ベジエ曲線のなめらかさの話 / Bézier Smoothness

社内勉強会で発表した内容です

デモ: https://mashabow.github.io/bezier-smoothness/
ソース: https://github.com/mashabow/bezier-smoothness
補足記事: https://shiromoji.hatenablog.jp/entry/2019/05/18/005232

Transcript

1. ϕδΤۂઢͷ

   ͳΊΒ͔͞ͷ࿩
   ೥݄೔
   !NBTIBCPX
   

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2. ͜Μʹͪ͸
   w !NBTIBCPX
   w ιʔγϟϧ1-64ϑϩϯτΤϯυΤϯδχΞ
   w લ৬Ͱ͸ϑΥϯτͷΤϯδχΞΛ΍͍ͬͯͨ
   w ࠓ೔͸ϕδΤۂઢʹ͍ͭͯޠΓ·͢
   
   
   

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3. 
   
   w ͍Ζ͍ΖͳਤܗฤूπʔϧͰݟΔΞϨ
   ϕδΤۂઢͬͯ
   

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4. 
   
   w ਓؒʹͱͬͯ΋ίϯϐϡʔλʹͱͬͯ΋ѻ͍΍͍͢
   w ϕΫλը૾ɾϑΥϯτɾඳը"1*ɾ$"%ͳͲͰར༻
   w ೥͝ΖɺγτϩΤϯͷ1BVMEF$BTUFMKBVͱ

   ϧϊʔͷ1JFSSF#É[JFS͕ಠཱʹ։ൃͨ͠Β͍͠
   w ࠓճ͸ฏ໘্ͷ࣍ϕδΤۂઢΛલఏʹͯ͠࿩͠·͢
   ϕδΤۂઢ
   

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5. ਺ֶతͳදݱ
   w ࢝఺ɹ੍ޚ఺ऴ఺ɹͷ఺Ͱఆٛ͞ΕΔ
   w ύϥϝʔλΛͷൣғͰ

   ಈ͔ͨ͠ͱ͖ͷي੻͕ɺ

   ීஈΑ͘ݟΔϕδΤۂઢ
   
   
   B(t) = (1 − t)3 p0
   + 3(1 − t)2t c0
   + 3(1 − t)t2 c1
   + t3 p1
   p0
   c0
   , c1
   p1
   t [0,1]
   p0
   = B(0)
   p1
   = B(1)
   c0
   c1
   B(0.6)
   

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6. ༡΂ΔσϞ
   w ϕδΤۂઢΛຊ༻ҙ͓͍ͯͨ͠ͷͰɺ৮ͬͯΈΔ
   w IUUQTNBTIBCPXHJUIVCJPCF[JFSTNPPUIOFTT
   
   
   

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7. ࠓճͷςʔϚ
   w ϕδΤۂઢͱϕδΤۂઢΛͳΊΒ͔ʹͭͳ͍͛ͨʂ
   w ׳Ε͍ͯͳ͍ͱҙ֎ͱΉ͔͍ͣ͠
   w ͳΊΒ͔ʹͱ͸
   
   
   
   

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8. ·ͣ͸ͦ΋ͦ΋
   w ͭͷϕδΤۂઢΛͭͳ͛Δඞཁ͕͋Δ
   w ۂઢ"ͷऴ఺ͱɺۂઢ#ͷ࢝఺ΛҰகͤ͞Δʢ؆୯ʣ
   w ɹ࿈ଓʢҐஔ࿈ଓʣͱ͍͏
   G0
   
   
   
   

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9. Ͱ΋·ͩંΕ͍ͯΔ
   w ࣍͸ɺͭͳ͛ͨ෦෼ͷ޲͖Λἧ͑Δඞཁ͕͋Γͦ͏
   w ۂઢͷ޲͖ͱ͸ʁˠ઀ઢͷ޲͖
   w ܭࢉͰٻΊΔͷͰ͋Ε͹ɺඍ෼͢Ε͹Α͍
   
   
   B′(t) = (−3t2 + 6t − 3) p0
   + (9t2 − 12t + 3) c0
   + (−9t2 + 6t) c1
   + 3t2 p1
   

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10. ઀ઢͱϋϯυϧ
    w ࣮͸ɺ࢝఺ʗऴ఺ʹ͓͚Δ઀ઢͷ޲͖͸ɺ

    ϋϯυϧͷ޲͖ͱҰக͍ͯ͠Δ
    w ΍ɹɹΛܭࢉͯ͠ΈΔͱΘ͔Δ
    
    
    B′(0) B′(1)
    

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11. ޲͖Λἧ͑Δ
    w ۂઢ" #ͷϋϯυϧΛҰ௚ઢʹฒ΂Ε͹Α͍
    w ࠓճ͸୯७ʹɺۂઢ#ͷϋϯυϧͷ֯౓Λม͑ͯௐ੔
    w ɹ࿈ଓʢ઀ઢ࿈ଓʣͱ͍͏
    
    
    G1
    
    

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12. Ͱ΋·ͩෆࣗવ
    w ۂ͕Γ۩߹͕ҧ͏Α͏ͳʜʁ
    w ۂ͕Γ۩߹ͱ͸
    
    
    ͜͜Β΁ΜͰٸʹ

    ௚ઢͬΆ͘ͳͬͯ
    ϋϦ͕ͳ͘ͳͬͨΑ͏ͳ
    

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13. ઀৮ԁ
    w ۂઢҰൠͩͱΑ͘Θ͔Βͳ͍ͷͰɺ਎ۙͳԁͰߟ͑ͯΈΔ
    w ۂઢͷ͘͝Ұ෦෼͚ͩʹ஫໨ͯ͠ɺԁހͰۙࣅ͢Δ
    w ͦͷۃݶͷԁΛ઀৮ԁͱ͍͏
    
    
    

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14. R
    w ઀৮ԁͷେ͖͞Ͱۂ͕Γ۩߹͕දݱͰ͖ͦ͏
    w ઀৮ԁͷ൒ܘΛۂ཰൒ܘͱݺͿ
    w ۂ཰൒ܘ͕େ͖͍˱Χʔϒ͕ΏΔ͍
    w ۂ཰൒ܘ͕খ͍͞˱Χʔϒ͕͖͍ͭ
    w ٸΧʔϒ஫ҙͷඪࣝʹ

    ॻ͍ͯ͋ͬͨΓ͢Δ
    ۂ཰൒ܘ
    
    
    R
    

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15. ۂ཰൒ܘͷܭࢉ
    w Ұൠͷύϥϝʔλۂઢͷۂ཰൒ܘ͸ɺ࣍ࣜͰܭࢉͰ͖Δ
    w ͨͩ͠ɺμογϡ͸ύϥϝʔλɹʹΑΔඍ෼Λද͢
    w ͜ͷࣜͰग़ͯ͘Δۂ཰൒ܘ͸ූ߸෇͖
    w ਖ਼ͳΒࠨճΓɺෛͳΒӈճΓ
    
    
    R =
    (x′2 + y′2
    )
    3
    2
    x′y′′− y′x′′
    t
    

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16. ໰୊෦෼ͷۂ཰൒ܘ
    w ۂ཰൒ܘ͕ͥΜͥΜҧ͏
    
    
    
    

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17. ۂ཰
    w ۂ཰൒ܘͷٯ਺ɹɹΛۂ཰ɹͱ͍͏
    w ۂ཰൒ܘ͕େ͖͍˱ۂ཰͕খ͍͞˱Χʔϒ͕ΏΔ͍
    w ۂ཰൒ܘ͕খ͍͞˱ۂ཰͕େ͖͍˱Χʔϒ͕͖͍ͭ
    w ͬͪ͜ͷํ͕Θ͔Γ΍͍͢ʁ
    
    
    1/R κ
    ۂ཰େ
    ۂ཰খ
    

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18. ۂ཰ͷࢹ֮Խ
    w DVSWBUVSFDPNCʢۂ཰ͷᷱʁʣ
    w ֤఺ʹ͓͚Δۂ཰Λɺ๏ઢํ޲ʹදࣔͨ͠΋ͷ
    w $"%ιϑτ΍ϑΥϯτΤσΟλʹ౥ࡌ͞Ε͍ͯΔ
    
    
    ۂ཰େ
    ۂ཰খ
    มۂ఺
    

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19. ໰୊෦෼ͷۂ཰ͷᷱ
    w ۂ཰ͷҧ͍͕Ұ໨ྎવ
    
    
    
    

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20. ۂ཰Λἧ͑Δ
    w ΄Ͳ୯७Ͱ͸ͳ͍͕ɺܭࢉͰͳΜͱ͔Ͱ͖Δ
    w ࠓճ͸ɺۂઢ#ͷϋϯυϧͷ௕͞Λม͑ͯௐ੔
    w ɹ࿈ଓʢۂ཰࿈ଓʣͱ͍͏
    
    
    G2
    G0, G1
    
    

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21. ϏϑΥʔΞϑλʔ
    w ͗ͪ͜ͳ͕͞ݮͬͯɺΑΓͳΊΒ͔ʹͳͬͨ
    
    
    
    ☺
    

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22. ·ͱΊ
    w ϕδΤۂઢɾ઀ઢɾۂ཰͋ͨΓΛͬ͘͟Γઆ໌ͨ͠
    w ɹɹɹɹɹ࿈ଓੑΛຬͨ͢Α͏ʹௐ੔͢Δ͜ͱͰɺ

    ϕδΤۂઢ͕ͳΊΒ͔ʹͳͬͨ
    w Ϣʔβʔʹͱͬͯ͸ɺ

    ࢹ֮Խ͕େ͖ͳख͕͔ΓʹͳΔ
    
    
    G0, G1, G2
    

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23. ͜ͷઌͷτϐοΫ
    w ɹ࿈ଓɺ͋Δ͍͸ͦΕҎ্͸ʁ
    w ɹ࿈ଓͳΒ͢΂ͯਓؒͷݟͨ໨ʹඒ͍͠ʁ
    w ࢥ͍ඳ͍͍ͯΔܗΛͳΔ΂่ͣ͘͞ʹௐ੔͢Δʹ͸ʁ
    w ΫϩιΠυۂઢʢผ໊ɿΦΠϥʔཐટɺ4QJSPʣ
    
    
    G3
    Gn
    

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