p> <p>This question is the second part of my attempt to find out <a href="https://stackoverflow.com/questions/6353337/mathematica-3d-wire-frames">how to make wire frames in <em>Mathematica</em> with hidden line removal</a>. Please feel free to add more appropriate tags.</p>
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    1. COI don't have Mma here, but what you want is to use standard linear algebra to represent each line as A.{x, y}=c, and find the point where the equations for both lines are true, using LinearSolve. Then, check that the solution is in between the ends of the two line segments given. If so, break up the lines at that point. As with my answer to your earlier question, you want to do this to Tuples[Sort[lines],{2}].
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    2. CO@Verbeia, What is `Tuples[Sort[lines], {2}]` supposed to do? lets assum that `lines` is defined as in my post.
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    3. COthe Tuples function is only necessary if you have a list of more than two lines that you are checking, and you want to check all possible pairs of lines. All it does is create a list of all possible pairs of elements from a list with more than two elements. ACL's answer below deals with the issue of duplicates, and might be a better solution than Tuples.
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