WebGL Safari WebGL Kageyama (Kobe Univ.) Visualization / 55
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1 WebGL WebGL X Kageyama (Kobe Univ.) Visualization / 55
2 WebGL Safari WebGL Kageyama (Kobe Univ.) Visualization / 55
3 Kageyama (Kobe Univ.) Visualization / 55
4 WebGL WebBL = HTML5 canvas JavaScript 3D CG API Kageyama (Kobe Univ.) Visualization / 55
5 WebGL Web GPU OpenGL UI Kageyama (Kobe Univ.) Visualization / 55
6 Kageyama (Kobe Univ.) Visualization / 55
7 homogeneous coordinates 3 x v 4 3 x x y z (1) 1 v v x v y v z 0 (2) Kageyama (Kobe Univ.) Visualization / 55
8 3 x v x y F (x). (3) Kageyama (Kobe Univ.) Visualization / 55
9 n u u = u u v = u i v j = u v cos ϕ v u φ Kageyama (Kobe Univ.) Visualization / 55
10 e i e j = δ ij v {e 0, e 1,..., e n 1 } v i v i = v e i Kageyama (Kobe Univ.) Visualization / 55
11 3 w = u v w i = ϵ ijk u j v k w u v w = u v sin ϕ u v = v u u (v w) = (u w) v (u v) w Kageyama (Kobe Univ.) Visualization / 55
12 M N M M ij (i, j = 0, 1,..., n 1) N N ij (i, j = 0, 1,..., n 1) L = MN n 1 L ij = M ik N kj = M ik N kj k=0 Kageyama (Kobe Univ.) Visualization / 55
13 (LM) N = L (MN) (L + M) N = LN + MN MI = IM = M I MN NM Kageyama (Kobe Univ.) Visualization / 55
14 M N MN = NM = I M 1 glmatrix.js 4 4 Kageyama (Kobe Univ.) Visualization / 55
15 det (M) det (I) = 1 det (MN) = det (M) det (N) det ( M t) = det (M) Kageyama (Kobe Univ.) Visualization / 55
16 M M ij (i, j = 0, 1,..., n 1) M t a M N (am) t = am t (M + N) t = M t + N t ( M t ) t = M (MN) t = N t M t Kageyama (Kobe Univ.) Visualization / 55
17 n 1 tr(m) = i=0 M ii Kageyama (Kobe Univ.) Visualization / 55
18 MM t = M t M = I M M t = M 1 det (M) = ±1 M t Mu = u u v = 0 (Mu) (Mv) = 0 Kageyama (Kobe Univ.) Visualization / 55
19 3 p u v x = p + s u + t v n u v/ u v n x + d = 0 n f(x) = n x + d f(x 0 ) = 0 x 0 f(x 0 ) > 0 x 0 p + n f(x 0 ) < 0 x 0 p n Kageyama (Kobe Univ.) Visualization / 55
20 3 p, q, r 3 S = 1 (p r) (q r) 2 x-y n S = 1 n 1 (x i y i+1 y i x i+1 ) = 1 n 1 {x i (y i+1 y i 1 )} 2 2 i=0 i=0 mod (n) Kageyama (Kobe Univ.) Visualization / 55
21 3 u, v, w 6 V = u (v w) = v (w u) = w (u v) w v u Kageyama (Kobe Univ.) Visualization / 55
22 3 = x x y = y z z 1 v x v y = v z v x v y v z 0 Kageyama (Kobe Univ.) Visualization / 55
23 M = M 00 M 01 M 02 0 M 10 M 11 M 12 0 M 20 M 21 M Kageyama (Kobe Univ.) Visualization / 55
24 T (t x, t y, t z ) = t x t y t z (4) Kageyama (Kobe Univ.) Visualization / 55
25 z R z (θ) = cos θ sin θ 0 0 sin θ cos θ (5) Kageyama (Kobe Univ.) Visualization / 55
26 S(s x, s y, s z ) = s x s y s z (6) Kageyama (Kobe Univ.) Visualization / 55
27 H xy (β) = 1 β (7) Kageyama (Kobe Univ.) Visualization / 55
28 M 1 M 2 M 2 M 1 Kageyama (Kobe Univ.) Visualization / 55
29 RT T R Kageyama (Kobe Univ.) Visualization / 55
30 RT T R Kageyama (Kobe Univ.) Visualization / 55
31 WebGL Safari Kageyama (Kobe Univ.) Visualization / 55
32 WebGL WebGL Kageyama (Kobe Univ.) Visualization / 55
33 WebGL WebGL Web アプリ HTML + CSS + JavaScript + シェーダソースコード + オブジェクトデータ WebGL JavaScript API 頂点シェーダ プリミティブ組み立て ラスタ化 フラグメントシェーダ シザーテスト マルチサンプリング ステンシルテスト デプステスト アルファブレンディング ディザリング WebGL 用描画バッファ cavas の他の画像 スクリーン Kageyama (Kobe Univ.) Visualization / 55
34 WebGL Web アプリ HTML + CSS + JavaScript + シェーダソースコード + オブジェクトデータ WebGL JavaScript API 頂点シェーダ プリミティブ組み立て ラスタ化 フラグメントシェーダ シザーテスト マルチサンプリング ステンシルテスト デプステスト アルファブレンディング ディザリング WebGL 用描画バッファ cavas の他の画像 スクリーン Kageyama (Kobe Univ.) Visualization / 55
35 WebGL Web アプリ HTML + CSS + JavaScript + シェーダソースコード + オブジェクトデータ WebGL JavaScript API 頂点シェーダ プリミティブ組み立て ラスタ化 フラグメントシェーダ Kageyama (Kobe Univ.) Visualization / 55
36 WebGL WebGL Web HTML + CSS + JavaScript WebGL HTML + CSS + JavaScript + OpenGL SL Kageyama (Kobe Univ.) Visualization / 55
37 WebGL n n Kageyama (Kobe Univ.) Visualization / 55
38 WebGL 頂点シェーダ用ソースコード (GLSL) 頂点 attribute 変数 ( 座標 色など ) 頂点シェーダ 組み込み変数 gl_position gl_frontfacing g_lpointsize ユーザ定義 uniform 変数 ( 変換行列 光源位置 ) ユーザ定義の varying 変数 Kageyama (Kobe Univ.) Visualization / 55
39 WebGL C OpenGL SL (Shading Language) 4 4 a t t r i b u t e vec3 avertexpos ; a t r r i b u t e vec4 a V e r t e x C o l o r ; u n i f o r m mat4 umvmatrix ; u n i f o r m mat4 upmatrix ; v a r y i n g vec4 v C o l o r ; void main ( ) { g l P o s i t i o n = upmatrix umvmatrix vec4 ( avertexpos, 1. 0 ) ; v C o l o r = a V e r t e x C o l o r ; } Kageyama (Kobe Univ.) Visualization / 55
40 WebGL a t t r i b u t e vec3 avertexpos ; a t r r i b u t e vec4 a V e r t e x C o l o r ; attribute RGBA 4 Kageyama (Kobe Univ.) Visualization / 55
41 WebGL u n i f o r m mat4 umvmatrix ; u n i f o r m mat4 upmatrix ; mat4 4 4 uniform Kageyama (Kobe Univ.) Visualization / 55
42 WebGL v a r y i n g vec4 v C o l o r ; varing (varying variable) varying gl Position gl FrontFacing gl PointSize Kageyama (Kobe Univ.) Visualization / 55
43 WebGL void main ( ) { g l P o s i t i o n = upmatrix umvmatrix vec4 ( avertexpos, 1. 0 ) ; v C o l o r = a V e r t e x C o l o r ; main varying gl Position varying vcolor Kageyama (Kobe Univ.) Visualization / 55
44 WebGL primitive assembly 3 OpenGL 1.x 3 Kageyama (Kobe Univ.) Visualization / 55
45 WebGL プリミティブ フラグメント Kageyama (Kobe Univ.) Visualization / 55
46 WebGL varying varying varying varyingvalue_1 varyingvalue_2 varyingvalue_3 Kageyama (Kobe Univ.) Visualization / 55
47 WebGL フラグメントシェーダ用 ソースコード (GLSL) 組み込み変数 gl_position gl_frontfacing g_lpointsize フラグメント シェーダ 組み込み変数 gl_fragcolor ユーザ定義 varying 変数 ユニフォーム変数 テクスチャ用サンプラ Kageyama (Kobe Univ.) Visualization / 55
48 WebGL p r e c i s i o n mediump f l o a t ; // p r e c i s i o n q u a l i f i e r v a r y i n g vec4 v C o l o r ; // void main ( ) { g l F r a g C o l o r = v C o l o r ; } Kageyama (Kobe Univ.) Visualization / 55
49 WebGL WebGL Web アプリ HTML + CSS + JavaScript + シェーダソースコード + オブジェクトデータ WebGL JavaScript API 頂点シェーダ プリミティブ組み立て ラスタ化 フラグメントシェーダ シザーテスト マルチサンプリング ステンシルテスト デプステスト アルファブレンディング ディザリング WebGL 用描画バッファ cavas の他の画像 スクリーン Kageyama (Kobe Univ.) Visualization / 55
50 WebGL scissors OpenGL Super Bible (2011, p.112) シザーテスト不合格 シザーテスト 合格 Kageyama (Kobe Univ.) Visualization / 55
51 WebGL OpenGL Super Bible (2011, p.382) Kageyama (Kobe Univ.) Visualization / 55
52 WebGL OpenGL Super Bible (2011, p.399) Kageyama (Kobe Univ.) Visualization / 55
53 WebGL Kageyama (Kobe Univ.) Visualization / 55
54 WebGL Kageyama (Kobe Univ.) Visualization / 55
55 WebGL Kageyama (Kobe Univ.) Visualization / 55
WebGL Kageyama (Kobe Univ.) Visualization / 39
WebGL *1 WebGL 2013.04.30 *1 X021 2013 LR301 Kageyama (Kobe Univ.) Visualization 2013.04.30 1 / 39 WebGL Kageyama (Kobe Univ.) Visualization 2013.04.30 2 / 39 3 1 PC ID Kageyama (Kobe Univ.) Visualization
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145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
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m v = mg + kv m v = mg k v v m v = mg + kv α = mg k v = α e rt + e rt m v = mg + kv v mg + kv = m v α + v = k m v (v α (v + α = k m ˆ ( v α ˆ αk v = m v + α ln v α v + α = αk m t + C v α v + α = e αk m
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II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
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