{"id":1101,"date":"2018-01-16T23:30:17","date_gmt":"2018-01-16T14:30:17","guid":{"rendered":"http:\/\/naochang.me\/?p=1101"},"modified":"2018-01-16T23:45:00","modified_gmt":"2018-01-16T14:45:00","slug":"%e3%83%99%e3%82%b8%e3%82%a7%e6%9b%b2%e7%b7%9abezier-curve","status":"publish","type":"post","link":"https:\/\/stacstar.jp\/blog\/?p=1101","title":{"rendered":"\u30d9\u30b8\u30a7\u66f2\u7dda(B\u00e9zier Curve)"},"content":{"rendered":"<p><script async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.0\/MathJax.js?config=TeX-AMS_CHTML\"><\/script><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/curve_bezier.gif\" alt=\"\" width=\"600\" height=\"275\" class=\"border_light aligncenter size-full wp-image-1095\" \/><br \/>\n<a href=\"https:\/\/github.com\/naochang\/Curve\" rel=\"noopener\" target=\"_blank\">github | naochang | Curve<\/a><br \/>\nIllustrator\u7b49\u3067\u3082\u304a\u306a\u3058\u307f\u306e\u30d9\u30b8\u30a7\u66f2\u7dda\u3002\u30d9\u30b8\u30a7\u66f2\u7dda\u306b\u306f\u6b21\u6570\u304c\u3042\u308a\u3001Illustrator\u3067\u306f3\u6b21\u3001Flash\u3067\u306f\u7d75\u3092\u63cf\u304f\u3068\u304d\u306f3\u6b21\u3060\u3051\u3069\u3001\u5b9f\u884c\u3059\u308b\u3068\u304d\u306f2\u6b21\u306b\u5909\u63db\u3055\u308c\u308b\u3002(FlashPlayer11\u4ee5\u964d\u306f3\u6b21\u3082\u30b5\u30dd\u30fc\u30c8)<br \/>\n\u6b21\u6570\u304c\u5927\u304d\u3044\u307b\u3069\u5236\u5fa1\u70b9\u304c\u591a\u304f\u3001\u5c11\u306a\u3044\u30a2\u30f3\u30ab\u30fc\u30dd\u30a4\u30f3\u30c8\u3067\u7dda\u3092\u8abf\u6574\u304c\u3067\u304d\u308b\u304c\u3001\u51e6\u7406\u3082\u91cd\u305f\u304f\u306a\u308b\u3002<\/p>\n<p>\u3069\u306e\u3088\u3046\u306a\u4ed5\u7d44\u307f\u304b\u306f\u3001<a href=\"http:\/\/blog.sigbus.info\/2011\/10\/bezier.html\" rel=\"noopener\" target=\"_blank\">\u3053\u3061\u3089<\/a>\u306e\u30a2\u30cb\u30e1\u30fc\u30b7\u30e7\u30f3gif\u3092\u898b\u308b\u3068\u308f\u304b\u308a\u3084\u3059\u3044\u3002<\/p>\n<p>3\u6b21\u306e\u5834\u5408\u306f\u70b9\u304c4\u3064\u5fc5\u8981\u306b\u306a\u308a\u3001\u901a\u308b\u306e\u306f\u59cb\u70b9\u3068\u7d42\u70b9\u306e\u307f\u3002\u6b8b\u308a\u306e2\u70b9\u306f\u66f2\u7dda\u3092\u5236\u5fa1\u3059\u308b\u305f\u3081\u306b\u4f7f\u308f\u308c\u308b\u3002Illustrator\u306a\u3069\u3067\u3044\u3046\u3068\u3053\u308d\u306e\u30cf\u30f3\u30c9\u30eb\u304c\u305d\u308c\u306b\u3042\u305f\u308b\u30023\u6b21\u306e\u5834\u5408\u306f4\u3064\u306e\u59cb\u70b9\u3068\u7d42\u70b9\u306e\u307f\u7e4b\u3044\u3067\u3044\u304f\u306e\u3067\u3001\u305d\u308c\u3092\u6700\u521d\u306e\u70b9\u304b\u3089\u6700\u5f8c\u306e\u70b9\u307e\u3067\u7e70\u308a\u8fd4\u3057\u3066\u3001\u8907\u6570\u306e\u70b9\u304c\u306a\u3081\u3089\u304b\u306b\u7e4b\u304c\u308b\u3088\u3046\u306b\u898b\u305b\u308b\u304c\u3001\u7dda\u304c\u6ed1\u3089\u304b\u306b\u306a\u308b\u3088\u3046\u306b\u70b9\u306b\u3042\u308b2\u3064\u306e\u30cf\u30f3\u30c9\u30eb\u304c\u70b9\u3092\u631f\u3093\u3067\u7dda\u5bfe\u79f0\u3068\u306a\u308b\u3088\u3046\u306b\u914d\u7f6e\u3059\u308b\u3002\u7dda\u5bfe\u79f0\u3067\u306a\u3044\u5834\u5408\u306f\u3001Illustrator\u3067\u3044\u3046\u30cf\u30f3\u30c9\u30eb\u3092\u6298\u3063\u3066\u3044\u308b\u72b6\u614b\u3067\u3001\u6ed1\u3089\u304b\u3067\u306f\u306a\u304f\u306a\u308b\u3002\u9006\u306b\u3053\u3046\u884c\u3063\u305f\u64cd\u4f5c\u304c\u5bb9\u6613\u306b\u3067\u304d\u308b\u30d9\u30b8\u30a7\u66f2\u7dda\u306f\u7279\u306b2D\u3092\u63cf\u304f\u306e\u306b\u5411\u3044\u3066\u3044\u308b\u3093\u3067\u3059\u306d\u3002<\/p>\n<p><!--more--><\/p>\n<p>\u66f2\u7dda\u3067\u306f\u3069\u306e\u3088\u3046\u306b\u6ed1\u3089\u304b\u306a\u66f2\u7dda\u3092\u4f5c\u308b\u304b\u3068\u3044\u3046\u30d6\u30ec\u30f3\u30c7\u30a3\u30f3\u30b0\u95a2\u6570\u304c\u3042\u308a\u3001\u30d9\u30b8\u30a7\u66f2\u7dda\u3067\u306f\u30d0\u30fc\u30f3\u30b9\u30bf\u30a4\u30f3\u57fa\u5e95\u95a2\u6570\u3068\u3044\u3046\u30d6\u30ec\u30f3\u30c7\u30a3\u30f3\u30b0\u95a2\u6570\u3092\u4f7f\u3063\u3066\u3044\u308b\u3088\u3046\u3067\u3059\u3002\u30d9\u30b8\u30a7\u66f2\u7dda\u306e\u5f0f\u3092\u898b\u3066\u898b\u308b\u3068\u3002<\/p>\n<p>\\begin{align}<br \/>\nP(t) = \\sum_{i=0}^{N-1}P_iB_{(N-1)i}(t)<br \/>\n\\end{align}<\/p>\n<p>\\(P_i\\)\u306f\u5236\u5fa1\u70b9\u3001\\(N\\)\u306f\u5236\u5fa1\u70b9\u306e\u6570\u3002\\(B_ni(t)\\)\u304c\u30d0\u30fc\u30f3\u30b9\u30bf\u30a4\u30f3\u57fa\u5e95\u95a2\u6570\u3002\u30d0\u30fc\u30f3\u30b9\u30bf\u30a4\u30f3\u57fa\u5e95\u95a2\u6570\u306f<br \/>\n\\begin{align}<br \/>\nB_{ni}(t) = { n \\choose i }t^i(1 &#8211; t)^{n-i}<br \/>\n\\end{align}<\/p>\n<p>\u3053\u3053\u3067\\({n \\choose i}\\)\u306f\u4e8c\u9805\u4fc2\u6570\u3067\u3001\u7d44\u307f\u5408\u308f\u305b\u3068\u540c\u3058\u3002<br \/>\n\\begin{align}<br \/>\n{n \\choose k} = \\frac{n!}{k!(n-k)!}<br \/>\n\\end{align}<\/p>\n<p>\u4e0a\u8a18\u306e\u5f0f\u3092\u307e\u3068\u3081\u308b\u3068<br \/>\n\\begin{align}<br \/>\nP(t) = \\sum_{i=0}^{N-1}P_i(\\frac{(N-1)!}{i!((N-1)-i)!})t^i(1 &#8211; t)^{(N-1)-i}<br \/>\n\\end{align}<br \/>\n\u5236\u5fa1\u70b9\u306e\u6570 &#8211; 1 \u304c\u6b21\u6570\u306b\u306a\u308b\u306e\u3067\u6b21\u6570\u3092n\u3068\u3057\u3066\u8868\u73fe\u3059\u308b\u3068\u3053\u3046\u306a\u308b\u3002<br \/>\n\\begin{align}<br \/>\nP(t) = \\sum_{i=0}^nP_i(\\frac{n!}{i!(n-i)!})t^i(1 &#8211; t)^{n-i}<br \/>\n\\end{align}<\/p>\n<p>\u3042\u3068\u306f\u3001\u3059\u3067\u306b<a href=\"https:\/\/qiita.com\/edo_m18\/items\/643512f27c2b083b47ac\" rel=\"noopener\" target=\"_blank\">\u3053\u3061\u3089<\/a>\u3067\u975e\u5e38\u306b\u308f\u304b\u308a\u3084\u3059\u304f\u30d7\u30ed\u30b0\u30e9\u30e0\u5316\u3055\u308c\u3066\u3044\u308b\u306e\u3067\u3001\u305d\u308c\u3092\u306a\u305e\u308b\u3060\u3051\u3002\u3088\u304f\u4f7f\u3044\u305d\u3046\u306a2\u6b21\u30683\u6b21\u3092\u3042\u3089\u304b\u3058\u3081\u5c55\u958b\u3057\u3066\u304a\u304f\u3068\u30012\u6b21\u306e\u65b9\u306f<br \/>\n\\begin{align}<br \/>\n      P(t) &#038;= P_0(\\frac{2!}{0! \\times 2!})t^0(1-t)^2\\\\<br \/>\n      &#038;+ P_1(\\frac{2!}{1! \\times 1!})t^1(1-t)^1\\\\<br \/>\n      &#038;+ P_2(\\frac{2!}{1!})t^2(1-t)^0\\\\<br \/>\n      &#038;= P_0(1-t)^2 + 2P_1t(1-t) + 2P_2t^2<br \/>\n\\end{align}<br \/>\n3\u6b21\u306e\u65b9\u306f<br \/>\n\\begin{align}<br \/>\n      P(t) &#038;= P_0(\\frac{3!}{0! \\times 3!})t^0(1-t)^3\\\\<br \/>\n      &#038;+ P_1(\\frac{3!}{1! \\times 2!})t^1(1-t)^2\\\\<br \/>\n      &#038;+ P_2(\\frac{3!}{2! \\times 1!})t^2(1-t)^1\\\\<br \/>\n      &#038;+ P_3(\\frac{3!}{3! \\times 0!})t^3(1-t)^0\\\\<br \/>\n      &#038;= P_0(1-t)^3 + 3P_1t(1-t)^2 + 3P_2t^2(1-t) + P_3t^3<br \/>\n\\end{align}<\/p>\n<p>\u30cf\u30f3\u30c9\u30eb\u306e\u4f4d\u7f6e\u304c\u30a2\u30f3\u30ab\u30fc\u30dd\u30a4\u30f3\u30c8\u306e\u70b9\u5bfe\u8c61\u306b\u3042\u308c\u3070\u6ed1\u3089\u304b\u306a\u30ab\u30fc\u30d6\u3068\u306a\u308b\u304c\u3001\u4f4d\u7f6e\u304c\u9055\u3046\u3068\u6298\u308c\u66f2\u304c\u3063\u305f\u30ab\u30fc\u30d6\u3068\u306a\u308b\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/curve_bezier_independent_handle.gif\" alt=\"\" width=\"600\" height=\"275\" class=\"border_dark aligncenter size-full wp-image-1178\" \/><\/p>\n<p>\u53c2\u8003\uff09<br \/>\n<a href=\"https:\/\/qiita.com\/edo_m18\/items\/643512f27c2b083b47ac\" rel=\"noopener\" target=\"_blank\">[\u6570\u5b66] 3D\u7a7a\u9593\u306b\u30d9\u30b8\u30a7\u66f2\u7dda\u3092\u63cf\u304f<\/a><br \/>\n<a href=\"http:\/\/blog.sigbus.info\/2011\/10\/bezier.html\" rel=\"noopener\" target=\"_blank\">\u4e2d\u5b66\u751f\u3067\u3082\u308f\u304b\u308b\u30d9\u30b8\u30a7\u66f2\u7dda<\/a><\/p>\n<p><script src=\"http:\/\/gist-it.appspot.com\/github\/naochang\/Curve\/blob\/master\/Assets\/Scripts\/Curve\/BezierCurve.cs\"><\/script><\/p>\n","protected":false},"excerpt":{"rendered":"<p>github | naochang | Curve Illustrator\u7b49\u3067\u3082\u304a\u306a\u3058\u307f\u306e\u30d9\u30b8\u30a7\u66f2\u7dda\u3002\u30d9\u30b8\u30a7\u66f2\u7dda\u306b\u306f\u6b21\u6570\u304c\u3042\u308a\u3001Illustrator\u3067\u306f3\u6b21\u3001Flash\u3067\u306f\u7d75\u3092\u63cf\u304f\u3068\u304d\u306f3\u6b21\u3060\u3051\u3069\u3001\u5b9f\u884c\u3059\u308b\u3068\u304d\u306f2\u6b21\u306b [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4,17,16,2],"tags":[],"class_list":["post-1101","post","type-post","status-publish","format-standard","hentry","category-c","category-curve","category-math","category-unity"],"_links":{"self":[{"href":"https:\/\/stacstar.jp\/blog\/index.php?rest_route=\/wp\/v2\/posts\/1101","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/stacstar.jp\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/stacstar.jp\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/stacstar.jp\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/stacstar.jp\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1101"}],"version-history":[{"count":63,"href":"https:\/\/stacstar.jp\/blog\/index.php?rest_route=\/wp\/v2\/posts\/1101\/revisions"}],"predecessor-version":[{"id":1515,"href":"https:\/\/stacstar.jp\/blog\/index.php?rest_route=\/wp\/v2\/posts\/1101\/revisions\/1515"}],"wp:attachment":[{"href":"https:\/\/stacstar.jp\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1101"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/stacstar.jp\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1101"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/stacstar.jp\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1101"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}