Masaya Nakamura
May 17, 2019
39k

ベジエ曲線のなめらかさの話 / Bézier Smoothness

May 17, 2019

Transcript

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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)

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

G1

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

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

ۂ཰খ มۂ఺

G0, G1

22. ·ͱΊ w ϕδΤۂઢɾ઀ઢɾۂ཰͋ͨΓΛͬ͘͟Γઆ໌ͨ͠ w ɹɹɹɹɹ࿈ଓੑΛຬͨ͢Α͏ʹௐ੔͢Δ͜ͱͰɺ  ϕδΤۂઢ͕ͳΊΒ͔ʹͳͬͨ w Ϣʔβʔʹͱͬͯ͸ɺ  ࢹ֮Խ͕େ͖ͳख͕͔ΓʹͳΔ 

 G0, G1, G2

 G3 Gn