%PDF-1.3
%
1 0 obj
<<
/Type /Pages
/Count 8
/Kids [ 3 0 R 4 0 R 5 0 R 6 0 R 7 0 R 8 0 R 9 0 R 10 0 R ]
>>
endobj
2 0 obj
<<
/Producer (PyPDF2)
/Title (Solving\040geometric\040problems\040with\040the\040rotating\040calipers)
/Author (Godfried\040T\056\040Toussaint)
/Subject (Shamos\040\1331\135\040recently\040showed\040that\040the\040diameter\040of\040a\040convex\040n\055sided\040polygon\040could\040be\040computed\040in\040O\050n\051\040time\040using\040a\040very\040elegant\040and\040simple\040procedure\040which\040resembles\040rotating\040a\040set\040of\040calipers\040around\040the\040polygon\040once\056\040In\040this\040paper\040we\040show\040that\040this\040simple\040idea\040can\040be\040generalized\040in\040two\040ways\072\040several\040sets\040of\040calipers\040can\040be\040used\040simultaneously\040on\040one\040convex\040polygon\054\040or\040one\040set\040of\040calipers\040can\040be\040used\040on\040several\040convex\040polygons\040simultaneously\056\040We\040then\040show\040that\040these\040generalizations\040allow\040us\040to\040obtain\040simple\040O\050n\051\040algorithms\040for\040solving\040a\040variety\040of\040problems\040defined\040on\040convex\040polygons\056\040Such\040problems\040include\040\0501\051\040finding\040the\040minimum\055area\040rectangle\040enclosing\040a\040polygon\054\040\0502\051\040computing\040the\040maximum\040distance\040between\040two\040polygons\054\040\0503\051\040performing\040the\040vector\055sum\040of\040two\040polygons\054\040\0504\051\040merging\040polygons\040in\040a\040convex\040hull\040finding\040algorithms\054\040and\040\0505\051\040finding\040the\040critical\040support\040lines\040between\040two\040polygons\056\040Finding\040the\040critical\040support\040lines\054\040in\040turn\054\040leads\040to\040obtaining\040solutions\040to\040several\040additional\040problems\040concerned\040with\040visibility\054\040collision\054\040avoidance\054\040range\040fitting\054\040linear\040separability\054\040and\040computing\040the\040Grenander\040distance\040between\040sets\056)
/Keywords (01\040Jan\0401983\054\040Rotating\040calipers\054\040Convex\040polygon\054\040Polygon\054\040Polygon\040covering\054\040Regular\040polygon)
>>
endobj
3 0 obj
<<
/Type /Page
/MediaBox [ 0 0 612 792 ]
/Rotate 0
/Parent 1 0 R
/Resources <<
/ProcSet [ /PDF /Text ]
/ExtGState 12 0 R
/Font 14 0 R
>>
/Contents 20 0 R
>>
endobj
4 0 obj
<<
/Type /Page
/MediaBox [ 0 0 612 792 ]
/Rotate 0
/Parent 1 0 R
/Resources <<
/ProcSet [ /PDF /Text ]
/ExtGState 21 0 R
/Font 22 0 R
>>
/Contents 25 0 R
>>
endobj
5 0 obj
<<
/Type /Page
/MediaBox [ 0 0 612 792 ]
/Rotate 0
/Parent 1 0 R
/Resources <<
/ProcSet [ /PDF /Text ]
/ExtGState 26 0 R
/Font 28 0 R
>>
/Contents 29 0 R
>>
endobj
6 0 obj
<<
/Type /Page
/MediaBox [ 0 0 612 792 ]
/Rotate 0
/Parent 1 0 R
/Resources <<
/ProcSet [ /PDF /Text ]
/ExtGState 30 0 R
/Font 31 0 R
>>
/Contents 32 0 R
>>
endobj
7 0 obj
<<
/Type /Page
/MediaBox [ 0 0 612 792 ]
/Rotate 0
/Parent 1 0 R
/Resources <<
/ProcSet [ /PDF /Text ]
/ExtGState 33 0 R
/Font 34 0 R
>>
/Contents 35 0 R
>>
endobj
8 0 obj
<<
/Type /Page
/MediaBox [ 0 0 612 792 ]
/Rotate 0
/Parent 1 0 R
/Resources <<
/ProcSet [ /PDF /Text ]
/ExtGState 36 0 R
/Font 37 0 R
>>
/Contents 39 0 R
>>
endobj
9 0 obj
<<
/Type /Page
/MediaBox [ 0 0 612 792 ]
/Rotate 0
/Parent 1 0 R
/Resources <<
/ProcSet [ /PDF /Text ]
/ExtGState 40 0 R
/Font 41 0 R
>>
/Contents 42 0 R
>>
endobj
10 0 obj
<<
/Type /Page
/MediaBox [ 0 0 612 792 ]
/Rotate 0
/Parent 1 0 R
/Resources <<
/ProcSet [ /PDF /Text ]
/ExtGState 43 0 R
/Font 44 0 R
>>
/Contents 45 0 R
>>
endobj
11 0 obj
<<
/Type /Catalog
/Pages 1 0 R
>>
endobj
12 0 obj
<<
/R7 13 0 R
>>
endobj
13 0 obj
<<
/Type /ExtGState
/TR [ /Identity /Identity /Identity /Identity ]
/OPM 1
>>
endobj
14 0 obj
<<
/R11 15 0 R
/R8 16 0 R
/R9 18 0 R
/R10 19 0 R
>>
endobj
15 0 obj
<<
/BaseFont /Symbol
/Type /Font
/Subtype /Type1
>>
endobj
16 0 obj
<<
/BaseFont /Times-Roman
/Type /Font
/Encoding 17 0 R
/Subtype /Type1
>>
endobj
17 0 obj
<<
/Type /Encoding
/Differences [ 210 /quotedblleft /quotedblright 213 /quoteright ]
>>
endobj
18 0 obj
<<
/BaseFont /Times-Bold
/Type /Font
/Subtype /Type1
>>
endobj
19 0 obj
<<
/BaseFont /Times-Italic
/Type /Font
/Subtype /Type1
>>
endobj
20 0 obj
<<
/Filter /FlateDecode
/Length 2233
>>
stream
xXɒ+6-`#Q;t(Ba!L;?~Y ؔbU˗l!MُC3е)} _m{ئ}9LIrys#ffL9y.ޭ׳N#x`亁hl ZZnڑ** \6J*_2tT͎e#^Qdj/_?~vз\:=R2uֵPEc.\xp|a2rC|6UdHViAeP8W^pQ0,JlNE&s_TL\V|w?֟?|\30iZxq ZzLbR*5*\#AH+LnEѐ(RU$2i[I:TQF"mk.
UD,CU^`l1W⥭,d325ǒ#\8v>RfM,B(vRSYbgo6绎 T;{]r CH/YmMMIKf1S]:l[VhKt]ּrSQ 'ݷ4ؽB:õ;IfExKp6Z@d[I֦
?R!}m z=+; lˋ;G0Us5G)
tzVTI0T5MY-jNw؝j[O@OkG}C|Dn$ ePoaџhu`2-V@y&E:x[y$":*GV*=V/*ϱ0'w8*z̩By%Bnfs p^TɩA]u5LS
~d3Xr+!ziyh:Lۄ;9ֳ`1P{