Problem 193. Smallest distance between a point and a rectangle

Created by Alfonso Nieto-Castanon

Solve Later

{"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>Given two points</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>x</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>y</w:t></w:r><w:r><w:t> placed at opposite corners of a rectangle, find the minimal euclidean distance between another point</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>z</w:t></w:r><w:r><w:t> and every point within this rectangle.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>For example, the two points</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[     x = [-1,-1];\n     y = [1,1];]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>define a square centered at the origin. The distance between the point</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[   z = [4,5];]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>and this square is</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[   d = 5;]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>(the closest point in the square is at [1,1])</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>The distance between the point z = [0,2] and this same square is d = 1 (closest point at [0,1])</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>The distance between the point z = [0,0] and this same square is d = 0 (inside the square)</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Notes:</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>you can always assume that</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>x</w:t></w:r><w:r><w:t> &lt;</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>y</w:t></w:r><w:r><w:t> (element-wise)</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>The function should work for points x,y,z in an arbitrary n-dimensional space (with n&gt;1)</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>"}]}

Solve

Solution Stats

341 Solutions
96 Solvers

Last Solution submitted on Sep 22, 2021

Last 200 Solutions

Problem Comments

1 Comment

1 Comment

Sam on 8 Oct 2014

For the n dimensional case it would be better to say that x and y lie on opposite vertices of the n-hypercuboid such that each edge is parallel to a coordinate axis.

Solution Comments

Solution 18090

1 Comment

1 Comment

Alfonso Nieto-Castanon on 31 Jan 2012

nice trick :)

Problem Recent Solvers96

Problem Tags

distance math

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Problem 193. Smallest distance between a point and a rectangle

Solution Stats

Last 200 Solutions

Problem Comments

Solution Comments

Problem Recent Solvers96

Suggested Problems

More from this Author38

Problem Tags

Community Treasure Hunt

Problem 193. Smallest distance between a point and a rectangle

Solution Stats

Last 200 Solutions

Problem Comments

Solution Comments

Problem Recent Solvers96

Suggested Problems

More from this Author38

Problem Tags

Community Treasure Hunt

Players