Robustness in Geometric Computations

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Robustness in Geometric Computations

• Christoph M. Hoffmann

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??sa????

• ? s?st? a??pt??? ????sµ???? p???p???te? t?? ???s?
  t?? ßas???? st?at?????? t?? d?ast??µ?t?s??
• ? st???? t?? d?ast??µ?t?s?? e??a? ?a d?a?e???ste?
  t?? p???p????t?ta ?a? ?a ep?t??e?
  epa?a???s?µ?p???s? t?? tµ?µ?t?? ??d??a ?a? ?ts?
  ?a µe??se? t?? p??a??t?ta ??????

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??sa????

• ??t? ? µ???d?? e??a? t?s? ßas??? ?ste ?? p??
  p????? p????aµµat?st?? de? t?? ?aµß????? ?a?????
  ?p ???? ?a? as??????ta? µe ta epa??µe?a
  ?????sµata ?p?? p???p????t?te? ?a? a????e???t?s?
  ??d??a

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??sa????

• St?? a??pt??? ?e?µet????? efa?µ????, ?st?s?,
  p??pe? ?a a?a?e???s??µe ta ßas??? tµ?µata
  a??pt???? ??d??a ?? ??µµ?t? t?? a??????µ????
  st?at?????? ?a? p??pe? ?a epa?e?et?s??µe
  ?p???se?? p?? ?e?????ta? p???? a?t?µata

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??p??? ??ast??µ?t?s?

• St?? pe??pt?s? t?? t?µ?? p????d???, t? tµ?µata
  ??d??a e?te????ta? ap? ta p??? ?e???? sta p???
  s???e???µ??a p?? a?af????ta? se s???e???µ?????
  t??p??? a?apa??stas?? ste?e??. ?a ep?peda ?a
  µp????sa? ?a e??a? ?? e???

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??p??? ??ast??µ?t?s?

• ??a?e???s? µ??µ?? ?a? ßas???? ?p??es?e?
  s?st?µat??. ?as???? d?µ?? ded?µ???? ?p?? ???f??
  ?a? ??ste?
• ?p??? p???e?? ?p?? p??s?es? ?a? afa??es? p??????
  es?te???? ????µe?? ?a? ta????µ?s? s????aµ????
  s?µe??? ?.?.?.
• S???ete? p???e?? ?p?? e??es? e??? ep?p?d?? ap?
  s?µe?a, ??e???? ??a t? a? ??a s?µe?? a???e? se
  ??a ep?ped?, t?µ? e??e?a? / ep?p?d?? ?.?.?.

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??p??? ??ast??µ?t?s?

• ?????te?e? e?????e? ?p?? t?µ? faces, d??s??s?
  ????? (boundary traversal), d?a?e???s? t?p?????a?
• ????e?? tµ?µ?t?? ?p?? t?µ? ?e??f?? ?a?, ?t????ta?
  p??? se a?t?, ? a??????µ?? t?µ?? ? ?d???
• ????e?ta? ?t? ta ?aµ???te?a ep?peda p??pe? ?a
  e??a? e? ????????? ?µp?sta

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??p??? ??ast??µ?t?s?

• ?? p??ß??µa e??a? ?t? de? e??a? t?s? e????? ?a
  ???s?µ?p????e? ? d?ast??µ?t?s? st??? ?e?µet??????
  ?p?????sµ???.

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??p??? ??ast??µ?t?s?

• ??t? ??at? ?p???e? ste?? a????ep?d?as? µeta?? t??
  a???µ?t???? ?p?????sµ?? ap? t? µ?a ?a? ap? t??
  ???? t?? d?µ?? ded?µ???? p?? a?apa??st??? ta
  s?µpe??sµata t?? ?e?µet????? ????sµ??.

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?? ???ß??µa ??? ????st?a?

• ?st? ?t? ?????µe ?a ?p?????s??µe t?? t?µ? t??
  pa?a??t? d?? p????d???

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?? ???ß??µa ??? ????st?a?
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?? ???ß??µa ??? ????st?a?

• ?? ?p???s??µe ?t? st?? p??? ep?f??e?a ? ????a
  e??a? µe???? ?a? st?? µp??st? e??a? µ????, t?te
  e??a? p??a?? ?a ?e????e? ap? t? s?st?µa ?t? ?
  t?µ? st?? p??? ep?f??e?a a???e? st? es?te???? t??
  ep?f??e?a?, e?? ? t?µ? st?? µp??st? ep?f??e?a ?a
  a???e? st?? a?µ? t??. ?ts? µp??e? ?a p?????e? t?
  e???

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?? ???ß??µa ??? ????st?a?
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?? ???ß??µa ??? ????st?a?

• ? µe???? d?s????a st? ?a ?????µe ??a s?st?µa
  e???st? ???e?ta? st? ?t? ßas???µaste se
  a???µ?t????? ?p?????sµ??? ??a ?a ?p?????s??µe
  ?e?µet???? ?e????ta. ? a?a???ße?a ?µ?? t??
  ?p?????sµ?? a?t?? ???e? ta s?µpe??sµata µ?
  a???p?sta.

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?? ???ß??µa ??? ????st?a?

• ?p?s?? t? ?d?? ?e?µet???? p??ß??µa µp??e? ?a
  p?????e? p????? f???? µe ???a ???µe?a ?a? ?a
  p????µe a?t?fat???? apa?t?se??.
• ? s?ßa??t?ta t?? p??ß??µat?? e??a? s????t?s? t??
  p???p????t?ta? t?? ?e?µet????? a?t??e?µ???? p??
  ???s?µ?p????µe.

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??p??e? ???se???se?? ??se??

• ?p e??e?a? µp????µe ?a a?a?????s??µe µe?????
  st?at??????
• 1. ???s? a???ß??? a???µ?t????
• 2. ???s? s?µß?????? ???????
• 3. ???s? a???p?st?? ?p?????sµ??

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????ß?? ????µ?t???

• ?d? ?a? pa?ap??? ap? 30 ?????a ? Macsyma
  ???p???se a???ß? a???µ?t??? ??a µp??e? ?a
  ???s?µ?p????e? ?e????. ?p?????? ?µ?? 2
  p??ß??µata
• 1. ???d?s? (Proliferation)
• 2. Irrationality

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Proliferation

• ?? ? e?s?d?? se µ?a ?e?µet???? p???? ??e?
  a???ße?a t?? k-bit t? ap?t??esµa µp??e? ?a
  apa?te? µe?a??te?? a???ße?a.
• G?a pa??de??µa a? fa?tast??µe t?? t?µ? d??
  e??????µµ?? tµ?µ?t?? ?p?? fa??eta? st? ep?µe??
  s??µa

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Proliferation
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Irrationality

• ?e????? p???e?? ?d????? se s??teta?µ??e? p?? de?
  ????? pepe?asµ??? a???ße?a.
• ??a pa??de??µa e??a? ??a? µ??ad?a??? ??ß?? p??
  t?? pe??st??f??µe ?at? 45 µ???e?.

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????ß?? ????µ?t???

• ?p???e? t??p?? ?a ???p????e? a???µ?t??? ?pe????
  a???ße?a? a??? e??a? p??? a???ß?? ??a ?a
  ???s?µ?p????e? sa? ?e????? t??p?? sta s?st?µata
  CAD.
• ?p?????? ?µ?? s???e???µ??e? efa?µ???? p??
  µe?????? t?? apa?t??µe?? a???ße?a

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??µ? ???????µµ?? ?µ?µ?t??

• ?st? ?t? ????µe t? tµ?µa S1 µe s?µe?a ta
  A1(x1,y1), B1(x2,y2) ?a? t? S2 µe s?µe?a ta
  A2(x3,y3), B2(x4,y4)

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??e???? ??µ??

• St?? p?? ap?? pe??pt?s? ep???µ??µe ?a d??µe ap???
  a? ta tµ?µata t?µ???ta? ????? ?a µa? e?d?af??e?
  t? s?µe?? t?µ??.
• ?a s?µe?a t?? S1 d????ta? pa?aµet???? ap? t??
  e??s?s? t1A1(1-t1)B1 ?p?? 0ltt1lt1. ?? ta
  tµ?µata t?µ???ta? t?te t? ??aµµ??? s?st?µa.
• t1A1(1-t1)B1t2A2(1-t2)B2
• ??e? ??s? µe 0ltt1,t2lt1.

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??e???? ??µ??

• ?? ?p???e? ??s?, t?te e??a? t?? µ??f??
• t1 D1/D0
• t2 D2/D0
• ?e D1 1 x2 y2
• 1 x3 y3
• 1 x4 y4

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??e???? ??µ??

• D2 1 x4 y4
• 1 x2 y2
• 1 x1 y1
• D0 1 x1 y1
• 1 x2 y2
• 1 x3 y3

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??e???? ??µ??

• G?a ?a d?ap?st?s??µe e?? ta t1, t2 ?????
  pepe?asµ??e? t?µ?? ?a? e??a? st? ?at??????
  d??st?µa, e?et????µe t? p??s?µ? t?? ??????s?? ?a?
  µe????? ?at?????e? d?af???? a?t??. ?ts?
• ?? D00, t?te ta tµ?µata e??a? pa??????a ?
  s????aµ???. ?? ?p?pe??pt?se?? e??a?

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??e???? ??µ??

• ?) ?? D1?0, t?te ta 2 tµ?µata de? e??a?
  s????aµ??? ?a? ??a de? t?µ???ta?.
• ?) ?? D10 ?a? x1x2, d??ad? a? ta tµ?µata e??a?
  ???eta, t?te ?p???e? t?µ? a?? y1?y3?y2 ? a??
  y1?y4?y2 ? a?? y3?y1 ?y4 ? a?? y3?y2?y4.
• G) ?? D10 ?a? ta tµ?µata de? e??a? ???eta, t?te
  e??????µe t?? x s??teta?µ??e? ?p?? st? ?) ?
  a????? t?? y s??teta?µ??e? a?????a, s?µf??a µe
  t?? ???s? t?? tµ?µ?t??.

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??e???? ??µ??

• ?? D0?0 t?te ta 2 tµ?µata de? e??a? pa??????a.
  ?p???t??ta? ?t? D0gt0, ?p????????µe t? d??st?µa
  t?? t1 ?? e???
• If (D1lt0) assert(t1lt0)
• else if (D1 0) assert(t1 0)
• else if (D1 D0 lt 0) assert( 0 lt t1 lt1 )
• else assert(t1gt1)

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??e???? ??µ??

• ?? D0lt0 t?te ? ??d??a? p??pe? ?a a??a??e?
  a?????a.
• ?e pa??µ??? t??p? ?p????????µe t? d??st?µa t??
  t2.
• ?a?at????µe ?t? ? d?ad??as?a ap?fas?? ??e?
  a?a??e? st? ?a ?p????????µe t? p??s?µ? t??
  ??????s?? ?a? t?? d?af???? t???.

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??e???? ??µ??

• ??a?t?a? t?? µ??f?? t?? ??????s?? a?t??, ???? ??
  ?p?????sµ?? p??s?µ?? e??a? t?? µ??f??
• Sxkyk Sxjyj.
• A? ?p???s??µe ?t? ?? s??teta?µ??e? e?s?d?? e??a?
  p?a?µat???? a???µ?? ap??? a???ße?a?, ta ????µe?a
  µp????? ?a ?p?????st??? a???ß?? µe a???µ?t???
  d?p??? a???ße?a?. ? a??????µ?? ESSA ?p??????e? t?
  p??s?µ? e??? t?t???? a????sµat?? a???ß??.

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? ???????µ?? ESSA

• ?p??????eta? t? p??s?µ? e??? a????sµat?? t??
  µ??f??
• s Sai - Sbi
• µe ta a ?a? b ?et??? µe a???µ?t??? d?p???
  a???ße?a?. ? ?d?a ???e?ta? st? ?a afa??e?ta? ?
  ?d?a p?s?t?ta ap? ta a ?a? b ?ts? ?ste ?a µ??
  a????e? t? p??s?µ? t?? s.

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? ???????µ?? ESSA

• ?? ???? se ???e ?????sµa ta????µ???ta? ap? t?
  µe?a??te?? st? µ????te??. ? p?s?t?ta u ep????eta?
  µe t?t??? t??p? ?ste ?? d?af???? a1-u ?a? b1-u ?a
  e??a? ?d?e?. ?p?s?? ??a? ap? t??? ????? µe???eta?
  st? 0 se ??a ? pe??ss?te?a ß?µata ?ts? te???? ta
  a????sµata µetas??µat????ta? se t?t??a µ??f? ?ste
  ??a ap? a?t? ?a e??a? ?de??. Se a?t? t? s?µe?? t?
  p??s?µ? t?? s e??a? ???st?. ???????e? ?
  a??????µ??.

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? ???????µ?? ESSA
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?p????????ta? ??? ??µ??

• ?? de? ?????µe ap?? ?a d??µe a? ?p???e? µ??? t?µ?
  a??? ?????µe ?a? ?a t?? ?p?????s??µe, t?te ? ESSA
  de? e??a? a??et??.
• ?? ?p???s??µe ?t? ta d?? tµ?µata t?µ???ta? se ??a
  µ??? s?µe?? ?a? a?t? e??a? t? p(x,y).

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?p????????ta? ??? ??µ??

• ??te t? p p??pe? ?a ??a??p??e? t?? pa?a??t?
  e??s?se??
• (y2-y1)x (x1-x2)y x1y2 x2y1
• (y4-y3)x (x3-x4)y x3y4 x4y3
• ? ??s? t?? s?st?µat?? e??a?
• xD1/D0
• yD2/D0

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?p????????ta? ??? ??µ??

• D1 x1y2 x2y1 x1 x2
• x3y4 x4y3 x3 x4
• D2 y2 y1 x1y2 x2y1
• y4 y3 x3y4 x4y3
• D0 y2 y1 x1 x2
• y4 y3 x3 x4

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?p????????ta? ??? ??µ??

• ?? ??????se? s??µat????ta? ap? a????sµata t??p???
  ????µ???? xixjyk ?a? xiyiyk. ???e ?????sµa ??e?
  µ???? ?a? 8 ?????. ?ts? a? ? a???ße?a e?s?d??
  e??a? t?? t???? t?? k bits, ta a????sµata
  apa?t??? 3k3 bits ??a ?a ?p?????s???? s?st? ?ta?
  d????ta? se a???a?a a???µ?t???. ?p?s?? p??pe? ?a
  p??s????µe ?t? ?t? t? a???a?? p???µa s?µe??? p??
  a?apa??sta?ta? e??a? ?µ???µ??f? e?? t? p???µa t??
  a?apa??st?µe??? p?a?µat???? a???µ?? de? e??a?.

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?p????????ta? ??? ??µ??

• S??µa 3

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?p????????ta? ??? ??µ??

• S??µa 4

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????ß?? ?s?te???? G???µe??

• S?µf??a µe t? IEEE ? p?a?µat???? a???µ?? ap???
  a???ße?a? ??e? ??a e??et??? µ???? t?? 8 bit ?a?
  ??a de?ad??? µ???? t?? 24 bit. ???te??eta? ?
  µetat??p? t?? se ??a? a???µ? µe de?ad??? µ????
  t?? 280 bit. ?? ep?t?????µe ?a? µe???? a??µa ??a
  ?a ?????? s?st? ?? p???ap?as?asµ?? ?a? ??
  p??s??se?? se ?????? p?a?s?a, ??a? s?ss??e?t??
  t?? 40 bytes µp??e? ?a ?p?????se? ??a es?te????
  ????µe?? µet???? µ????? a???ß?? s?ed?? se ??e?
  t?? pe??pt?se??. ? t?µ? p?? ??ate?ta? µp??e? ?a
  t?p??et??e? se ??a? p?a?µat??? a???µ? ap???
  a???ße?a? µe st???????p???s?.

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?e?????? ?p? ??? ??µ? ?µ?µ?t??

• ?? te?????? p?? pa???s??st??a? de? s?et????ta? µe
  t?? ?e???? st?at????? p?? ???s?µ?p??e?ta? ?ta?
  p???? tµ?µata t?µ???ta?. ??af????ta? µ??? se 2
  ?a? e??a? µ?a te????? p?? ?a ???p????se ??p????
  se µ?a ß?ß??????? ?a? ?a t? ?e????se ??t?te. ??
  ??st?? ?µ?? e??a? a??et? ?????.

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?e?????? ?p? ??? ??µ? ?µ?µ?t??

• ?? a??????µ?? t?µ?? tµ?µ?t?? se ß?ß??a apa?t???
  t?? ??e??? t?? ?s?t?ta? t?? s??teta?µ???? t??
  s?µe??? t?µ?? a??? ?e????? ?t? t? test e??a?
  tet??µµ??? t? ?p??? ?µ?? ?p?? e?daµe e??a?
  ?a??asµ??? p??s????s?.

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??µ? ?????d???

• ? t?µ? 2 p????d??? e??a? µ?a ?a?? pe??pt?s? ??a
  ?a µe?et??e? t? proliferation µ?a? ?a? ? t?µ?
  t??? e??a? p??? ??a p???ed?? ?a??? ?a? t?
  irrationality p?? p????pte? ap? t?? pe??st??f??.
• ?p?s??, ??a? ?a??? a??????µ?? t?µ?? p????d???
  µp??e? ?a epe?ta?e? e????a ?a? st?? t?µ? ste?e??
  µe ?aµp??e? ep?f??e?e?

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? ????d?? ??? Sugihara

• ?a ?e?µet???? st???e?a a?apa??sta?ta? ?????
  p?e??asµ??? ????ta? µ??? t??? s??te?est?? t??
  e??s?se?? t?? ep?p?d??. ?? s??teta?µ??e? t??
  vertices ?p????????ta? ?ta? ??e?????ta? ?a
  ?p?????st??? µe t?? ?d?? t??p? p?? ?p????????ta?
  st?? t?µ? tµ?µ?t??.

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? ????d?? ??? Sugihara

• ? ?e???? e??s?s? t?? ep?p?d?? e??a? axbyczd0
• d1 b1 c1
• Dx- d2 b2 c2
• d3 b3 c3

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? ????d?? ??? Sugihara

• a1 d1 c1
• Dy - a2 d2 c2
• a3 d3 c3
• a1 b1 d1
• Dz - a2 b2 d2
• a3 b3 d3

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? ????d?? ??? Sugihara

• a1 b1 c1
• Dw a2 b2 c2
• a3 b3 c3
• ?? s??teta?µ??e? t?? t?µ?? t?? t???? ep?p?d??
  e??a?
• (Dx / Dw, Dy / Dw, Dz / Dw)

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? ????d?? ??? Sugihara

• ?p?? ?a? ?p???t??µe ?t? Dw ? 0 d??ad? ta normals
  t?? ep?p?d?? e??a? ??aµµ??? a?e???t?ta.
• ?a a?apa??st?µe?a p???ed?a de? ??e???eta? ?a
  e??a? ???t? µ?a? ?a? ?? s??teta?µ??e? t?? s?µe???
  ?p????????ta? ?ta? ??e?????ta? ?? t?µ? ?e?t??????
  ep?p?d?? p?? ??????ta? ap? ta faces.

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? ????d?? ??? Sugihara

• ?ts? ?? a?µ?? µp????? ?a ???st??? t?p???????, ap?
  ta ???µata t?? s???e???µ???? s?µe???, ?a? ??
  ß????? a?µ?? p?? ??????? ??a face µp????? ?a
  ???st??? ?a?a?? t?p??????? ????a ?aµ?a a????? ??a
  s???e???µ??? p????f???a s??teta?µ????.
• ??e? ?? s??teta?µ??e? µp????? ?a ?p?????s???? ap?
  t?? e??s?se?? t?? ep?p?d?? ?p?? ??e???eta?.

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? ????d?? ??? Sugihara

• ?? ?e???s??µe ?t? ???s?µ?p????µe a???a????
  s??te?est?? t?? B bits, t?te ??e?????ta? pe??p??
  3B bits ??a t?? s??teta?µ??e? t?? s?µe??? ?????
  d?a??es?.
• ?a??te?a ?µ?? ?a ???s?µ?p???s??µe s??te?est?? d
  p?? ????? 2B bits µ?a? ?a? ?d????? se ??a p??
  ?µ???µ??f? p???µa ap? a?apa??st?µe?a ep?peda. Se
  a?t?? t?? pe??pt?s? ??e?a??µaste s??teta?µ??e?
  tet?ap??? a???ße?a?.

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? ????d?? ??? Sugihara

• ?? ?a? ??e?a??µaste tet?ap?? a???ße?a ??a t???
  ?p?????sµ??? µa?, t? te???? p???ed?? ???????µe?
  ??e? t?? ep?f??e??? t?? ap? ta a????? p???ed?a.
• ?p?te, t? te???? p???ed?? de? ??e???eta?
  µe?a??te?? a???ße?a ap? ta a????? ??a ?a
  pa?asta?e?.
• ?ts?, de? ????µe proliferation se a?t??es? µe t??
  t?µ? tµ?µ?t??.

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? ????d?? ??? Sugihara

• ?ts?, a? ?a? fa??eta? ?t? ? a?apa??stas? µe
  vertices ?a a??a?e e??et??? t?? a???ße?a ?p?? ?
  t?µ? e??????µµ?? tµ?µ?t??, ? a???s? t?? a???ße?a?
  s?µßa??e? µ??? µ?a f??? ?a? p?t? ?a??.

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??apa??stas? ?et? ?p? ?e??st??f?

• ? t?µ? d?? p????d??? µp??e? ?a ???e? s?st? µe
  a???ß? a???µ?t??? a? ta p???ed?a e?s?d?? e??a?
  s?st?.
• ?? pe??st?????µe ??a? ap?? ??ß? (µe e????ste?
  apa?t?se?? se a???ße?a) t?te ??te ?? s??te?est??
  t?? ??aµµ?? ??te ?? s??teta?µ??e? t?? s?µe???
  e??a? ??????? a???µ??.

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??apa??stas? ?et? ?p? ?e??st??f?

• ??a ap?? p???? st???????p???s?? ?a µp????se ?a
  µetat???e? ??a p???ed?? se ??a ?a??asµ??? µe
  a?t?teµ??µe?a st???e?a. ??t? s?µßa??e? ?ta?
  ?p?????? p??? µ???? ?a?a?t???st???.

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??apa??stas? ?et? ?p? ?e??st??f?

• Ta µp????saµe ?a ???s?µ?p???s??µe pe??st??f?? p??
  d?at????? t?? ??????t?ta t?? s??te?est?? ?a? t??
  s??teta?µ????.
• ??t??e? pe??st??f?? a?t?st?????? sta ??????
  s?µe?a t?? µ??ad?a??? ?????? p?? pa?????ta? ap?
  ??????? t?µ?? t?? t
• x(t)(1-t²)/(1t²)
• y(t)(2t)/(1t²)

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??apa??stas? ?et? ?p? ?e??st??f?

• O Sugihara ?a? ? Iri ap?fe????? t? p??ß??µa
  ?e????ta? ?t? ??a ste?e? a?apa??stata? ap? 2
  a?apa?ast?se??
• 1. ??apa??stas? ?????
• 2.CSG
• ???te????? ?a ?????ta? ?? pe??st??f?? sta 3ed?a
  s??µata p?? a?apa??st??? t? p???ed?? ta ?p??a
  pa?aµ????? s?st? (de? ????? µ???? st???e?a ?a?
  e??a? µe?a??te?a ap? ??a e????st? µ??e???), ?a?
  µet? ?a ???s?µ?p????µe t? CSG ??a ?a
  d?µ??????s??µe t? ste?e?.

57
? ????d?? ??? Fortune

• ?a?? ? ????a a?apa??stas? e??a? ? e??s?s?
  ep?p?d?? ax by cz d 0, ?p?? ??
  s??te?est?? e??a? a???a??? t?? B bit e?t?? ap? t?
  d t?? ?p???? ? a???ße?a e??a? 2B.
• Se a?t??es? µe t??? Sugihara ?a? Iri, ? Fortune
  a???eta? ta a?t?teµ??µe?a st???e?a sta p???ed?a
  ?? t? ap?t??esµa ????s?? ??aµpt?? s?µat??, a???
  te???? afa??e? t?? p????a p?? e??a? p??ta ??a
  s?st? p???ed??.

58
? ????d?? ??? Fortune
59
? ????d?? ??? Fortune
60
? ????d?? ??? Fortune
61
? ????d?? ??? Fortune

• ??a a???µ?t??? p???? ???µ??eta? ?at?????µa a? ?
  t?µ? t?? ep??e??e? t?? ??? e?t??es?? e???
  a??????µ?? ?a? ?atas?e?ast??(constructor) a? ?
  t?µ? t?? ???s?µ?p??e?ta? ??a ?a ???se? ?e?µet????
  ded?µ??a

62
? ????d?? ??? Fortune

• ? ap?t?µ?s? t?? ?at?????µ?t?? s?????? apa?te? t??
  ?a????sµ? p??s?µ?? a??? p??pe? ?a e??a? a???ß??.
• ? ap?t?µ?s? t?? ?atas?e?ast?? µp??e? ?a a?e??e?
  µe???? ???? e? ??e? t?? s??µat?? st???????p???s??
  p?? ap?sp? t?? p????a.

63
? ????d?? ??? Fortune

• ?a? ta ?at?????µata ?a? ?? ?atas?e?ast?? e??a?
  p??????µa a?e?a???. ??a? p??epe?e??ast?? e?et??e?
  ???e p??????µ? ?a? ?a?????e? p?te µ?a ap?t?µ?s?
  ??e? a??et? a???ße?a ??a ?a ap?t?µ??e? t?
  p??????µ? µe ßeßa??t?ta.
• ?as???µe??? se a?t?? t?? ?a????sµ?, ?
  p??epe?e??ast?? pa???e? ??d??a C p?? ???p??e?
  t?? ap?t?µ?s?.

64
?aµp??e? ?p?f??e?e?

• St?? pe??pt?se?? t?? e?????aµµ?? tµ?µ?t??
  tet?ap?? a???µ?t??? a???ße?a e??a? a??et? ??a ?a
  ap?t?µ????? a?t? p?? ??e?a??µaste.
• St?? de?te??ß??µ?e? ep?f??e?e?, a???ß? test
  s?µe??? / ep?fa?e?a? ?????ta? µ? p?a?µat?p???s?µa.

65
?aµp??e? ?p?f??e?e?

• ?? ?e???s??µe t?? de?te??ß??µ?a ep?f??e?a
• ax² by² cz² dxy eyz fzx gx hy iz
  j 0
• ?p?? ?? s??te?est?? e??a? a???a??? ?a?
• -L a,b,c,d,e,f L
• -L² g,h,i L²
• -L³ j L³

66
?aµp??e? ?p?f??e?e?

• ?????ta? ?p?????sµ??? ß??s???µe ?t? ??e?a??µaste
  L720 a???ße?a ??a ?a µp????µe ?a d?a????s??µe 2
  t?µ?? de?te??ß??µ??? ep?fa?e???. ??t? p??
  a???ste?e? t?? st?at????? t?? a???ß???
  a???µ?t????.
• ?p?????? ?µ?? ?a? µe????? st?at?????? p??
  ep?t??p??? t? ???s? p?? ap?d?t???? a???ß??
  µe??d??.

67
S?µß?????? ????sµ??

• ??e? pa?at????e? ?t? ta ?at?????µata apa?t??? se
  ?e?µet????? e??t?se??.
• ?ts? t? p??ß??µa t?? a?a?????s?? ?t? µ?a ap?fas?
  ?p???e?ta? ap? p??????µe?e? ap?f?se?? p?? ?????
  pa??e? ß??s?eta? st? ped?? t?? s?µß??????
  ???????.
• ?p?s?? µp????µe ?a a?e?t??µe µ?a ?a??asµ???
  ap?fas? se µ?a ???a?? pe??pt?s? ?s? e??a? s??ep??
  µe ??e? t?? ???e? ap?f?se??.

68
?p?f?se?? St?? ??µ? ?????d???

• ? t?µ? p????d??? ßas??eta? st?? ??e??? t??
  ?at?????µ?t?? ?t? 4 ep?peda t?µ???ta? se ??a
  s?µe??.
• ?? ?a?ad??µe t? s??µa 1, ?ta? ?? d?? a?µ?? e??a?
  p??? ???t? µp????µe ?a ap?fas?s??µe e?te t?µ?
  e?te µ? t?µ?.
• ?s? ? ap?fas? e??a? s??ep??, ?a ?d??e? se
  s??epe?? d?µ?? ded?µ????.

69
?p?f?se?? St?? ??µ? ?????d???
70
?p?f?se?? St?? ??µ? ?????d???

• ?? ?a p????µe s??epe?? ap?f?se?? apa?te? t? ?a
  e??µe??s??µe ??a t?? ap?fas? µa? ??e? t??
  ep??ea??µe?e? t?µ??.
• ?ts?, a? ap?fas?s??µe st? s??µa 1 ?t? ? t?µ? t??
  tet??ed??? µe t?? µp??st? ep?f??e?a ???eta? st??
  a?µ? t?? ??ß??, p??pe? ?a e??µe??s??µe ?a? t??
  ???? ep?f??e?a t?? ??ß??.

71
?p?f?se?? St?? ??µ? ?????d???

• ?µ?? de? ????µe ??e??? st?? se??? µe t?? ?p??a
  e??????ta? ?? ep?f??e?e?.
• ?ts? a?t? ?a st?????µe ap?f?se?? µ? t?µ?? ??t?
  p?? µe???e? t?? ap?d?s?, p??pe? ?a ß?e?e? ??a?
  t??p?? ?a a???????µe t?? ap?f?se?? µ? t?µ??.

72
?p?f?se?? St?? ??µ? ?????d???

• ??a ???? s???? e??a? ?a e?a??st?p????e? ?
  eµf???s? t?? ?s?t?µ?? ?at?????µ?t?? ?a? ?a
  µe?????? ??e? ?? ap?f?se?? se ?s? t? d??at??
  µ????te?? set ap? ?at?????µata ???eta?.
• ??t? µe???e? t?? e??a???e? ??a as???pe?e? p?? de?
  a?a????????ta?. ?ts? st?? ??sta ap?f?se?? p??
  a??????e? µe????? ap? a?t?? e??a? ????e? ?a?
  ?p????????ta? ap? ta ?e?µet???? ded?µ??a

73
?p?f?se?? St?? ??µ? ?????d???

• 1. u n f ? ??µß?? u p??pe? ?a e??a? a??et?
  ???t? st? ep?ped? p?? ????e? ? p?e??? f .
  ??t? e??a? µ?a ? - ???s?
• 2. e n e ?a e ?a? e p??pe? ?a e??a? a??et?
  ???t?. ??µ???µe t? e µe ??a ep?ped? ?e?t?????
  t?? e ?a? ep?ßeßa?????µe ?t? ? t?µ? ß??s?eta?
  se ??a ta ???a ep?peda ?e?t????? t?? e

74
?p?f?se?? St?? ??µ? ?????d???

• ?p?s?? ?p?????? ?a? de?te?e???te? ap?f?se??.
• 3. u n e f e??a? µ?a p?e??? ?e?t????? st? e. ??
  s?µe?? u p??pe? ?a e??a? p??? st? ep?ped?
  ap? ???e ?e?t????? p?e??? f.
• 4. u n u ? ??µß?? u p??pe? ?a ß??s?eta? p???
  se ???e a?µ? p?? ß??s?eta? ? u. ???ad? ? u
  p??pe? ?a ß??s?eta? se ???e ep?ped? p?e????
  p?? ß??s?eta? ? u ?a? a?t?st??fa

75
?p?f?se?? St?? ??µ? ?????d???

• 5. e on f ???? ?? ??µß?? st?? a?µ? p??pe? ?a
  ß??s???ta? st? ep?ped? p?e???? f
• 6. e s????aµ??? µe e ?st? f µ?a p?e???
  ?e?t????? st? e. ?? ??µß?? t?? e p??pe? ?a
  ß??s???ta? st? ep?ped? p?e???? ap? ???e
  ?e?t????? p?e??? f. ??t?st??fa, ?? ??µß?? st? e
  p??pe? ?a ß??s???ta? se ???e ep?ped? p?e???? f
  p?? ß??s?eta? st? e

76
?p?f?se?? St?? ??µ? ?????d???

• 7. e n f ??e???? a? ? a?µ? t?µ?e? t? ep?ped?
  p?e???? e????s?a, d??ad?, a? ta s?µe?a t?? e
  ß??s???ta? se a?t??ete? µe???? t?? f ? a? ??p???
  ap? a?t? ß??s?eta? p??? st? f e?? ??p??? ????
  ???.
• 8. f n f ???e s?µe?? st? f p??pe? ?a ß??s?eta?
  p??? st? ep?ped? ep?fa?e?a? t?? f ?a? ???e
  s?µe?? t?? f p??pe? ?a ß??s?eta? p??? st?
  ep?ped? ep?fa?e?a? t?? f.

77
?p?f?se?? St?? ??µ? ?????d???

• ?a??? p??st??e?ta? d?a???? s?µe?a ?a? ep?f??e?e?,
  p??pe? epa?a??pt??? ?a e?te???µe a?t??? t???
  e??????? ??a ?a ?p???e? s???pe?a.
• ?p?s?? ??a ?a ?p???e? s?µµet??a, p??pe? ?a ??????
  µe???? test µe p???ap??? ? - ???se??.

78
?p?f?se?? St?? ??µ? ?????d???

• ?p?????? e?de??e?? ?t? ? ???s? t?? pa?ap???
  ?d??e? se s??µata µe ??a ?a?? µ?t?? s???pe?a?,
  ?µ?? de? ?p???e? ??p??a ap?de??? ?t? ? te????
  ???p???s? ?a e??a? s?st?, ?a? e??a? ?e???? ???st?
  ?t? ??t? t?t??? de? ?s??e?.
• ? ?e?t???? d?s????a st?? p??s????s? t??
  ?e?µet????? ????sµ?? e??a? ?t? ?? ap?f?se??
  ßas????ta? st?? ?????a t?? e???t?ta?. G? a?t?
  ßas????ta? se pe??e????? ?a???e? ???????.

79
?p?f?se?? St?? ??µ? ?????d???
80
?p?f?se?? St?? ??µ? ?????d???
81
?p?????sµ?? ??ast?µ?t??

• St??? ?p?????sµ??? µe ??a? ?p?????st?, de?
  ???s?µ?p????µe p?t? a???ß?? a???µ??? a??? p??ta
  p??se???se??.
• ?? a???a??? ????? t??e?a a???ße?a a??? ????? t?
  p??ß??µa t?? ?pe??e???s??
• ? a???ß?? a???µ?t??? e??a? a???ß?? µ?a? ?a?
  p?????eta? ap? a???a?a a???µ?t??? a??? ?
  apa?t??µe?? a???ße?a e??a? p??? µe???? ?a?
  ?p?????? ?a? µ? ?????? ap?te??sµata.

82
?p?????sµ?? ??ast?µ?t??

• ?p? t?? f?s? t???, ?? p?a?µat???? a???µ?? e??a?
  pe??e???? ??????? a???µ?? µe ??a pe??e??? e????
  a?apa??stas?? ?p?? fa??eta? ?a? st? s??µa.
• G?a ?a d???????µe µe p?a?µat????? a???µ???
  ??????µ??? ???s?µ?p????µe ta p?a?µat???
  d?ast?µata (floating-point interval).

83
?p?????sµ?? ??ast?µ?t??

• ??a p?a?µat??? d??st?µa s?µß????eta? ??
• xx , x
• ?p?? x , x p?a?µat???? a???µ??.
• ?? d??st?µa a?t? pe???aµß??e? ????? t???
  p?a?µat????? a???µ??? µeta?? x , x
• xr ? R x ? r ? x

84

• ?as??? ??? ??ast?µ?t??

85
??a?µat??? ??ast?µata

• ? d??µet??? e??? d?ast?µat?? x e??a? d(x) , ?
  a?t??a e??a? r(x) ?a? t? µ?s? e??a? m(x) ?p??
• d(x) x x r(x) (x x) / 2
• m(x) (x x) / 2

86
??a?µat??? ??ast?µata

• ? e????st? ?a? ? µ???st? ap???t? t?µ? e???
  d?ast?µat?? e??a?
• ltxgt min x ? x
• x max x x ? x max (x,x)
• ?? 0 ? x a?? ltxgt 0

87
??a?µat??? ??ast?µata

• ? s?et??? d??µet??? µa? de???e? t?? p???t?ta µ?a?
  p??s????s?? ap? ??a d??st?µa. ????eta? ??
• Drel d(x)/ltxgt a? 0 ? x
• d(x) a?????

88
??a?µat??? ??ast?µata

• ?as??? a???µ?t???
• x y x y , x y
• x y x y , x y
• x y min(xy,xy,xy,xy), max(xy,xy,xy,
  xy)
• S??a?t?se?? F(x) F(x) x ? x

89
??a?µat??? ??ast?µata

• x2 ltxgt2 , x2
• e x e x , e x
• log(x) log(x) , log(x)
• xn xn , xn a? 0 lt x ? n
  pe??tt??
• 0, xn a? 0 ? x ?a? n
  ??t???
• xn, xn a? x lt 0 ?a? n
  ??t???

90
??a?µat??? ??ast?µata

• ?? ßas??? p??ß??µa µe t?? a???µ?t??? d?ast?µ?t??
  e??a? ? ?p?????sµ?? t?? e????? t?? ep??tas?? µ?a?
  s????t?s?? f se ??a d??st?µa x. ?? p????µe ??a
  p??????µ?
• f(x) a0 a1x a2x2 anxn
• ??te ? ap?t?µ?s? t?? d?ast?µat?? ?a ?ta?
• f(x) a0 a1x a2x2
  anxn

91
??a?µat??? ??ast?µata

• ?a?at????µe t?? e???e?sµ? f(x)? f(x)
• ??t? e??a? ??a p??ß??µa. S?????? t? p?s? µ????
  e??a? ? ??e?st?t?ta e??a? de??t?? p???t?ta?
  p????f???a?. ?a?a??? ??e?st?t?te? d????? ????
  p????f???a.

92
??a?µat??? ??ast?µata

• ??a ?e???? te????? ??a ?a µe???e? a?t? ?
  ?pe?e?t?µ?s? e??a? ? ?e?t???? µ??f? (centered
  form) p?? p?????eta? ap? t? ?e???µa t?? µ?s??
  t?µ??.
• ?? ? f e??a? pa?a????s?µ? st? D ?a? c,x ?
  x t?te
• f(x) ? f(c) f (x)(x-c)

93
???tetaµ??a ??ast?µata

• ?pe?te????µe t?? ?????a t?? d?ast?µat?? ??a ?a
  ep?t?????µe ?pe??a ???a. ??ast?µata µe ??a ? ?a?
  ta d?? ???a ?pe??a ?????ta? p??tetaµ??a
  d?ast?µata (extended intervals). ?? ßas????
  p???e?? ?????ta? ?a??????. St?? pe??pt?s? t??
  d?a??es?? ????µe

94
???tetaµ??a ??ast?µata

• x / y -8 , 8 if x lt 0 lt x or
  x 0 or y 0
• x/y, 8 if x 0 and y lt y 0
• -8 , x/y ? x/y , 8 if x0
  and ylt0lty
• -8, x/y if x0 and 0ylty
• -8, x/y if 0x and
  ylty0
• -8, x/y ? x/y , 8
  if 0x and ylt0lty
• x/y, 8 if 0x and 0ylty

95
???tetaµ??a ??ast?µata

• St?? e?d??? pe??pt?s? ?p?? t? x e??a? ??a thin
  interval (??t? ?a? ??? ???? ta?t????ta?), ?a? t?
  y ??e? t??????st?? ??a ?pe??? ????, ????µe
• x y -8 , 8 if y -8 , 8
• -8 , x y if y y,
  8
• x y , 8 if y -8 ,
  y

96
????µ?t??? ??? ??a?µat????

• ???p????µe t??? ?p?????sµ??? t?? d?ast?µ?t??
  ???s?µ?p????ta? floating point a???µ???.
• G?a ?a pa?aµe??e? ?µ?? ? ??e?st?t?ta p??pe? ??
  st???????p???se?? ?a ?????ta? s?st?.

97
????µ?t??? ??? ??a?µat????

• ? st???????p???s? p??? ta p??? (rounding up) e???
  a???µ?? r e??a? ? p?? ???t???? a???µ?? p?? µp??e?
  ?a a?apa?asta?e? ap? ?p?????st? ?(r)r p?? ?a µ??
  e??a? µ????te??? t?? r.
• ??t?st???a ????eta? ?a? ? st???????p???s? p??? ta
  ??t? (rounding down) ?? ?(r).

98
????µ?t??? ??? ??a?µat????

• ?? s?µß???s??µe µe ? ?a? ? t?? p??s??se?? p??
  st???????p????? t? ap?t??esµa p??? ta p??? ?a?
  ??t? a?t?st???a ?a? µe -? ?a? -? ?µ??a ??a t??
  afa???se?? ????µe
• x y x ? y , x ? y
• x y x -? y , x -? y

99
????µ?t??? ??? ??a?µat????

• ? ??ass???? t??p?? st???????p???s?? µetaß???e? t?
  µ??e??? t?? a???µ??.
• ??t? e??a? ????st? ??a t??? ?p?????sµ???
  d?ast?µ?t?? ??at? a?a??e? t?? ?d??t?ta t??
  ??e?st?t?ta? ?a? epa?e?s??e? t?? aßeßa??t?ta.
• ?e????? FPU ????? t?? ??a??t?ta ?a ?????? s?st??
  st???????p???se?? a??? apa?te?ta? ??a? e?d????
  assembler ??a ?a d?se? t?? ?at?????e? e?t???? ?a?
  ??a? compiler p?? ?a µ?? t?? d?a????e? ?? tµ?µa
  ße?t?st?p???s??.

100
Univariate Root Finding

• Ge???? t? p??ß??µa ??e? ???e? a??? ?? a??????µ??
  e??a? p???p?????.
• G?a p??????, ?? ?e?t??e?e? epa?a???e?? e??a? µ?a
  ?a?? ?a? ap?? ??s? p?? a? a?????p????e? ?at?????a
  ?a ß?e? µ?a ???a.
• ?d??ate? ?µ?? ?a ?e?t?????se? ?ta? ?p??????
  p????? ???e?. ?p?s??, ????ta? ß?e? µ?a ???a x,
  epa?a?aµßa??µe?e? d?a???se?? t?? p??????µ?? µe t?
  (x-x) s?ss??e???? a?a???ße?? s??te?est??.

101
?p?t?µ?s? ???????µ??

• ?? s??µa Horner e??a? µ?a ap?d?t??? ?a? sta?e??
  d?ad??as?a ??a t?? ap?t?µ?s? p??????µ?? µe
  ???sµata p?a?µat????? a???µ???.
• ?? p??????µ? f(x) a0 a1x a2x2 anxn
• µetat??peta? se
• (((anx an-1)x an-2)x a1 ) x a0

102
?p?t?µ?s? ???????µ??

• ?e? e??a? d??at?? ?a pe?????s??µe t?? ??e?st?t?ta
  e??? d?ast?µat?? se µ?a ???a ?????? t????
  ???s?µ?p????ta? ap?t?µ?s? µ?sa st?? ?e?t???? t??
  ???a?, ??te µp????µe ?a ap?µ???s??µe a??et??
  ???e? µe µ???? d?a????sµ?.
• ?? p????µe ??a pa??de??µa t?? ap?t?µ?s? t?? f(x)
  x4 8x3 24x2 32x 16 (x 2)4 ???t?
  st?? tet?ap?? ???a 2 ?a??? ?a? t?? f(x) (1-x)3 .

103
?p?t?µ?s? ???????µ??
104
Interval Newton

• Te????µe f(x)0 ?p?? ? f e??a? s??e???
  pa?a????s?µ?. ?p? t? ?e???µa t?? µ?s?? t?µ??
  pa?????µe
• f(m(x)) f(x) f(?) (m(x) x)
• µe x, ? ? x
• ?p???t??µe ?t? x e??a? ???a t?? f(x).

105
Interval Newton

• ??????µe
• N(x) m(x) f(m(x)) / f(x)
• ??te
• x m(x) f(m(x))/f(?) ? N(x)

106
Interval Newton

• S??ep?? t? x ß??s?eta? ?a? st? x ?a? st?
  N(x) ?a? ??a st?? t?µ? t?? d?? d?ast?µ?t??.
• ?e?????ta? ap? ??a a????? d??st?µa x0 ?????µe
  epa?a???e??
• x(k1) x(k) n N(x(k)), 0 ? f(x(k))
  and k 0, 1, 2

107
Interval Newton

• ?f?? ??a ta d?ast?µata pe??????ta? st? a?????
  d??st?µa ? µ???d?? de? µp??e? ?a ap?????e?.
• Ge?µet???? ? µ???d?? e??a? a?t?st???? µe t??
  ??ass??? µ???d? t?? ?e?t??a.
• ?p???t??µe ?t? fg,g ?a? ?t? ?? ??aµµ?? µe
  a?t?? t?? ???se?? t?µ???? t?? ????a x sta l, r.
  ??t? e??a? ?a? t? d??st?µa N(x(k)) l, r .
  ?? ? t?µ? e??a? ?de?a t?te ?????µe s?????a ?t?
  de? ?p???e? ???a st? x.

108
Interval Newton
109
Interval Newton

• ???s?µ?p????ta? p??tetaµ??a d?ast?µata µp????µe
  ?a ?ef????µe ap? t?? pe?????sµ? ?t? 0 ?
  f(x(k)) . ?? t? d??st?µa f(x(k)) pe????e?
  t? 0, pa?????µe p??tetaµ??a d?ast?µata ??a t? ?
• N(x(k)) -8 , r ? l, 8
• ??µ? µe t? x d??e? ?? ap?t??esµa d??
  pepe?asµ??a s????a
• x(k1) x(k), r ? l, x(k)

110
Interval Newton
111
Interval Newton

• S?µpe?asµat??? t? a??????? e??a? a?????
• f D ? R ? R s??e??? pa?a????s?µ? s????t?s?
• x ? D ??a d??st?µa st? ped?? t?? f. ??te t?
• N(x) m(x) f(m(x)) / f(x)
• ??e? t?? e??? ?d??t?te?

112
Interval Newton

• 1. ???e x ? x ??a??p??e? t? x? N(x)
• 2. ?? N(x) n x 0 t?te de? ?p???e? ???a t??
  f st? x
• 3. ?? t? N(x) pe????eta? st? x, t?te ?p???e?
  µ?a µ??ad??? ???a t?? f st? x ?a? ??a st?
  N(x)

113
??ast?µata st??? Ge?µet?????? ?p?????sµ???

• St?? pe??pt?s? t?? Newton Intervals ?a µa?
  e?d??fe?e ?a d??µe t? p?? µetaß?ß??eta? t? sf??µa
  ?e?µet???? se d??f??e? ?atas?e???.
• ??a ?a?? pa??de??µa ??a µe??t? e??a? ? ap?t?µ?s?
  t?? ?aµp???? bezier C(t).

114
???t? S????a

• ??a s????? K ???eta? ???t? (convex) a? ??a ???e
  d?? s?µe?a t??, t? e?????aµµ? tµ?µa p?? ta
  s??d?e? e??a? µ?sa st? ?.
• ?? ep?ped? E e??a? t? ep?ped? ?p?st??????
  (support plane) t?? ? a? ta ? ?a? ? ????? ??a
  ????? s?µe?? ?a? t? ? a???e? se ??a? ap? t???
  ?µ??????? p?? ????e? t? ?.

115
???t? S????a
116
???t? S????a

• ? s????t?s? ?p?st?????? sK e??a? µ?a s????t?s?
  p?? ape??????e? ???e µ??ad?a?? d????sµa n st?
  ep?ped? sk(n) p?? e??a? ????????? st? n ?a? e??a?
  ep?ped? ?p?st?????? t?? K.
• ?? a???µ?t???? p???e?? ?????ta? ?p?? pe??p?? ?a?
  µe ta d?ast?µata. G?a ??a? a???µ? t ??????µe t?
  tK ?? s????? s?µe??? tx ?p?? t? x e??a? st? ?.

117
???t? S????a

• ?? s????? (1-t)K1 tK2 e??a? ??a? s?et????
  s??d?asµ?? (affined combination) t?? ?1 ?a? ?2.
  ?? 0 t 1 t?te e??a? ??a? ???t?? s??d?asµ??
  (convex combination).
• ?? s(n) s????t?s? ?p?st?????? t?? ?, t?te
  s(n)ts(n) e??a? ? s????t?s? ?p?st?????? t??
  ?tK.
• S???e???µ??a s(n)s(-n) e??a? ? s????t?s?
  ?p?st?????? t?? ? -?.

118
???t? S????a

• ?? s1 ?a? s2 e??a? s??a?t?se?? ?p?st?????? t?? K1
  ?a? K2 ?a? 0 t 1 t?te ? ???t?? s??d?asµ??
  ?(1-t)K1 tK2 e??a? ep?s?? ???t?? ?a? ??e?
  s????t?s? ?p?st?????? s(1-t)s1 ts2. ?? ??e?d?
  e??a? ?t? ?? s??te?est?? t ?a? 1-t e??a? µ?
  a???t????.
• ??? ?e????
• KStjKj Stjkj StjKj S(-tj)(-Kj)
• tjgt0 tjlt0 tjgt0 tjlt0
  

119
???t? S????a

• ?ts? ? s????t?s? ?p?st?????? e??a?
• s(n)Stjsj(n) S(-tj)sj(-n)
• ? ap?stas? µeta?? d?? ep?p?d?? ?p?st?????? µe
  normals n ?a? n e??a? ? d??µet??? dn e??? ???t??
  s??????.

120
?aµp??e? Bezier

• C(t) SPiBin(t)
• ?a s?µe?a t?? ?aµp???? s?????? ?p????????ta? ap?
  t?? a??????µ? t?? DeCasteljau

121
?aµp??e? Bezier
122
?aµp??e? Bezier

• ?? p??s?µ? t?? s??a?t?se?? Bernstein d??eta? ap?
  t?? t?p?
• sgn(Bin(t)) sgn(t)n-isgn(1-t)i
• ? s????t?s? ?p?st?????? t?? ???t?? s?µat??
  ?SBin(t)?i e??a?
• s(n) SBin(t)si(n)

123
?aµp??e? Bezier

• ?p?? si(n) si(n) if tlt0, n-i even, or tgt1,
  i even, or 0t1
• -si(-n) if tlt0, n-i odd, or tgt1, i
  odd
• ?p????µe ?a ap?t?µ?s??µe t?? t?µ? µ?a? s????t?s??
  a????? d????t?? t?? t?µ?? t?? s?µe??? e?????? µe
  a????.

124
?aµp??e? Bezier
125
?aµp??e? Bezier
126
?aµp??e? Bezier
127
Interval control points
128
?ept?µ??e?e?

• ??s?ast???, d????t?? e??? ???t?? s?????? ? ?a?
  e??? s?µe??? e??????, ???e e???e?d?? s?µe?? st?
  s????? µp??e? ?a e??a? s?µe?? e??????.
• ??a ? ?aµp??? a????? e??a? ? ???s? ???? t??
  ?aµp????.