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

1. ϕδΤۂઢͷ
 ͳΊΒ͔͞ͷ࿩ ೥݄೔
   !NBTIBCPX
2. ͜Μʹͪ͸ w !NBTIBCPX
   w ιʔγϟϧ1-64
   
   ϑϩϯτΤϯυΤϯδχΞ
   w લ৬Ͱ͸ϑΥϯτͷΤϯδχΞΛ΍͍ͬͯͨ
   w ࠓ೔͸ϕδΤۂઢʹ͍ͭͯޠΓ·͢
   
3. 
    w ͍Ζ͍ΖͳਤܗฤूπʔϧͰݟΔΞϨ ϕδΤۂઢͬͯ
4. 
    w ਓؒʹͱͬͯ΋ίϯϐϡʔλʹͱͬͯ΋ѻ͍΍͍͢
   w ϕΫλը૾ɾϑΥϯτɾඳը
   "1*ɾ$"%
   ͳͲͰར༻
   w ೥͝ΖɺγτϩΤϯͷ
   1BVM
   EF
   $BTUFMKBV
   ͱ
 ϧϊʔͷ
   1JFSSF
   #É[JFS
   ͕ಠཱʹ։ൃͨ͠Β͍͠
   w ࠓճ͸ฏ໘্ͷ
   
   ࣍ϕδΤۂઢΛલఏʹͯ͠࿩͠·͢ ϕδΤۂઢ
5. ਺ֶతͳදݱ w ࢝఺ɹ
   
   
   ੍ޚ఺
   
   
   
   
   
   
   
   
   
   ऴ఺ɹ
   
   ͷ
   
   ఺Ͱఆٛ͞ΕΔ
   w ύϥϝʔλ
   
   
   Λ
   
   
   
   
   
   
   
   
   
   ͷൣғͰ
 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)
6. ༡΂ΔσϞ w ϕδΤۂઢΛ
   
   ຊ༻ҙ͓͍ͯͨ͠ͷͰɺ৮ͬͯΈΔ
   w IUUQTNBTIBCPXHJUIVCJPCF[JFSTNPPUIOFTT
   
7. ࠓճͷςʔϚ w ϕδΤۂઢͱϕδΤۂઢΛͳΊΒ͔ʹͭͳ͍͛ͨʂ
   w ׳Ε͍ͯͳ͍ͱҙ֎ͱΉ͔͍ͣ͠
   w ͳΊΒ͔ʹ
   ͱ͸
   
8. ·ͣ͸ͦ΋ͦ΋ w 
   ͭͷϕδΤۂઢΛͭͳ͛Δඞཁ͕͋Δ
   w ۂઢ
   "
   ͷऴ఺ͱɺۂઢ
   #
   ͷ࢝఺ΛҰகͤ͞Δʢ؆୯ʣ
   w ɹ
   
   ࿈ଓʢҐஔ࿈ଓʣͱ͍͏ G0
   
9. Ͱ΋·ͩંΕ͍ͯΔ w ࣍͸ɺͭͳ͛ͨ෦෼ͷ޲͖Λἧ͑Δඞཁ͕͋Γͦ͏
   w ۂઢͷ޲͖ͱ͸ʁ
   ˠ
   ઀ઢͷ޲͖
   w ܭࢉͰٻΊΔͷͰ͋Ε͹ɺඍ෼͢Ε͹Α͍
    B′(t) = (−3t2 + 6t − 3) p0 + (9t2 − 12t + 3) c0 + (−9t2 + 6t) c1 + 3t2 p1
10. ઀ઢͱϋϯυϧ w ࣮͸ɺ࢝఺ʗऴ఺ʹ͓͚Δ઀ઢͷ޲͖͸ɺ
 ϋϯυϧͷ޲͖ͱҰக͍ͯ͠Δ
    w
    
    
    
    
    
    
    
    
    ΍ɹɹ
    
    
    Λܭࢉͯ͠ΈΔͱΘ͔Δ
     B′(0) B′(1)
11. ޲͖Λἧ͑Δ w ۂઢ
    "
    #
    ͷϋϯυϧΛҰ௚ઢʹฒ΂Ε͹Α͍
    w ࠓճ͸୯७ʹɺۂઢ
    #
    ͷϋϯυϧͷ֯౓Λม͑ͯௐ੔
    w ɹ
    
    ࿈ଓʢ઀ઢ࿈ଓʣͱ͍͏
     G1
12. Ͱ΋·ͩෆࣗવ w ۂ͕Γ۩߹͕ҧ͏Α͏ͳʜʁ
    w ۂ͕Γ۩߹
    ͱ͸
     ͜͜Β΁ΜͰٸʹ
 ௚ઢͬΆ͘ͳͬͯ
    ϋϦ͕ͳ͘ͳͬͨΑ͏ͳ
    
13. ઀৮ԁ w ۂઢҰൠͩͱΑ͘Θ͔Βͳ͍ͷͰɺ਎ۙͳԁͰߟ͑ͯΈΔ
    w ۂઢͷ͘͝Ұ෦෼͚ͩʹ஫໨ͯ͠ɺԁހͰۙࣅ͢Δ
    w ͦͷۃݶͷԁΛ઀৮ԁͱ͍͏
    
14. R w ઀৮ԁͷେ͖͞Ͱۂ͕Γ۩߹͕දݱͰ͖ͦ͏
    w ઀৮ԁͷ൒ܘΛۂ཰൒ܘ
    
    
    
    ͱݺͿ
    w ۂ཰൒ܘ͕େ͖͍
    ˱
    Χʔϒ͕ΏΔ͍
    w ۂ཰൒ܘ͕খ͍͞
    ˱
    Χʔϒ͕͖͍ͭ
    w ٸΧʔϒ஫ҙͷඪࣝʹ
 ॻ͍ͯ͋ͬͨΓ͢Δ ۂ཰൒ܘ
     R
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
16. ໰୊෦෼ͷۂ཰൒ܘ w ۂ཰൒ܘ͕ͥΜͥΜҧ͏
    
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
18. ۂ཰ͷࢹ֮Խ w DVSWBUVSF
    DPNCʢۂ཰ͷᷱʁʣ
    w ֤఺ʹ͓͚Δۂ཰Λɺ๏ઢํ޲ʹදࣔͨ͠΋ͷ
    w $"%
    ιϑτ΍ϑΥϯτΤσΟλʹ౥ࡌ͞Ε͍ͯΔ
     ۂ཰େ ۂ཰খ มۂ఺
19. ໰୊෦෼ͷۂ཰ͷᷱ w ۂ཰ͷҧ͍͕Ұ໨ྎવ
    
20. ۂ཰Λἧ͑Δ w
    
    
    
    
    
    
    
    
    
    
    ΄Ͳ୯७Ͱ͸ͳ͍͕ɺܭࢉͰͳΜͱ͔Ͱ͖Δ
    w ࠓճ͸ɺۂઢ
    #
    ͷϋϯυϧͷ௕͞Λม͑ͯௐ੔
    w ɹ
    
    ࿈ଓʢۂ཰࿈ଓʣͱ͍͏
     G2 G0, G1
21. ϏϑΥʔΞϑλʔ w ͗ͪ͜ͳ͕͞ݮͬͯɺΑΓͳΊΒ͔ʹͳͬͨ
     ☺
22. ·ͱΊ w ϕδΤۂઢɾ઀ઢɾۂ཰͋ͨΓΛͬ͘͟Γઆ໌ͨ͠
    w ɹɹɹɹɹ࿈ଓੑΛຬͨ͢Α͏ʹௐ੔͢Δ͜ͱͰɺ
 ϕδΤۂઢ͕ͳΊΒ͔ʹͳͬͨ
    w Ϣʔβʔʹͱͬͯ͸ɺ
 ࢹ֮Խ͕େ͖ͳख͕͔ΓʹͳΔ
     G0, G1, G2
23. ͜ͷઌͷτϐοΫ w ɹ
    ࿈ଓɺ͋Δ͍͸ͦΕҎ্͸ʁ
    w ɹ
    ࿈ଓͳΒ͢΂ͯਓؒͷݟͨ໨ʹඒ͍͠ʁ
    w ࢥ͍ඳ͍͍ͯΔܗΛͳΔ΂่ͣ͘͞ʹௐ੔͢Δʹ͸ʁ
    w ΫϩιΠυۂઢʢผ໊ɿΦΠϥʔཐટɺ4QJSPʣ
     G3 Gn