二次元アニメーションの制作に適した三次元物理シミュレーション手法
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(2) Vol.2014-CG-154 No.17 2014/2/21. ሗฎ⌮Ꮫ◊✲ሗ࿌ IPSJ SIG Technical Report ⓗࡣ㸪ࢩ࣑࣮ࣗࣞࢩࣙࣥࣇ࣮࣒ࣞẖḟࡢࡘࡢࢫࢸࢵ. 2.3 ḟඖࢽ࣓࣮ࢩࣙࣥⓗ࡞ᙇ⾲⌧. ࣉࢆ⧞ࡾ㏉ࡍ㸬 (1) Goal Position ィ⟬. ḟඖࢽ࣓࣮ࢩࣙࣥࡣ㸪ࢫࢺ࣮࣮ࣜࢆᵓᡂࡍࡿࡓࡵ㸪. ࡲࡎ㸪ࠕኚᙧ๓ࡢඖᙧ≧ࠖࢆࠕኚᙧᚋࡢ⌧ᅾᙧ≧ࠖ࡞. ⌧ᐇࡢୡ⏺ࡽ࢟ࣕࣛࢡࢱ࣮ࡢጼ㸪ືస㸪⫼ᬒ࡞ᢳฟࡋ㸪. ࡿࡃࡨࡗࡓࡾྜࢃࡏࡿ๛యኚ (ᖹ⾜⛣ື) ࢆィ⟬ࡍࡿ㸬. ࡑࡋ࡚ࡑࢀࡽࢆ⤠ࡾ㎸ࡳ㸪ࡲࡓᙇࡍࡿࡇࡼࡗ࡚㸪ࡑ. ᅗ 1 ࡢ (c) ࡣඖᙧ≧ࢆᖹ⾜⛣ືࡢࡳ࡛⌧ᅾᙧ≧㔜ࡡࡓࡶ. ࡢୡ⏺ほࢆ⾲⌧ࡍࡿ㸬ࡋࡓࡀࡗ࡚㸪ఏ࠼ࡓ࠸ࡇࢆ⌧ᐇࡼ. ࡢ࡛㸪ᅗ 1 (d) ࡣ᭦ᅇ㌿ࢆຍ࠼ࡓࡶࡢ࡛࠶ࡿ㸬ᅗ 1 (e) ࡢ. ࡾࡶᙉࡃほᐈ༳㇟ࡅࡿࡇࡀ࡛ࡁࢀࡤ㸪ࡑࡢ┠ⓗࢆ㐩. ㏻ࡾ㸪⌧ᅾᙧ≧ࡢྛ㡬Ⅼࢆ㸪๛యኚࡉࢀࡓඖᙧ≧㡬Ⅼ. ᡂࡋࡓࡇ࡞ࡿ㸬. ⛣ືࡉࡏࢀࡤ㸪⌧ᅾᙧ≧ࢆඖᙧ≧ᘬࡁᡠࡿࡇࡀ࡛ࡁࡿ㸬 (2) 㡬Ⅼࡢ㏿ᗘ⨨ࡢ᭦᪂. ࡢ⾲⌧㸪࠾ࡼࡧከᩘࡢࢯࣇࢺ࣎ࢹࡢࠕ⾪✺ࠖࡢࢩ࣑ࣗࣞ. ḟ㸪ྛ㡬Ⅼ ܠ ( ࡢࡑࡀ )ݐGoal Position ྥࡗ࡚⛣ື ࡍࡿࡼ࠺㸪⌧ᅾࡢ้ࡿࡅ࠾ݐ㡬Ⅼࡢ⨨ ܠ ()ݐ㸪㏿ᗘ ܞ (ࢆ )ݐ᭦᪂ࡍࡿ㸬 ܞ ( ݐ+ ݄) = ܞ ( )ݐ+ ߙ. ࣮ࢩࣙࣥࢆᑟධࡍࡿ㸬 ࠕ㉸ᙎຊࠖࡣ Shape Matching ἲࡢࠕࢇ࡞⃭ࡋ࠸ኚᙧ ࡋ࡚ࡶ㸪ᚲࡎඖࡢᙧᡠࡿࠖ࠸࠺≉ᚩࢆ⏝ࡋ㸪ᐇࡋ. െ ܠ ()ݐ ௫௧ +݄ ݉ ݄. ܠ ( ݐ+ ݄) = )ݐ( ܑ ܠ+ ݄ܞ ( ݐ+ ݄) ௫௧. ࡑࡇ࡛㸪ࡇࢀࢯࣇࢺ࣎ࢹࡼࡿࠕ㉸ᙎຊࠖࠕศゎࠖ. ࡓຠᯝ࡛࠶ࡿ㸬ᘧ (1)(2) ࢆࡶ㸪ලయⓗ࡞ฎ⌮ᡭ㡰ࢆᅗ (1). 2 ♧ࡍ㸬 ࠕศゎࠖࡣ๓㏙ࡢࠕ㉸ᙎຊࠖࡢᐇࡶ࡙࠸࡚ᥦࡍ. (2). ࡣ ܠ (ࡿࡅ )ݐእຊ㸪݄ ࡣᚤᑠ㛫㸪ߙ ( א0,1] ࡣ. ◳ࡉࢆ⾲ࡍಀᩘ࡛࠶ࡿ㸬ߙ ࡢ್ࡀᑠࡉࡅࢀࡤ㸪㡬Ⅼࡀ Goal. ࡿຠᯝ࡛࠶ࡿ㸬ࣔࢹࣝᘬࡁఙࡤࡍຊࢆຍ࠼ࡓ㸪࣎ࢡࢭ ࣝྠኈ㛫ࡢ㊥㞳ࡀ㝈⏺Ⅼࢆ㉸࠼ࡓ㉁Ⅼ㛫ࡣඖຊࢆኻ࠸㸪 ࣔࢹࣝࡀᅗ 3 ࡢࡼ࠺ศࡉࢀࡿ㸬 ከᩘࢯࣇࢺ࣎ࢹࡢࠕ ࠕ⾪✺ࠖࡣ㸪ࣜࢪࢵࢺ࣎ࢹ࠾ࡅ. Position ฿㐩ࡍࡿࡲ࡛㛫ࡀࡾ㸪≀యࡣᰂࡽ࠸ ༳㇟࡞ࡿ㸬㏫㸬ߙ ࡢ್ࡀࡁࡅࢀࡤ㸪≀యࡣ◳࠸༳㇟ ࡞ࡿ㸬 2.2 ୕ḟඖࡢᣑᙇ ḟඖࡢᙧ≧ࢆ㉁Ⅼࡢ㞟ྜ࡛ࣔࢹࣝࡋࡓࡇྠᵝ 㸪୕ḟඖࡢᙧ≧ࡣ࣎ࢡࢭࣝࡢ㞟ྜࡋ࡚ࣔࢹࣝࡍࡿ㸬 ࡇࢀࢆඖ Shape-Matching ᡭἲࡼࡿࢯࣇࢺ࣎ࢹࢩ࣑ࣗ ࣮ࣞࢱࢆᐇࡍࡿ㸬 ලయⓗ㸪ࣔࢹࣝࡢ」㞧࡞⾲㠃ᵓ㐀ࢆ⪃࠼ࡎ㸪࣎ࢡࢭࣝ ࡢ㞟ྜࢆ⏝࠸࡚ィ⟬ࡍࡿ㸬ࡇࢀ࡛ィ⟬㔞ࡢ๐ῶࡍࡿࡇࡀ ᐇ⌧ࡋ㸪㧗㏿ࡀྍ⬟࡛࠶ࡿ㸬 ᚋࡇࡢィ⟬ࡼࡾ㸪ᚓࡽࢀࡓ࣎ࢡࢭࣝ࠾ࡅࡿ㉁Ⅼࡢ ⨨㏿ᗘࢆࢫ࢟ࢽࣥࢢᡭἲࡼࡿ≀యࣔࢹࣝࢆఏ࠼ࢀࡤ㸪. ᅗ 2 ḟඖᅗᙧᑐࡍࡿࠕ㉸ᙎຊࠖࡢຠᯝ Figure 2 The super elasticity effect that is applied to the 2-dimensional object.. ồࡵࡓຠᯝࡀᐇ⌧࡛ࡁࡿ㸬. ᅗ 3 ḟඖᅗᙧᑐࡍࡿࠕศゎࠖࡢຠᯝ ᅗ 1 Goal Position ࢆồࡵࡿᡭ㡰. Figure 3 The breakdown effect. Figure 1 The procedure for calculating goal position.. that is applied to the 2-dimensional object.. ⓒ 2014 Information Processing Society of Japan. 2.
(3) Vol.2014-CG-154 No.17 2014/2/21. ሗฎ⌮Ꮫ◊✲ሗ࿌ IPSJ SIG Technical Report ࡿ⾪✺ࡢ⌧ᡭἲࢆ⏝ࡋ࡚㸪≀యయࡢ⨨㏿ᗘࢆỴ. ࡿࡇࡀ࡛ࡁࡓ㸬. ᐃࡍࡿ㸬≀యࡢኚᙧࡢ⌧ࡣ Shape Matching ἲࢆ⏝࠸ࡿ㸬. 3.2 ୕ḟඖᙧ≧ࡢ㐺⏝ ୕ḟඖᙧ≧࡛ࡣ≀యࡢᙧ≧ࢆ࣎ࢡࢭ࡛ࣝ㏆ఝࡋࡓࣔࢹ. 3. ホ౯ᐇ㦂. ࣝࢆసᡂࡋ㸪ࡑࢀᑐࡋ࡚ Shape Matching ἲࡼࡿኚᙧ. ᮏ◊✲࡛ᥦࡋࡓᙇຠᯝࡢ⏕ᡂᡭἲࢆᐇࡋ㸪ࡑࡢ᭷. ࢆồࡵࡓᚋ㸪ࡑࡢኚᙧࢆ࣎ࢡࢭࣝࡀໟྵࡍࡿ≀యࡢ㒊ศᙧ. ⏝ᛶࢆ᳨ドࡋࡓ㸬ᥦᡭἲࢆ୕ḟඖᙧ≧㐺⏝ࡍࡿ๓㸪. ≧ᫎࡍࡿ㸬ࡇࡇ࡛ࡣᅗ 4 ♧ࡍ࢘ࢧࢠࡢࣔࢹࣝࢆ⏝࠸. ⡆༢࡞ḟඖᙧ≧㐺⏝ࡋࡓ (ᅗ 2㸪ᅗ 3)㸬. ࡓ㸬ࡇࡢࣔࢹࣝࡣᆅ㠃ᅛᐃࡋ࡚࠸ࡿ㸬ࡇࢀຊࢆຍ࠼ࡓ. 3.1 ḟඖᙧ≧ࡢ㐺⏝. ࡁ㸪ࡑࢀࡼࡿኚᙧຠᯝࡀᚓࡽࢀࡓ㸬ࡲࡓ㸪ࡇࡢຊࢆᾘ. ࡇࡇ࡛ࡣ㉁Ⅼࡢ㞟ྜ࡛⾲ࡋࡓேయᙧ≧ࡢࣔࢹࣝࢆ⏝࠸. ࡋࡓᚋࡣ㸪≀యᙧ≧ 㐠ືࡸࡑࡢῶ⾶ࡀⓎ⏕ࡋࡓ㸬. ࡓ㸬ࡇࡢࣔࢹࣝࡣຠᯝࢆ᫂☜⾲⌧ࡍࡿࡓࡵ㸪ࣔࢹࣝࡢ㢌. ࠕศゎࠖࡢຠᯝࡣ㸪ࡇࡢࠕ㉸ᙎຊࠖࡢຠᯝࡢୖᐇࡋ. ᕥᡭᕥ㊊ࢆᅛᐃࡋ࡚࠸ࡿ㸬. ࡓ㸬ࡇࢀࡣࠕ㉸ᙎຊࠖࡢຠᯝ࠾࠸࡚࣎ࢡࢭࣝ㛫ࡢ㊥㞳. ࡇࡢᐇ࠾࠸࡚㸪ࣔࢹࣝຊࢆຍ࠼ࡓࡇࡼࡾ㸪ኚ. 㝈⏺Ⅼࢆタᐃࡋ㸪ࡑࢀࢆ㉸࠼ࡓࡁ㞄᥋ࡍࡿ࣎ࢡࢭࣝ㛫. ᙧࡀⓎ⏕ࡋࡓ㸬ࡇࡢࡁ㸪ᅗ 2 ࡢ A ♧ࡍࡼ࠺㸪ຊࡢ. ࢆศ㞳ࡍࡿࡇࡼࡗ࡚ᐇ⌧ࡋࡓ㸬ࡇࡇ࡛ࡣᅗ 5 ♧ࡍቨ. ᙉࡉࡼࡾ㸪␗࡞ࡿኚᙧࡢᙇຠᯝࡀᚓࡽࢀࡓ㸬 እຊࡀᾘ࠼ࢀࡤ㸪ࣔࢹࣝࡢᙧ≧ඖᡠࡿ㸬ࡑࡢ㐣⛬࡛ࡣ㸪 ࣔࢹࣝࢆᵓᡂࡍࡿ㉁Ⅼࡣ 㐠ືࢆ⾜࠸㸪ᅗ 2 ࡢ B, C, D ♧ࡍࠕ㉸ᙎຊࠖࡢຠᯝࢆᚓࡿࡇࡀ࡛ࡁࡓ㸬ࡲࡓ㸪ᢠ ຊࢆタᐃࡍࡿࡇࡼࡾ㸪ࡇࡢ 㐠ືࡣῶ⾶ࡋ㸪᭱⤊ⓗ ඖࡢᙧ≧ᡠࡿࡇࡀ☜ㄆࡉࢀࡓ㸬 ࡲࡓ㸪ᅗ 3 ♧ࡍࡼ࠺㸪ᘬࡁఙࡤࡍຊࢆ୍ᐃ௨ୖࡁ ࡃࡋࡓࡁᙧ≧ࡣ⮬ືⓗศࡉࢀ㸪 ࠕศゎࠖࡢຠᯝࢆᚓ. ᅗ 4 ୕ḟඖࣔࢹࣝᑐࡍࡿࠕ㉸ᙎຊࠖຠᯝࡢ㐺⏝. ᅗ 5 ୕ḟඖࣔࢹࣝᑐࡍࡿࠕศゎࠖຠᯝࡢ㐺⏝. Figure 4 Applying the super elasticity effect. Figure 5 Applying the breakdown effect. to the 2-dimensional object.. ⓒ 2014 Information Processing Society of Japan. to the 3-dimensional object.. 3.
(4) Vol.2014-CG-154 No.17 2014/2/21. ሗฎ⌮Ꮫ◊✲ሗ࿌ IPSJ SIG Technical Report ࡢࣔࢹࣝࢆ⏝࠸࡚ࢩ࣑࣮ࣗࣞࢩࣙࣥࢆ⾜ࡗࡓ㸬 ࡇࢀࡲ࡛ྠᵝ㸪ࣔࢹࣝእຊࡅ㸪ኚᙧࡉࡏࡓ ⤖ᯝ㸪㝈⏺Ⅼࢆ㉸࠼ࡓࡇࢁ࡛ࠕศゎࠖࡢຠᯝࡀᚓࡽࢀࡓ㸬 ḟ㸪ከᩘࢯࣇࢺ࣎ࢹࡢࠕ⾪✺ࠖࡢຠᯝࡘ࠸࡚ᐇ㦂 ࢆ⾜ࡗࡓ㸬ᅗ 6 ࡢࡼ࠺ከᩘࡢࢯࣇࢺ࣎ࢹࢆ㔜ࡡ࡚㓄⨨. ࡍࡿࢩ࣮ࣥ࠾࠸࡚㸪┦స⏝ࢆ᭷ຠࡋࡓࡁ㸪ᅗ 7 ࠾ࡼࡧᅗ 8 ♧ࡍ⇿Ⓨⓗ࡞⾪✺ຠᯝࡀᚓࡽࢀࡓ㸬. 4. ࠾ࢃࡾ ᮏ◊✲࡛ࡣ㸪ࢯࣇࢺ࣎ࢹࡢ㐠ືࢩ࣑࣮ࣗࣞࢩࣙࣥࡶ ࡙࠸࡚㸪ḟඖࢽ࣓࣮ࢩࣙࣥⓗ࡞ᙇ⾲⌧ࢆ⌧ࡍࡿ ᡭἲࢆ㛤Ⓨࡋࡓ㸬 ࡋࡋ㸪ḟඖࢽ࣓࣮ࢩࣙࣥⓗ࡞ᙇ⾲⌧ࡣ㸪ᮏ◊✲ ࠾࠸࡚ᐃࡋࡓࡶࡢ௨እከᵝ࡞ࡶࡢࡀ࠶ࡿ㸬ࡇࢀᑐ ࡋ㸪ᮏ◊✲࡛ᐇ࡛ࡁࡓࡶࡢࡣࢃࡎ࡛࠶ࡾ㸪⾲⌧ྍ⬟࡞ ຠᯝࡀࡲࡔᑡ࡞࠸㸬ࡲࡓ㸪ᥦᡭἲࡣࡲࡔࢭࣝࢩ࢙࣮ࢹ ࣥࢢ࡞ NPR ᡭἲࡼࡗ࡚ࣞࣥࢲࣜࣥࢢࡉࢀࡿࡶࡢ ⏝ࡋ࡚࠸࡞࠸ࡓࡵ㸪⌧≧࡛ࡣᮏ◊✲ࡢ᭷⏝ᛶࢆ♧ࡍࡇࡀ ࡛ࡁ࡚࠸࡞࠸㸬ᚋࡣࡼࡾከᵝ࡞⾲⌧ࡢᐇ㸪ḟඖ ࢽ࣓࣮ࢩࣙࣥࡢᛂ⏝ࡘ࠸࡚◊✲ࢆ㐍ࡵࡿᚲせࡀ࠶ࡿ㸬. ཧ⪃ᩥ⊩ ᅗ 6 ᅗᙧྠኈࡢ┦స⏝ࡀ↓ຠࡢሙྜ Figure 6 When the interaction force between objects was disabled.. 1) Nobuki KOYAMA, Kenshi TAKAYAMA, Nobuyuki UMETANI Takeo IGARASHI, Real-Time Example-Based Elastic Deformation, SCA 2012 2) ᑼᩗ, Shape Matching Dynamics . Computer Graphics Gems.2012. ᅗ 7 ᅗᙧྠኈࡢ┦స⏝ࢆ᭷ຠࡋࡓ┤ᚋ Figure 7 Immediately after the interaction force between objects was enabled.. ᅗ 8 ┦స⏝ࢆ᭷ຠࡋ࡚ࡽ୍ᐃ㛫⤒㐣ᚋ Figure 8 After the fixed time has passed the interaction force was enabled.. ⓒ 2014 Information Processing Society of Japan. 4.
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