{"id":180,"date":"2024-05-16T23:55:51","date_gmt":"2024-05-16T14:55:51","guid":{"rendered":"https:\/\/techwalkin.net\/?p=180"},"modified":"2024-05-16T23:58:04","modified_gmt":"2024-05-16T14:58:04","slug":"bezier-clipping-2","status":"publish","type":"post","link":"https:\/\/techwalkin.net\/?p=180","title":{"rendered":"Bezier Clipping (2)"},"content":{"rendered":"\n

\u524d\u56de (BezierClipping(1))<\/a> \u306fBezier Clipping\u6cd5\u306e\u8a71\u306b\u5165\u308b\u524d\u306e\u5c0e\u5165\u3092\u3057\u307e\u3057\u305f\u3002\u4eca\u65e5\u306fBezier(\u30d9\u30b8\u30a8)\u66f2\u7dda<\/strong>\u306b\u3064\u3044\u3066\u8a71\u3092\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

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\\( \\int^{x_n}_{x_1} {f^{\\prime \\prime} (x) dx} \\)<\/p>\n\n\n\n

\u304c\u6700\u5c0f\u306b\u306a\u308b\u3088\u3046\u306b\u66f2\u7dda\u3092\u4f5c\u308b\u3053\u3068\u3082\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306b\u6ed1\u3089\u304b\u306a\u66f2\u7dda\u3092\u63cf\u304f\u3068\u3044\u3063\u3066\u3082\u8272\u3005\u306a\u8868\u73fe\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

<\/div>\n\n\n\n
<\/div>\n\n\n\n

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<\/div>\n\n\n\n

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\u30d9\u30b8\u30a7\u66f2\u7dda<\/strong>\uff08\u30d9\u30b8\u30a7\u304d\u3087\u304f\u305b\u3093\u3001B\u00e9zier Curve\uff09\u307e\u305f\u306f\u30d9\u30b8\u30a8\u66f2\u7dda<\/strong>\u3068\u306f\u3001N<\/em> \u500b\u306e\u5236\u5fa1\u70b9\u304b\u3089\u5f97\u3089\u308c\u308b N<\/em> \u2212 1 \u6b21\u66f2\u7dda<\/a>\u3067\u3042\u308b\u3002\u30d5\u30e9\u30f3\u30b9<\/a>\u306e\u81ea\u52d5\u8eca<\/a>\u30e1\u30fc\u30ab\u30fc\u3001\u30b7\u30c8\u30ed\u30a8\u30f3<\/a>\u793e\u306e\u30c9\u30fb\u30ab\u30b9\u30c6\u30ea\u30e7<\/a>\uff08\u82f1\u8a9e\u7248<\/a>\uff09 \u3068\u30eb\u30ce\u30fc<\/a>\u793e\u306e\u30d4\u30a8\u30fc\u30eb\u30fb\u30d9\u30b8\u30a7<\/a>\u306b\u3088\u308a\u72ec\u7acb\u306b\u8003\u6848\u3055\u308c\u305f\u3002\u30c9\u30fb\u30ab\u30b9\u30c6\u30ea\u30e7\u306e\u65b9\u304c\u5148\u3093\u3058\u3066\u3044\u305f\u304c\u3001\u305d\u306e\u8ad6\u6587\u304c\u516c\u77e5\u3068\u306a\u3089\u306a\u304b\u3063\u305f\u305f\u3081\u30d9\u30b8\u30a7\u306e\u540d\u304c\u51a0\u3055\u308c\u3066\u3044\u308b[1]<\/a><\/sup>\u3002\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u30fc<\/a>\u4e0a\u3067\u6ed1\u3089\u304b\u306a\u66f2\u7dda\u3092\u63cf\u304f\u306e\u306b2\u6b21\u30d9\u30b8\u30a7\u66f2\u7dda (Quadratic B\u00e9zier curve) \u30843\u6b21\u30d9\u30b8\u30a7\u66f2\u7dda (Cubic B\u00e9zier curve) \u306a\u3069\u304c\u5e83\u304f\u5229\u7528\u3055\u308c\u3066\u3044\u308b\u3002<\/p>\u30d9\u30b8\u30a7\u66f2\u7dda (\u30d9\u30b8\u30a7\u66f2\u7dda – Wikipedia<\/a>)<\/cite><\/blockquote><\/figure>\n\n\n\n

\u3088\u304f\u308f\u304b\u3089\u306a\u3044\u306e\u3067\u3001\u4ee5\u4e0b\u3067\u306f\u5177\u4f53\u7684\u306a\u6570\u5f0f\u3068\u30b0\u30e9\u30d5\u3067\u8aac\u660e\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

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<\/div>\n\n\n\n

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\"\"<\/figure>\n<\/div><\/div>\n\n\n\n
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\u30d9\u30b8\u30a8\u66f2\u7dda\u3067\u306f\u59cb\u70b9\u3068\u7d42\u70b9\u306f\u5fc5\u305a\u901a\u308a\u307e\u3059\u3002\u306a\u306e\u3067\u66f2\u7dda\u304c\u901a\u3089\u306a\u3044\u5236\u5fa1\u70b9\u306f \\( 3-2=1 \\) \u70b9\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u308c\u3092\u70b9\\(A\\)\u3068\u547c\u3076\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u56f3\u3092\u307f\u3066\u308f\u304b\u308b\u3088\u3046\u306b\u3001\u59cb\u70b9\u3068\u70b9 \\(A\\) \u3092\u7d50\u3076\u7dda\u306f\u59cb\u70b9\u3067\u306e\u66f2\u7dda\u306e\u63a5\u7dda<\/strong>\u306b\u306a\u3063\u3066\u3044\u307e\u3059\u3002\u307e\u305f\u7d42\u70b9\u3068\u70b9 \\(A\\) \u3092\u7d50\u3076\u7dda\u306f\u7d42\u70b9\u3067\u306e\u66f2\u7dda\u306e\u63a5\u7dda<\/strong>\u306b\u306a\u3063\u3066\u3044\u307e\u3059\u3002\u3064\u307e\u308a\u3001\u3053\u306e\u70b9\\( A \\) \u306f\u59cb\u70b9\u3068\u7d42\u70b9\u306e\u63a5\u7dda\u3092\u6c7a\u3081\u308b\u70b9\u3067\u3042\u308b<\/strong>\u3068\u3044\u3048\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n\n\n\n

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\"\"<\/figure>\n<\/div><\/div>\n\n\n\n
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\u66f2\u7dda\u304c\u901a\u3089\u306a\u3044\u5236\u5fa1\u70b9\u304c\\( 4-2=2 \\) \u70b9\u306b\u306a\u308a\u307e\u3057\u305f\u3002\u3059\u308b\u3068\u3084\u306f\u308a\u3001\u59cb\u70b9\u3068 \\( A \\) \u3092\u7d50\u3076\u7dda\u306f\u59cb\u70b9\u306e\u63a5\u7dda\u306b\u3001\u7d42\u70b9\u3068\\( B \\) \u3092\u7d50\u3076\u7dda\u306f\u7d42\u70b9\u306e\u63a5\u7dda\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

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<\/div>\n\n\n\n

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<\/div>\n\n\n\n

\u30d9\u30b8\u30a8\u66f2\u7dda\u3092\u6570\u5f0f\u3067\u8868\u3057\u307e\u3059\u3002\\(N\\) \u500b\u306e\u5236\u5fa1\u70b9\u3092 \\( b_i \\) \u3068\u7f6e\u304d\u307e\u3059\u3002\u3053\u3053\u3067\u3001Python\u306eindex\u304c0-\u3067\u3042\u308b\u3053\u3068\u304b\u3089, \\( (i=0, 1, …, N-1) \\) \u3068\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u57fa\u5e95\u95a2\u6570 \\( J^{n}_{k} (t) \\) \u3068\u5b9f\u6570\u306e\u5a92\u4ecb\u5909\u6570 \\( t \\in [0, 1] \\) \u3092\u7528\u3044\u3066\u3092\u30d9\u30b8\u30a8\u66f2\u7dda \\( P(t) \\) \u306f\u4e0b\u8a18\u306e\u3088\u3046\u306b\u66f8\u3051\u307e\u3059\u3002<\/p>\n\n\n\n

\\( P(t) = \\sum_{i=0}^{N-1} b_i \\cdot J^{N-1}_{i} (t) \\)<\/p>\n\n\n\n

\u3053\u306e\u57fa\u5e95\u95a2\u6570\u306f\u30d0\u30fc\u30f3\u30b9\u30bf\u30a4\u30f3\u57fa\u5e95\u95a2\u6570\u3068\u3088\u3070\u308c\u3001<\/p>\n\n\n\n

\\( J^{n}_{k} (t) = {}_n \\mathrm{C}_k \\cdot t^k (1-t)^{n-k} \\)<\/p>\n\n\n\n

\u3067\u3059\u3002\u3053\u306e\u66f2\u7dda\u306e\u610f\u5473\u3092\u8003\u3048\u308b\u4e0a\u3067\u3053\u306e\u57fa\u5e95\u95a2\u6570\u3092\u6df1\u5800\u308a\u3057\u3066\u3082\u9762\u767d\u3044\u306e\u3067\u3059\u304c\u3001\u3053\u3053\u3067\u306f\u3042\u3068\u306eBezier Clipping\u6cd5\u306b\u3064\u306a\u304c\u308b\u3088\u3046\u306b\u3001\u30c9\u30fb\u30ab\u30b9\u30c6\u30ea\u30e7\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0<\/strong>\u3092\u7d39\u4ecb\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n\n
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\u30d9\u30b8\u30a8\u66f2\u7dda\u306f \\( t \\) \u3092 0 \u304b\u3089 1 \u307e\u3067\u52d5\u304b\u3057\u3066\u59cb\u70b9\u304b\u3089\u7d42\u70b9\u307e\u3067\u306e\u66f2\u7dda\u3092\u4f5c\u6210\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\u3053\u306e\u3068\u304d\u3001\\(t = t_0 \\) \u306e\u3068\u304d\u306e\u66f2\u7dda\u4e0a\u306e\u70b9\u306e\u4f4d\u7f6e\u306f\u4e0b\u8a18\u306e\u3088\u3046\u306b\u3057\u3066\u4f5c\u56f3\u3067\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n\n\n\n

\n

\u96a3\u540c\u58eb\u306e\u5236\u5fa1\u70b9 (\u8d64\u8272<\/span>) \u3092\u7d50\u3093\u3067 \\( t : (1-t) \\) \u306b\u5185\u5206\u3057\u305f\u70b9 (\u7dd1\u8272<\/span>)\u3092\u66f8\u304d\u307e\u3059\u3002\u6b21\u306b\u305d\u306e\u96a3\u308a\u5408\u3046\u5185\u5206\u70b9 (\u7dd1\u8272<\/span>) \u3092\u7d50\u3093\u3060\u7dda\u5206\u3092 \\( t : (1-t) \\) \u306b\u5185\u5206\u3057\u305f\u70b9 (\u9ec4\u8272<\/span>) \u3092\u66f8\u304d\u307e\u3059\u3002\u3053\u308c\u3092 1 \u70b9\u306b\u306a\u308b\u307e\u3067\u7e70\u308a\u8fd4\u3057\u305f\u3068\u304d\u3001\u6700\u5f8c\u306e\u70b9\u304c\u66f2\u7dda\u4e0a\u306e\u70b9\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<\/div><\/div>\n\n\n\n

\u3053\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3067\u4f5c\u3063\u305f\u70b9\u3068\u5148\u306b\u7d39\u4ecb\u3057\u305f\u30d9\u30b8\u30a8\u66f2\u7dda\u306e\u65b9\u7a0b\u5f0f\u304c\u4e00\u81f4\u3059\u308b\u3053\u3068\u3092 \\(N=3\\) \u3092\u4f8b\u306b\u7c21\u5358\u306b\u624b\u8a08\u7b97\u3057\u3066\u793a\u3057\u307e\u3059\u3002\\(b_0, b_1, b_2 \\) \u306b\u3088\u3063\u3066\u4f5c\u3089\u308c\u308b2\u500b\u306e\u5185\u5206\u70b9\u306e\\( x \\)\u5ea7\u6a19\u306f\u305d\u308c\u305e\u308c\u3001<\/p>\n\n\n\n

\\( \\left\\{ \\begin{align*} b^{1}_{0} = (1-t) \\cdot b_0 + t\\cdot{b_1} \\\\ b^{1}_{1} = (1-t) \\cdot b_1 + t\\cdot b_2 \\end{align*} \\right. \\)<\/p>\n\n\n\n

\u3053\u3053\u3067\u53f3\u4e0a\u306b\u5185\u5206\u70b9\u304c\u5236\u5fa1\u70b9\u304b\u3089\u4f55\u4e16\u4ee3\u76ee\u304b\u306e\u6dfb\u3048\u5b57\u3092\u66f8\u304d\u307e\u3057\u305f\u3002\u540c\u69d8\u306b\u3053\u308c\u3089\u304b\u3089\u66f2\u7dda\u4e0a\u306e\u70b9\u306e \\(x\\) \u5ea7\u6a19\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\\( \\begin{split} b^2_0 &= (1-t) \\cdot b^1_0 + t \\cdot b^1_1 \\\\ &= (1-t) \\cdot \\{(1-t) \\cdot b_0 + t\\cdot b_1\\} + t \\cdot \\{ (1-t) \\cdot b_1 + t\\cdot b_2 \\} \\\\ &= (1-t)^2 b_0 + 2 t (1-t) b_1 + t^2 b_2 \\end{split} \\)<\/p>\n\n\n\n

\u3068\u306a\u3063\u3066\u305f\u3057\u304b\u306b\u516c\u5f0f\u901a\u308a\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

<\/div>\n\n\n\n
<\/div>\n\n\n\n

\u6700\u5f8c\u306b\u3053\u306e\u30d9\u30b8\u30a8\u66f2\u7dda\u306e\u7279\u5fb4\u3092\u7d39\u4ecb\u3057\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\u307e\u305a\u3001\\( N \\) \u500b\u306e\u5236\u5fa1\u70b9\u306b\u5bfe\u3057\u3066 \\( N-1 \\) \u6b21\u5143\u306e\u591a\u9805\u5f0f\u306b\u306a\u308b\u3053\u3068<\/strong>\u3067\u3059\u3002\u3053\u308c\u306f\u81ea\u7531\u5ea6\u3092\u4e0a\u3052\u3088\u3046\u3068\u3059\u308b\u3068\u6b21\u5143\u6570\u3082\u5897\u3048\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059\u3002\u4e00\u822c\u306b\u9ad8\u6b21\u5143\u306e\u66f2\u7dda\u306f\u3046\u306d\u308a\u304c\u751f\u3058\u3084\u3059\u304f3\u6b21\u5143\u524d\u5f8c\u306e\u591a\u9805\u5f0f\u304c\u3088\u304f\u4f7f\u308f\u308c\u307e\u3059\u3002\u3059\u308b\u3068\u5236\u5fa1\u70b9\u30924\u500b\u307e\u3067\u3057\u304b\u8a2d\u5b9a\u3067\u304d\u306a\u304f\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306a\u8ab2\u984c\u304b\u3089\u4f7f\u3044\u52dd\u624b\u306e\u826f\u3044 B-\u30b9\u30d7\u30e9\u30a4\u30f3\u95a2\u6570 <\/strong>\u3068\u3044\u3046\u767a\u60f3\u304c\u3067\u3066\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\u6b21\u306b\u3001\u30d9\u30b8\u30a8\u66f2\u7dda\u306f\u5e38\u306b\u5236\u5fa1\u70b9\u306e\u51f8\u5305\u306e\u4e2d\u306b\u3042\u308b<\/strong>\u3068\u3044\u3046\u91cd\u8981\u306a\u6027\u8cea\u304c\u3042\u308a\u307e\u3059\u3002\u51f8\u5305\u306b\u3064\u3044\u3066\u306f\u6b21\u56de\u7d39\u4ecb\u3057\u307e\u3059\u304c\u3001Bezier Clipping\u6cd5\u306e\u52b9\u7387\u6027\u306b\u8ca2\u732e\u3059\u308b\u91cd\u8981\u306a\u6027\u8cea\u3067\u3059\u3002<\/p>\n\n\n\n

\u6700\u5f8c\u306b\u3001\u30d9\u30b8\u30a8\u66f2\u7dda\u306f\u4efb\u610f\u306e\u7b87\u6240\u3067\u5206\u5272\u3067\u304d\u3001\u305d\u306e\u5206\u5272\u5f8c\u3082\u307e\u305f\u30d9\u30b8\u30a8\u66f2\u7dda\u306b\u306a\u308a\u307e\u3059<\/strong>\u3002\u5206\u5272\u5f8c\u306e\u5236\u5fa1\u70b9\u306e\u6c7a\u5b9a\u6cd5\u3082\u6b21\u56de\u4ee5\u964d\u306b\u8aac\u660e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

<\/div>\n\n\n\n

\u3053\u308c\u3067\u307f\u306a\u3055\u3093\u3082\u4eba\u751f\u306b\u304a\u3044\u3066\u6ed1\u3089\u304b\u306a\u66f2\u7dda\u306b\u56f0\u3063\u305f\u3068\u304d\u3082\u63cf\u3051\u307e\u3059\u3002\u697d\u3057\u3044\u6ed1\u3089\u304b\u66f2\u7dda\u30e9\u30a4\u30d5\u3092\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u524d\u56de (BezierClipping(1)) \u306fBezier Clipping\u6cd5\u306e\u8a71\u306b\u5165\u308b\u524d\u306e\u5c0e\u5165\u3092\u3057\u307e\u3057\u305f\u3002\u4eca\u65e5\u306fBezier(\u30d9\u30b8\u30a8)\u66f2\u7dda\u306b\u3064\u3044\u3066\u8a71\u3092\u3057\u3066\u3044\u304d\u307e\u3059\u3002 \u307f\u306a\u3055\u3093\u3082\u4e00\u751f\u306b\u4e00\u56de\u304f\u3089\u3044\u306f\u300c\u6ed1\u3089\u304b\u306a\u66f2\u7dda\u3092\u63cf\u304d\u305f […]<\/p>\n","protected":false},"author":1,"featured_media":242,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[14],"tags":[15,17,16,18],"class_list":["post-180","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-computing","tag-bezierclipping","tag-cagd","tag-16","tag-18"],"_links":{"self":[{"href":"https:\/\/techwalkin.net\/index.php?rest_route=\/wp\/v2\/posts\/180","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/techwalkin.net\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/techwalkin.net\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/techwalkin.net\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/techwalkin.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=180"}],"version-history":[{"count":55,"href":"https:\/\/techwalkin.net\/index.php?rest_route=\/wp\/v2\/posts\/180\/revisions"}],"predecessor-version":[{"id":243,"href":"https:\/\/techwalkin.net\/index.php?rest_route=\/wp\/v2\/posts\/180\/revisions\/243"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/techwalkin.net\/index.php?rest_route=\/wp\/v2\/media\/242"}],"wp:attachment":[{"href":"https:\/\/techwalkin.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=180"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/techwalkin.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=180"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/techwalkin.net\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=180"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}