ベジエ曲線のなめらかさの話 / Bézier Smoothness

by Masaya Nakamura

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ϕδΤۂઢͷ  ͳΊΒ͔͞ͷ࿩ ೥݄೔ !NBTIBCPX

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

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w ͍Ζ͍ΖͳਤܗฤूπʔϧͰݟΔΞϨ ϕδΤۂઢͬͯ

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w ਓؒʹͱͬͯ΋ίϯϐϡʔλʹͱͬͯ΋ѻ͍΍͍͢ w ϕΫλը૾ɾϑΥϯτɾඳը"1*ɾ$"%ͳͲͰར༻ w ೥͝ΖɺγτϩΤϯͷ1BVMEF$BTUFMKBVͱ  ϧϊʔͷ1JFSSF#É[JFS͕ಠཱʹ։ൃͨ͠Β͍͠ w ࠓճ͸ฏ໘্ͷ࣍ϕδΤۂઢΛલఏʹͯ͠࿩͠·͢ ϕδΤۂઢ

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਺ֶ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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༡΂ΔσϞ w ϕδΤۂઢΛຊ༻ҙ͓͍ͯͨ͠ͷͰɺ৮ͬͯΈΔ w IUUQTNBTIBCPXHJUIVCJPCF[JFSTNPPUIOFTT

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ࠓճͷςʔϚ w ϕδΤۂઢͱϕδΤۂઢΛͳΊΒ͔ʹͭͳ͍͛ͨʂ w ׳Ε͍ͯͳ͍ͱҙ֎ͱΉ͔͍ͣ͠ w ͳΊΒ͔ʹͱ͸

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

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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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઀ઢͱϋϯυϧ w ࣮͸ɺ࢝఺ʗऴ఺ʹ͓͚Δ઀ઢͷ޲͖͸ɺ  ϋϯυϧͷ޲͖ͱҰக͍ͯ͠Δ w ΍ɹɹΛܭࢉͯ͠ΈΔͱΘ͔Δ B′(0) B′(1)

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

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Ͱ΋·ͩෆࣗવ w ۂ͕Γ۩߹͕ҧ͏Α͏ͳʜʁ w ۂ͕Γ۩߹ͱ͸ ͜͜Β΁ΜͰٸʹ  ௚ઢͬΆ͘ͳͬͯ ϋϦ͕ͳ͘ͳͬͨΑ͏ͳ

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

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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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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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໰୊෦෼ͷۂ཰൒ܘ w ۂ཰൒ܘ͕ͥΜͥΜҧ͏

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

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໰୊෦෼ͷۂ཰ͷᷱ w ۂ཰ͷҧ͍͕Ұ໨ྎવ

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

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ϏϑΥʔΞϑλʔ w ͗ͪ͜ͳ͕͞ݮͬͯɺΑΓͳΊΒ͔ʹͳͬͨ ☺

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·ͱΊ w ϕδΤۂઢɾ઀ઢɾۂ཰͋ͨΓΛͬ͘͟Γઆ໌ͨ͠ w ɹɹɹɹɹ࿈ଓੑΛຬͨ͢Α͏ʹௐ੔͢Δ͜ͱͰɺ  ϕδΤۂઢ͕ͳΊΒ͔ʹͳͬͨ w Ϣʔβʔʹͱͬͯ͸ɺ  ࢹ֮Խ͕େ͖ͳख͕͔ΓʹͳΔ G0, G1, G2

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