Problem 1456. Beads on a Necklace (Convex Hulls)

Created by Ned Gulley

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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>We may describe a</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://en.wikipedia.org/wiki/Convex_hull\"><w:r><w:t>convex hull</w:t></w:r></w:hyperlink><w:r><w:t> as a rubber band stretched around a list of points. Some of the points will be inside the rubber band, and some will define vertices of (or lie along the edge of) a</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://en.wikipedia.org/wiki/Convex_and_concave_polygons\"><w:r><w:t>convex polygon</w:t></w:r></w:hyperlink><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Given an n-by-2 list of points xy where each row describes a point, your job is to determine if all the points belong to the convex hull for those points. In the matrix xy, the x coordinate is the first column and the y coordinate is the second column.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>So, for example, the points below form a single convex hull.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ * *\n\n * *]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>In contrast, the points below do not, since the convex hull is a triangle that includes an interior point. Any polygon that includes all the points must necessarily be concave.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ *\n * \n * *]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Thus if</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ xy = [1 1;1 2;2 2;2 1]]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>then</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ allConvex(xy) => true]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Whereas</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ xy = [1 1;3 1;2 2;2 3]\n allConvex(xy) => false]]></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>"}]}

We may describe a <a href = "http://en.wikipedia.org/wiki/Convex_hull">convex hull</a> as a rubber band stretched around a list of points. Some of the points will be inside the rubber band, and some will define vertices of (or lie along the edge of) a <a href = "http://en.wikipedia.org/wiki/Convex_and_concave_polygons">convex polygon</a>.Given an n-by-2 list of points xy where each row describes a point, your job is to determine if all the points belong to the convex hull for those points. In the matrix xy, the x coordinate is the first column and the y coordinate is the second column.So, for example, the points below form a single convex hull.<pre> * *</pre><pre> * * </pre>In contrast, the points below do not, since the convex hull is a triangle that includes an interior point. Any polygon that includes all the points must necessarily be concave.<pre> *
 * 
 * *</pre>Thus if<pre> xy = [1 1;1 2;2 2;2 1]</pre>then<pre> allConvex(xy) => true</pre>Whereas<pre> xy = [1 1;3 1;2 2;2 3]
 allConvex(xy) => false</pre>

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Problem 1456. Beads on a Necklace (Convex Hulls)

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