Problem 45220. Find triangles from edge

Created by Mathworlds

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{"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","targetMode":"","relationshipId":"rId1","target":"/matlab/document.xml"},{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/output","targetMode":"","relationshipId":"rId2","target":"/matlab/output.xml"}],"parts":[{"partUri":"/matlab/document.xml","relationship":[],"contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>First input is T, a Triplet list of indices -whom each row actually contains the three indices of a triangle vertices-. size(T) = [nb_triangles, 3]. Second input is e = [e1, e2], a row vector, couple of indices. Output S is the triplet list of indices which Share the edge e. For instance, if</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[e = [2, 4]]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>and T a tetrahedron</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[T = [1, 2, 3;...\n 1, 3, 4;...\n 1, 2, 4;...\n 2, 3, 4]]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>then the output of the function is</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[S = [1, 2, 4;...\n 2, 3, 4]]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>since both triangles [1, 2, 4] and [2, 3, 4] contain the edge [2, 4].</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Conditions :</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>If the edge is not part of any triangle in the list, the function must of course return the empty set, [].</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>Edges are symmetric :</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>[e1, e2] is the same edge as [e2, e1]</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Order of rows / edges in the output doesn't matter.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Triangle indices are assumed always to be sorted in ascending order, T = [t1, t2, t3] with t1 &lt; t2 &lt; t3.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>Every</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>indices are positive, distinct integers.</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

First input is T, a Triplet list of indices -whom each row actually contains the three indices of a triangle vertices-. size(T) = [nb_triangles, 3]. Second input is e = [e1, e2], a row vector, couple of indices. Output S is the triplet list of indices which Share the edge e. For instance, if<pre class="language-matlab">e = [2, 4]
</pre>and T a tetrahedron<pre class="language-matlab">T = [1, 2, 3;...
 1, 3, 4;...
 1, 2, 4;...
 2, 3, 4]
</pre>then the output of the function is<pre class="language-matlab">S = [1, 2, 4;...
 2, 3, 4]
</pre>since both triangles [1, 2, 4] and [2, 3, 4] contain the edge [2, 4].Conditions :<ul><li>If the edge is not part of any triangle in the list, the function must of course return the empty set, [].</li><li>Edges are symmetric : [e1, e2] is the same edge as [e2, e1]</li><li>Order of rows / edges in the output doesn't matter.</li><li>Triangle indices are assumed always to be sorted in ascending order, T = [t1, t2, t3] with t1 &lt; t2 &lt; t3.</li><li>Every indices are positive, distinct integers.</li></ul>

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Last Solution submitted on Oct 14, 2020

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Mathworlds on 3 Dec 2019

Info : T is actually a triangulation -list of triangles-, in which each index corresponding to the row index of a vertex in another list -a vertices list-, it is a widely used technique used to store and write triangular meshes in mesh processing. Here below the example is a tetrahedron -4 facets-.

Mathworlds on 3 Dec 2019

I actually realize this problem looks like a lot to my second problem, "Find a common edge" ( https://fr.mathworks.com/matlabcentral/cody/problems/45218-find-a-common-edge ) ...

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3d edge face facet find geometry index indices mathematics mesh mesh processing surface triangle triangulation trimesh vertex vertices

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Problem 45220. Find triangles from edge

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