{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-04-06T14:01:22.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2026-04-06T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":44817,"title":"Wrecked Angles?","description":"It's time for some simple geometry fun to start off the new year.\r\n\r\nYou will be given the perimeter P and the area A of a rectangle.  With these two values, calculate the area of the circle that circumscribes this rectangle.  This means we're looking for the area of the circle that touches this rectangle only at the four vertices of the rectangle.\r\n\r\nGood luck, and Happy 2019!","description_html":"\u003cp\u003eIt's time for some simple geometry fun to start off the new year.\u003c/p\u003e\u003cp\u003eYou will be given the perimeter P and the area A of a rectangle.  With these two values, calculate the area of the circle that circumscribes this rectangle.  This means we're looking for the area of the circle that touches this rectangle only at the four vertices of the rectangle.\u003c/p\u003e\u003cp\u003eGood luck, and Happy 2019!\u003c/p\u003e","function_template":"function y = wrecked_angles(P,A)\r\n  y = x;\r\nend","test_suite":"%%\r\nP=14;\r\nA=12;\r\ny=wrecked_angles(P,A);\r\ny_correct = 20.63495408493621;\r\njunk=abs(y-y_correct);\r\nassert(junk-1\u003c1e-10);\r\n%%\r\nP=34;\r\nA=60;\r\ny=wrecked_angles(P,A);\r\ny_correct = 131.7322896141688;\r\njunk=abs(y-y_correct);\r\nassert(junk-1\u003c1e-10);\r\n%%\r\nP=62;\r\nA=168;\r\ny=wrecked_angles(P,A);\r\ny_correct = 590.8738521234052;\r\njunk=abs(y-y_correct);\r\nassert(junk-100\u003c1e-10);\r\n%%\r\ns1=100;\r\ntotalsum=zeros(1,s1);\r\nfor s2=1:s1\r\n    P=2*(s1+s2);\r\n    A=s1*s2;\r\n    totalsum(s2)=wrecked_angles(P,A);\r\nend\r\ns=sum(totalsum);\r\ns_correct=1051137.631982975;\r\ns_junk=abs(s-s_correct);\r\nassert(s_junk\u003c1e-8);\r\n\r\nd=max(totalsum)-min(totalsum);\r\nd_correct=7853.196235811095;\r\nd_junk=abs(d-d_correct);\r\nassert(d_junk\u003c1e-8);\r\n%%\r\ns1=wrecked_angles(32,64);\r\ns2=wrecked_angles(72,288);\r\nP=2*(s1+s2);\r\nA=s1*s2;\r\ny=wrecked_angles(P,A);\r\ny_correct=259088.4479405854;\r\njunk=abs(y-y_correct);\r\nassert(junk\u003c1e-10);","published":true,"deleted":false,"likes_count":0,"comments_count":5,"created_by":1615,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":21,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":72,"created_at":"2019-01-03T14:27:57.000Z","updated_at":"2026-01-21T13:24:31.000Z","published_at":"2019-01-03T14:27:57.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIt's time for some simple geometry fun to start off the new year.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYou will be given the perimeter P and the area A of a rectangle. With these two values, calculate the area of the circle that circumscribes this rectangle. This means we're looking for the area of the circle that touches this rectangle only at the four vertices of the rectangle.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGood luck, and Happy 2019!\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":44386,"title":"Circumscribed Pentagon?","description":"Building off of \u003chttps://www.mathworks.com/matlabcentral/cody/problems/44368 Problem 44368\u003e, your function will be provided with the five vertices of a regular pentagon (p) as well as the center point (cp) and radius (r) of a circle. The function should return one of the following values:\r\n\r\n  0: the pentagon is completely enclosed within the circle but is not inscribed\r\n  1: the pentagon is inscribed in the circle (within ±0.02)\r\n  2: the vertices of the pentagon extend beyond the circle, but its edges still cross back into the circle\r\n  3: the pentagon circumscribes the circle (within ±0.02)\r\n  4: the pentagon completely encloses, and does not touch, the circle\r\n\r\nPoints will be rounded to the nearest hundredth. See the test cases for examples.","description_html":"\u003cp\u003eBuilding off of \u003ca href = \"https://www.mathworks.com/matlabcentral/cody/problems/44368\"\u003eProblem 44368\u003c/a\u003e, your function will be provided with the five vertices of a regular pentagon (p) as well as the center point (cp) and radius (r) of a circle. The function should return one of the following values:\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003e0: the pentagon is completely enclosed within the circle but is not inscribed\r\n1: the pentagon is inscribed in the circle (within ±0.02)\r\n2: the vertices of the pentagon extend beyond the circle, but its edges still cross back into the circle\r\n3: the pentagon circumscribes the circle (within ±0.02)\r\n4: the pentagon completely encloses, and does not touch, the circle\r\n\u003c/pre\u003e\u003cp\u003ePoints will be rounded to the nearest hundredth. See the test cases for examples.\u003c/p\u003e","function_template":"function y = circumscribed_pentagon(p,cp,r)\r\n  y = 0;\r\nend","test_suite":"%%\r\np = [0,5; 4.76,1.55; 2.94,-4.05; -2.94,-4.05; -4.76,1.55];\r\ncp = [0,0];\r\nr = 5;\r\ny_correct = 1;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [0,5.61; 5.40,1.69; 3.34,-4.66; -3.34,-4.66; -5.40,1.69];\r\ncp = [0,0];\r\nr = 5;\r\ny_correct = 2;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [0,6.18; 5.88,1.91; 3.63,-5.00; -3.63,-5.00; -5.88,1.91];\r\ncp = [0,0];\r\nr = 5;\r\ny_correct = 3;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [0,4.55; 4.28,1.44; 2.65,-3.59; -2.65,-3.59; -4.28,1.44];\r\ncp = [0,0];\r\nr = 5;\r\ny_correct = 0;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\ncp = [20,8];\r\np = [0,5; 4.76,1.55; 2.94,-4.05; -2.94,-4.05; -4.76,1.55] + repmat(cp,[5,1]);\r\nr = 5;\r\ny_correct = 1;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [20,13.61; 25.40,9.69; 23.34,3.34; 16.66,3.34; 14.60,9.69];\r\ncp = [20,8];\r\nr = 5;\r\ny_correct = 2;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [20,14.18; 25.88,9.91; 23.63,3.00; 16.37,3.00; 14.12,9.91];\r\ncp = [20,8];\r\nr = 5;\r\ny_correct = 3;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\ncp = [20,8];\r\np = [0,6.58; 6.42,1.92; 3.97,-5.63; -3.97,-5.63; -6.42,1.92] + repmat(cp,[5,1]);\r\nr = 5;\r\ny_correct = 4;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\ncp = [20,8];\r\np = [0,4.55; 4.28,1.44; 2.65,-3.59; -2.65,-3.59; -4.28,1.44] + repmat(cp,[5,1]);\r\nr = 5;\r\ny_correct = 0;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [23.66,11.42; 24.37,5.58; 19.05,3.10; 15.04,7.40; 17.89,12.54];\r\ncp = [20,8];\r\nr = 5;\r\ny_correct = 1;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [25.01,12.47; 25.98,4.58; 18.78,1.23; 13.37,7.03; 17.22,13.97];\r\ncp = [20,8];\r\nr = 5;\r\ny_correct = 4;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [23.27,11.12; 23.92,5.87; 19.12,3.63; 15.52,7.50; 18.08,12.13];\r\ncp = [20,8];\r\nr = 5;\r\ny_correct = 0;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [30.94,36.26; 35.61,27.09; 28.34,19.82; 19.17,24.49; 20.78,34.65];\r\ncp = [26.97,28.45];\r\nr = 8.75;\r\ny_correct = 1;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [32.54,38.78; 38.84,26.41; 29.02,16.59; 16.65,22.89; 18.83,36.61];\r\ncp = [26.97,28.45];\r\nr = 8.75;\r\ny_correct = 4;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [30.49,35.54; 34.69,27.29; 28.14,20.74; 19.89,24.95; 21.34,34.09];\r\ncp = [26.97,28.45];\r\nr = 8.75;\r\ny_correct = 0;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [26.97,34.06; 32.37,30.14; 30.31,23.79; 23.63,23.79; 21.57,30.14];\r\ncp = [26.97,28.45];\r\nr = 5;\r\ny_correct = 2;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [31.35,32.83; 32.49,25.64; 26.00,22.34; 20.85,27.48; 24.16,33.97];\r\ncp = [26.97,28.45];\r\nr = 5.01;\r\ny_correct = 3;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))","published":true,"deleted":false,"likes_count":2,"comments_count":4,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":64,"test_suite_updated_at":"2017-12-08T15:45:11.000Z","rescore_all_solutions":false,"group_id":35,"created_at":"2017-10-13T20:03:45.000Z","updated_at":"2025-11-04T13:12:51.000Z","published_at":"2017-10-16T01:51:02.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"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\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBuilding off of\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/44368\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 44368\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, your function will be provided with the five vertices of a regular pentagon (p) as well as the center point (cp) and radius (r) of a circle. The function should return one of the following values:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[0: the pentagon is completely enclosed within the circle but is not inscribed\\n1: the pentagon is inscribed in the circle (within ±0.02)\\n2: the vertices of the pentagon extend beyond the circle, but its edges still cross back into the circle\\n3: the pentagon circumscribes the circle (within ±0.02)\\n4: the pentagon completely encloses, and does not touch, the circle]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePoints will be rounded to the nearest hundredth. See the test cases for examples.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":1651,"title":"Circumcircle Points","description":"Determine the radius of the minimum sized circle that encompasses all the points.\r\n\r\nPer \u003chttp://www.inf.ethz.ch/personal/gaertner/texts/own_work/esa99_final.pdf Smallest Sphere Paper\u003e this question was first addressed by Sylvester in 1857, Megiddo in 1982, and optimized by Emo Welzl in 1991.\r\n\r\n*Input:* Points  (eg [0 0;0 1;2 0])  Minimum of 3 points\r\n\r\n*Output:* r  (radius of optimally centered circle)\r\n\r\n*Example:*  [0 0;0 1;2 0]  yields [xc,yc,r] [1 .5 1.118] Output r=1.118\r\n\r\n*Theory:*\r\nBest Circumcircle may occur in two ways:\r\n\r\n1) Center of Line connecting pair with maximum separation, or\r\n\r\n2) A circle utilizing three points from the set\r\n\r\n*Related Challenges:*\r\n\r\n\u003chttp://www.mathworks.com/matlabcentral/cody/problems/1336-geometry-find-circle-given-3-non-colinear-points/solutions/map Circle from 3 Points\u003e\r\n\r\n\u003chttp://www.mathworks.com/matlabcentral/cody/problems/554-is-the-point-in-a-circle/solutions/map Are Points in Circle\u003e\r\n\r\n*Warning:* Rounding Errors may cause solution errors. Usage of 10*eps(r) may be appropriate.\r\n\r\n\r\n","description_html":"\u003cp\u003eDetermine the radius of the minimum sized circle that encompasses all the points.\u003c/p\u003e\u003cp\u003ePer \u003ca href = \"http://www.inf.ethz.ch/personal/gaertner/texts/own_work/esa99_final.pdf\"\u003eSmallest Sphere Paper\u003c/a\u003e this question was first addressed by Sylvester in 1857, Megiddo in 1982, and optimized by Emo Welzl in 1991.\u003c/p\u003e\u003cp\u003e\u003cb\u003eInput:\u003c/b\u003e Points  (eg [0 0;0 1;2 0])  Minimum of 3 points\u003c/p\u003e\u003cp\u003e\u003cb\u003eOutput:\u003c/b\u003e r  (radius of optimally centered circle)\u003c/p\u003e\u003cp\u003e\u003cb\u003eExample:\u003c/b\u003e  [0 0;0 1;2 0]  yields [xc,yc,r] [1 .5 1.118] Output r=1.118\u003c/p\u003e\u003cp\u003e\u003cb\u003eTheory:\u003c/b\u003e\r\nBest Circumcircle may occur in two ways:\u003c/p\u003e\u003cp\u003e1) Center of Line connecting pair with maximum separation, or\u003c/p\u003e\u003cp\u003e2) A circle utilizing three points from the set\u003c/p\u003e\u003cp\u003e\u003cb\u003eRelated Challenges:\u003c/b\u003e\u003c/p\u003e\u003cp\u003e\u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/1336-geometry-find-circle-given-3-non-colinear-points/solutions/map\"\u003eCircle from 3 Points\u003c/a\u003e\u003c/p\u003e\u003cp\u003e\u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/554-is-the-point-in-a-circle/solutions/map\"\u003eAre Points in Circle\u003c/a\u003e\u003c/p\u003e\u003cp\u003e\u003cb\u003eWarning:\u003c/b\u003e Rounding Errors may cause solution errors. Usage of 10*eps(r) may be appropriate.\u003c/p\u003e","function_template":"function r = Circumcircle_radius(pts)\r\n  r=0;\r\nend","test_suite":"%%\r\npts=[0 0;5 0;1.8 2.4]; % 3 4 5 triangle\r\nr_exp=2.5;\r\nr = Circumcircle_radius(pts);\r\nassert(abs(r-r_exp)\u003c.001)\r\n%%\r\npts=[0 0;6 0;1.8 2.4]; % 3 x 6 triangle\r\nr_exp=3; % Two Point Solver\r\nr = Circumcircle_radius(pts);\r\nassert(abs(r-r_exp)\u003c.001)\r\n%%\r\npts=[0 0;0 1;1 2;3 0]; % r^2=2.5\r\nr_exp=sqrt(2.5);\r\nr = Circumcircle_radius(pts);\r\nassert(abs(r-r_exp)\u003c.001)\r\n%%\r\npts=[0 1; 0 3; 0 4; 2 6; 3 0; 4 5]; % r2 9.2820069 \r\nr_exp=sqrt(9.2820069 );\r\nr = Circumcircle_radius(pts);\r\nassert(abs(r-r_exp)\u003c.001)\r\n%%\r\npts=[0,2;0,6;1,1;3,0;3,3;4,10;5,10;7,2;9,7]; % r2 26.6919 \r\nr_exp=sqrt(26.6919420552286 );\r\nr = Circumcircle_radius(pts);\r\nassert(abs(r-r_exp)\u003c.001)\r\n%%\r\npts=[0,19;1,25;1,30;1,34;3,11;4,30;8,17;9,6;11,44;12,45;15,46;21,0;21,9;21,48;22,42;26,11;31,40;34,27;37,44;39,34;41,8;43,9;43,10;46,16;46,35;48,23]; % r2  exp 608.7807\r\nr_exp=sqrt(608.780718525455);\r\nr = Circumcircle_radius(pts);\r\nassert(abs(r-r_exp)\u003c.001)\r\n%%\r\n% Random case to avoid hard coders\r\nxc=rand;\r\nyc=rand;\r\nr=.5+rand;\r\npts=[];\r\n% Equilateral points\r\npts(1,:)=[xc+r,yc];\r\npts(2,:)=[xc+r*cos(2*pi/3),yc+r*sin(2*pi/3)];\r\npts(3,:)=[xc+r*cos(-2*pi/3),yc+r*sin(-2*pi/3)];\r\nfor i=4:10\r\n rnew=rand*r;\r\n theta=randi(360)*pi/180;\r\n pts(i,:)=[xc+rnew*cos(theta),yc+rnew*sin(theta)];\r\nend\r\npts=pts(randperm(size(pts,1)),:);\r\nr_exp=r;\r\nr = Circumcircle_radius(pts);\r\nassert(abs(r-r_exp)\u003c.001)\r\n%%\r\n% Random case to avoid hard coders\r\nxc=rand;\r\nyc=rand;\r\nr=.5+rand;\r\npts=[];\r\n% Equilateral points\r\npts(1,:)=[xc+r,yc];\r\npts(2,:)=[xc+r*cos(2*pi/3),yc+r*sin(2*pi/3)];\r\npts(3,:)=[xc+r*cos(-2*pi/3),yc+r*sin(-2*pi/3)];\r\nfor i=4:30\r\n rnew=rand*r;\r\n theta=randi(360)*pi/180;\r\n pts(i,:)=[xc+rnew*cos(theta),yc+rnew*sin(theta)];\r\nend\r\npts=pts(randperm(size(pts,1)),:);\r\nr_exp=r;\r\nr = Circumcircle_radius(pts);\r\nassert(abs(r-r_exp)\u003c.001)\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":3097,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2013-06-17T00:11:46.000Z","updated_at":"2013-06-17T00:27:11.000Z","published_at":"2013-06-17T00:27:11.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"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\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDetermine the radius of the minimum sized circle that encompasses all the points.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePer\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.inf.ethz.ch/personal/gaertner/texts/own_work/esa99_final.pdf\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eSmallest Sphere Paper\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e this question was first addressed by Sylvester in 1857, Megiddo in 1982, and optimized by Emo Welzl in 1991.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eInput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e Points (eg [0 0;0 1;2 0]) Minimum of 3 points\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eOutput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e r (radius of optimally centered circle)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e [0 0;0 1;2 0] yields [xc,yc,r] [1 .5 1.118] Output r=1.118\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eTheory:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e Best Circumcircle may occur in two ways:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e1) Center of Line connecting pair with maximum separation, or\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e2) A circle utilizing three points from the set\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eRelated Challenges:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/1336-geometry-find-circle-given-3-non-colinear-points/solutions/map\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCircle from 3 Points\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/554-is-the-point-in-a-circle/solutions/map\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eAre Points in Circle\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eWarning:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e Rounding Errors may cause solution errors. Usage of 10*eps(r) may be appropriate.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":44817,"title":"Wrecked Angles?","description":"It's time for some simple geometry fun to start off the new year.\r\n\r\nYou will be given the perimeter P and the area A of a rectangle.  With these two values, calculate the area of the circle that circumscribes this rectangle.  This means we're looking for the area of the circle that touches this rectangle only at the four vertices of the rectangle.\r\n\r\nGood luck, and Happy 2019!","description_html":"\u003cp\u003eIt's time for some simple geometry fun to start off the new year.\u003c/p\u003e\u003cp\u003eYou will be given the perimeter P and the area A of a rectangle.  With these two values, calculate the area of the circle that circumscribes this rectangle.  This means we're looking for the area of the circle that touches this rectangle only at the four vertices of the rectangle.\u003c/p\u003e\u003cp\u003eGood luck, and Happy 2019!\u003c/p\u003e","function_template":"function y = wrecked_angles(P,A)\r\n  y = x;\r\nend","test_suite":"%%\r\nP=14;\r\nA=12;\r\ny=wrecked_angles(P,A);\r\ny_correct = 20.63495408493621;\r\njunk=abs(y-y_correct);\r\nassert(junk-1\u003c1e-10);\r\n%%\r\nP=34;\r\nA=60;\r\ny=wrecked_angles(P,A);\r\ny_correct = 131.7322896141688;\r\njunk=abs(y-y_correct);\r\nassert(junk-1\u003c1e-10);\r\n%%\r\nP=62;\r\nA=168;\r\ny=wrecked_angles(P,A);\r\ny_correct = 590.8738521234052;\r\njunk=abs(y-y_correct);\r\nassert(junk-100\u003c1e-10);\r\n%%\r\ns1=100;\r\ntotalsum=zeros(1,s1);\r\nfor s2=1:s1\r\n    P=2*(s1+s2);\r\n    A=s1*s2;\r\n    totalsum(s2)=wrecked_angles(P,A);\r\nend\r\ns=sum(totalsum);\r\ns_correct=1051137.631982975;\r\ns_junk=abs(s-s_correct);\r\nassert(s_junk\u003c1e-8);\r\n\r\nd=max(totalsum)-min(totalsum);\r\nd_correct=7853.196235811095;\r\nd_junk=abs(d-d_correct);\r\nassert(d_junk\u003c1e-8);\r\n%%\r\ns1=wrecked_angles(32,64);\r\ns2=wrecked_angles(72,288);\r\nP=2*(s1+s2);\r\nA=s1*s2;\r\ny=wrecked_angles(P,A);\r\ny_correct=259088.4479405854;\r\njunk=abs(y-y_correct);\r\nassert(junk\u003c1e-10);","published":true,"deleted":false,"likes_count":0,"comments_count":5,"created_by":1615,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":21,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":72,"created_at":"2019-01-03T14:27:57.000Z","updated_at":"2026-01-21T13:24:31.000Z","published_at":"2019-01-03T14:27:57.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIt's time for some simple geometry fun to start off the new year.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYou will be given the perimeter P and the area A of a rectangle. With these two values, calculate the area of the circle that circumscribes this rectangle. This means we're looking for the area of the circle that touches this rectangle only at the four vertices of the rectangle.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGood luck, and Happy 2019!\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":44386,"title":"Circumscribed Pentagon?","description":"Building off of \u003chttps://www.mathworks.com/matlabcentral/cody/problems/44368 Problem 44368\u003e, your function will be provided with the five vertices of a regular pentagon (p) as well as the center point (cp) and radius (r) of a circle. The function should return one of the following values:\r\n\r\n  0: the pentagon is completely enclosed within the circle but is not inscribed\r\n  1: the pentagon is inscribed in the circle (within ±0.02)\r\n  2: the vertices of the pentagon extend beyond the circle, but its edges still cross back into the circle\r\n  3: the pentagon circumscribes the circle (within ±0.02)\r\n  4: the pentagon completely encloses, and does not touch, the circle\r\n\r\nPoints will be rounded to the nearest hundredth. See the test cases for examples.","description_html":"\u003cp\u003eBuilding off of \u003ca href = \"https://www.mathworks.com/matlabcentral/cody/problems/44368\"\u003eProblem 44368\u003c/a\u003e, your function will be provided with the five vertices of a regular pentagon (p) as well as the center point (cp) and radius (r) of a circle. The function should return one of the following values:\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003e0: the pentagon is completely enclosed within the circle but is not inscribed\r\n1: the pentagon is inscribed in the circle (within ±0.02)\r\n2: the vertices of the pentagon extend beyond the circle, but its edges still cross back into the circle\r\n3: the pentagon circumscribes the circle (within ±0.02)\r\n4: the pentagon completely encloses, and does not touch, the circle\r\n\u003c/pre\u003e\u003cp\u003ePoints will be rounded to the nearest hundredth. See the test cases for examples.\u003c/p\u003e","function_template":"function y = circumscribed_pentagon(p,cp,r)\r\n  y = 0;\r\nend","test_suite":"%%\r\np = [0,5; 4.76,1.55; 2.94,-4.05; -2.94,-4.05; -4.76,1.55];\r\ncp = [0,0];\r\nr = 5;\r\ny_correct = 1;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [0,5.61; 5.40,1.69; 3.34,-4.66; -3.34,-4.66; -5.40,1.69];\r\ncp = [0,0];\r\nr = 5;\r\ny_correct = 2;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [0,6.18; 5.88,1.91; 3.63,-5.00; -3.63,-5.00; -5.88,1.91];\r\ncp = [0,0];\r\nr = 5;\r\ny_correct = 3;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [0,4.55; 4.28,1.44; 2.65,-3.59; -2.65,-3.59; -4.28,1.44];\r\ncp = [0,0];\r\nr = 5;\r\ny_correct = 0;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\ncp = [20,8];\r\np = [0,5; 4.76,1.55; 2.94,-4.05; -2.94,-4.05; -4.76,1.55] + repmat(cp,[5,1]);\r\nr = 5;\r\ny_correct = 1;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [20,13.61; 25.40,9.69; 23.34,3.34; 16.66,3.34; 14.60,9.69];\r\ncp = [20,8];\r\nr = 5;\r\ny_correct = 2;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [20,14.18; 25.88,9.91; 23.63,3.00; 16.37,3.00; 14.12,9.91];\r\ncp = [20,8];\r\nr = 5;\r\ny_correct = 3;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\ncp = [20,8];\r\np = [0,6.58; 6.42,1.92; 3.97,-5.63; -3.97,-5.63; -6.42,1.92] + repmat(cp,[5,1]);\r\nr = 5;\r\ny_correct = 4;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\ncp = [20,8];\r\np = [0,4.55; 4.28,1.44; 2.65,-3.59; -2.65,-3.59; -4.28,1.44] + repmat(cp,[5,1]);\r\nr = 5;\r\ny_correct = 0;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [23.66,11.42; 24.37,5.58; 19.05,3.10; 15.04,7.40; 17.89,12.54];\r\ncp = [20,8];\r\nr = 5;\r\ny_correct = 1;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [25.01,12.47; 25.98,4.58; 18.78,1.23; 13.37,7.03; 17.22,13.97];\r\ncp = [20,8];\r\nr = 5;\r\ny_correct = 4;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [23.27,11.12; 23.92,5.87; 19.12,3.63; 15.52,7.50; 18.08,12.13];\r\ncp = [20,8];\r\nr = 5;\r\ny_correct = 0;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [30.94,36.26; 35.61,27.09; 28.34,19.82; 19.17,24.49; 20.78,34.65];\r\ncp = [26.97,28.45];\r\nr = 8.75;\r\ny_correct = 1;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [32.54,38.78; 38.84,26.41; 29.02,16.59; 16.65,22.89; 18.83,36.61];\r\ncp = [26.97,28.45];\r\nr = 8.75;\r\ny_correct = 4;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [30.49,35.54; 34.69,27.29; 28.14,20.74; 19.89,24.95; 21.34,34.09];\r\ncp = [26.97,28.45];\r\nr = 8.75;\r\ny_correct = 0;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [26.97,34.06; 32.37,30.14; 30.31,23.79; 23.63,23.79; 21.57,30.14];\r\ncp = [26.97,28.45];\r\nr = 5;\r\ny_correct = 2;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))\r\n\r\n%%\r\np = [31.35,32.83; 32.49,25.64; 26.00,22.34; 20.85,27.48; 24.16,33.97];\r\ncp = [26.97,28.45];\r\nr = 5.01;\r\ny_correct = 3;\r\nassert(isequal(circumscribed_pentagon(p,cp,r),y_correct))","published":true,"deleted":false,"likes_count":2,"comments_count":4,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":64,"test_suite_updated_at":"2017-12-08T15:45:11.000Z","rescore_all_solutions":false,"group_id":35,"created_at":"2017-10-13T20:03:45.000Z","updated_at":"2025-11-04T13:12:51.000Z","published_at":"2017-10-16T01:51:02.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"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\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBuilding off of\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/44368\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 44368\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, your function will be provided with the five vertices of a regular pentagon (p) as well as the center point (cp) and radius (r) of a circle. The function should return one of the following values:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[0: the pentagon is completely enclosed within the circle but is not inscribed\\n1: the pentagon is inscribed in the circle (within ±0.02)\\n2: the vertices of the pentagon extend beyond the circle, but its edges still cross back into the circle\\n3: the pentagon circumscribes the circle (within ±0.02)\\n4: the pentagon completely encloses, and does not touch, the circle]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePoints will be rounded to the nearest hundredth. See the test cases for examples.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":1651,"title":"Circumcircle Points","description":"Determine the radius of the minimum sized circle that encompasses all the points.\r\n\r\nPer \u003chttp://www.inf.ethz.ch/personal/gaertner/texts/own_work/esa99_final.pdf Smallest Sphere Paper\u003e this question was first addressed by Sylvester in 1857, Megiddo in 1982, and optimized by Emo Welzl in 1991.\r\n\r\n*Input:* Points  (eg [0 0;0 1;2 0])  Minimum of 3 points\r\n\r\n*Output:* r  (radius of optimally centered circle)\r\n\r\n*Example:*  [0 0;0 1;2 0]  yields [xc,yc,r] [1 .5 1.118] Output r=1.118\r\n\r\n*Theory:*\r\nBest Circumcircle may occur in two ways:\r\n\r\n1) Center of Line connecting pair with maximum separation, or\r\n\r\n2) A circle utilizing three points from the set\r\n\r\n*Related Challenges:*\r\n\r\n\u003chttp://www.mathworks.com/matlabcentral/cody/problems/1336-geometry-find-circle-given-3-non-colinear-points/solutions/map Circle from 3 Points\u003e\r\n\r\n\u003chttp://www.mathworks.com/matlabcentral/cody/problems/554-is-the-point-in-a-circle/solutions/map Are Points in Circle\u003e\r\n\r\n*Warning:* Rounding Errors may cause solution errors. Usage of 10*eps(r) may be appropriate.\r\n\r\n\r\n","description_html":"\u003cp\u003eDetermine the radius of the minimum sized circle that encompasses all the points.\u003c/p\u003e\u003cp\u003ePer \u003ca href = \"http://www.inf.ethz.ch/personal/gaertner/texts/own_work/esa99_final.pdf\"\u003eSmallest Sphere Paper\u003c/a\u003e this question was first addressed by Sylvester in 1857, Megiddo in 1982, and optimized by Emo Welzl in 1991.\u003c/p\u003e\u003cp\u003e\u003cb\u003eInput:\u003c/b\u003e Points  (eg [0 0;0 1;2 0])  Minimum of 3 points\u003c/p\u003e\u003cp\u003e\u003cb\u003eOutput:\u003c/b\u003e r  (radius of optimally centered circle)\u003c/p\u003e\u003cp\u003e\u003cb\u003eExample:\u003c/b\u003e  [0 0;0 1;2 0]  yields [xc,yc,r] [1 .5 1.118] Output r=1.118\u003c/p\u003e\u003cp\u003e\u003cb\u003eTheory:\u003c/b\u003e\r\nBest Circumcircle may occur in two ways:\u003c/p\u003e\u003cp\u003e1) Center of Line connecting pair with maximum separation, or\u003c/p\u003e\u003cp\u003e2) A circle utilizing three points from the set\u003c/p\u003e\u003cp\u003e\u003cb\u003eRelated Challenges:\u003c/b\u003e\u003c/p\u003e\u003cp\u003e\u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/1336-geometry-find-circle-given-3-non-colinear-points/solutions/map\"\u003eCircle from 3 Points\u003c/a\u003e\u003c/p\u003e\u003cp\u003e\u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/554-is-the-point-in-a-circle/solutions/map\"\u003eAre Points in Circle\u003c/a\u003e\u003c/p\u003e\u003cp\u003e\u003cb\u003eWarning:\u003c/b\u003e Rounding Errors may cause solution errors. Usage of 10*eps(r) may be appropriate.\u003c/p\u003e","function_template":"function r = Circumcircle_radius(pts)\r\n  r=0;\r\nend","test_suite":"%%\r\npts=[0 0;5 0;1.8 2.4]; % 3 4 5 triangle\r\nr_exp=2.5;\r\nr = Circumcircle_radius(pts);\r\nassert(abs(r-r_exp)\u003c.001)\r\n%%\r\npts=[0 0;6 0;1.8 2.4]; % 3 x 6 triangle\r\nr_exp=3; % Two Point Solver\r\nr = Circumcircle_radius(pts);\r\nassert(abs(r-r_exp)\u003c.001)\r\n%%\r\npts=[0 0;0 1;1 2;3 0]; % r^2=2.5\r\nr_exp=sqrt(2.5);\r\nr = Circumcircle_radius(pts);\r\nassert(abs(r-r_exp)\u003c.001)\r\n%%\r\npts=[0 1; 0 3; 0 4; 2 6; 3 0; 4 5]; % r2 9.2820069 \r\nr_exp=sqrt(9.2820069 );\r\nr = Circumcircle_radius(pts);\r\nassert(abs(r-r_exp)\u003c.001)\r\n%%\r\npts=[0,2;0,6;1,1;3,0;3,3;4,10;5,10;7,2;9,7]; % r2 26.6919 \r\nr_exp=sqrt(26.6919420552286 );\r\nr = Circumcircle_radius(pts);\r\nassert(abs(r-r_exp)\u003c.001)\r\n%%\r\npts=[0,19;1,25;1,30;1,34;3,11;4,30;8,17;9,6;11,44;12,45;15,46;21,0;21,9;21,48;22,42;26,11;31,40;34,27;37,44;39,34;41,8;43,9;43,10;46,16;46,35;48,23]; % r2  exp 608.7807\r\nr_exp=sqrt(608.780718525455);\r\nr = Circumcircle_radius(pts);\r\nassert(abs(r-r_exp)\u003c.001)\r\n%%\r\n% Random case to avoid hard coders\r\nxc=rand;\r\nyc=rand;\r\nr=.5+rand;\r\npts=[];\r\n% Equilateral points\r\npts(1,:)=[xc+r,yc];\r\npts(2,:)=[xc+r*cos(2*pi/3),yc+r*sin(2*pi/3)];\r\npts(3,:)=[xc+r*cos(-2*pi/3),yc+r*sin(-2*pi/3)];\r\nfor i=4:10\r\n rnew=rand*r;\r\n theta=randi(360)*pi/180;\r\n pts(i,:)=[xc+rnew*cos(theta),yc+rnew*sin(theta)];\r\nend\r\npts=pts(randperm(size(pts,1)),:);\r\nr_exp=r;\r\nr = Circumcircle_radius(pts);\r\nassert(abs(r-r_exp)\u003c.001)\r\n%%\r\n% Random case to avoid hard coders\r\nxc=rand;\r\nyc=rand;\r\nr=.5+rand;\r\npts=[];\r\n% Equilateral points\r\npts(1,:)=[xc+r,yc];\r\npts(2,:)=[xc+r*cos(2*pi/3),yc+r*sin(2*pi/3)];\r\npts(3,:)=[xc+r*cos(-2*pi/3),yc+r*sin(-2*pi/3)];\r\nfor i=4:30\r\n rnew=rand*r;\r\n theta=randi(360)*pi/180;\r\n pts(i,:)=[xc+rnew*cos(theta),yc+rnew*sin(theta)];\r\nend\r\npts=pts(randperm(size(pts,1)),:);\r\nr_exp=r;\r\nr = Circumcircle_radius(pts);\r\nassert(abs(r-r_exp)\u003c.001)\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":3097,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2013-06-17T00:11:46.000Z","updated_at":"2013-06-17T00:27:11.000Z","published_at":"2013-06-17T00:27:11.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"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\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDetermine the radius of the minimum sized circle that encompasses all the points.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePer\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.inf.ethz.ch/personal/gaertner/texts/own_work/esa99_final.pdf\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eSmallest Sphere Paper\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e this question was first addressed by Sylvester in 1857, Megiddo in 1982, and optimized by Emo Welzl in 1991.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eInput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e Points (eg [0 0;0 1;2 0]) Minimum of 3 points\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eOutput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e r (radius of optimally centered circle)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e [0 0;0 1;2 0] yields [xc,yc,r] [1 .5 1.118] Output r=1.118\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eTheory:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e Best Circumcircle may occur in two ways:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e1) Center of Line connecting pair with maximum separation, or\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e2) A circle utilizing three points from the set\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eRelated Challenges:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/1336-geometry-find-circle-given-3-non-colinear-points/solutions/map\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCircle from 3 Points\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/554-is-the-point-in-a-circle/solutions/map\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eAre Points in Circle\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle 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