{"id":349,"date":"2017-11-28T22:46:50","date_gmt":"2017-11-28T14:46:50","guid":{"rendered":"http:\/\/www.whudj.cn\/?p=349"},"modified":"2017-11-28T23:01:46","modified_gmt":"2017-11-28T15:01:46","slug":"nurbs%e5%85%88%e5%af%bc%e9%9b%86-%e5%8f%82%e6%95%b0%e6%9b%b2%e7%ba%bf%ef%bc%8c%e5%88%87%e7%ba%bf%ef%bc%8c%e6%9b%b2%e7%8e%87%e4%b8%8e%e8%bf%9e%e7%bb%ad%e6%80%a7","status":"publish","type":"post","link":"http:\/\/www.whudj.cn\/?p=349","title":{"rendered":"NURBS\u5148\u5bfc\u96c6: \u53c2\u6570\u66f2\u7ebf\uff0c\u5207\u7ebf\uff0c\u66f2\u7387\u4e0e\u8fde\u7eed\u6027"},"content":{"rendered":"

\u4ece\u672c\u7bc7\u5f00\u59cb\uff0c\u6211\u5c06\u9646\u7eed\u4ecb\u7ecdNURBS(Non-Uniform Relational B-Spline\u00a0\u975e\u5747\u5300\u6709\u7406B\u6837\u6761)\u6709\u5173\u7684\u7b97\u6cd5\u4ee5\u53ca\u5b9e\u73b0\u3002\u4f5c\u4e3a\u80cc\u666f\uff0c\u6211\u4eec\u6709\u5fc5\u8981\u4e86\u89e3\u53c2\u6570\u66f2\u7ebf\u7684\u77e5\u8bc6\u3002<\/p>\n

\u53c2\u6570\u66f2\u7ebf\u7684\u5f62\u5f0f<\/h3>\n

\u6211\u4eec\u5728\u63d0\u5230\u6570\u5b66\u51fd\u6570\u65f6\uff0c\u6700\u5148\u60f3\u5230\u7684\u662f\\(y=f(x)\\),\\(z=f(x,y)\\)\u6216\u8005\u66f4\u4e00\u822c\u7684\\(f(x,y,z,…)=0\\)\u7684\u5f62\u5f0f\u3002\u4f46\u662f\u8fd9\u79cd\u5f62\u5f0f\u5e76\u4e0d\u4fbf\u4e8e\u8ba1\u7b97\u673a\u7f16\u7a0b\u5b9e\u73b0\u3002\u4f7f\u7528\u53c2\u6570\u66f2\u7ebf\u5f62\u5f0f\u66f4\u597d\u3002\u53c2\u6570\u66f2\u7ebf\u662f\u5c06\u66f2\u7ebf\uff08\u65b9\u7a0b\uff09f\u00a0\u4ee5 \\(x=f_x (t) ,y=f_y(t)\\)\u7684\u5f62\u5f0f\u6765\u8868\u8fbe\u3002\u6bd4\u5982\u76f4\u7ebf\uff1a$$\\begin{equation}\\left\\{\\begin{aligned}
\n\\overset{.}
\nx &=x_0+v_x t\\\\
\ny&=y_0 + v_y t\\\\
\nz&=z_0 + v_z t\\\\
\n\\end{aligned}
\n\\right.
\n\\end{equation}$$
\n\u5706\uff1a$$\\begin{equation}\\left\\{\\begin{aligned}
\n\\overset{.}
\nx &=a + cos(\\theta)r\\\\
\ny&=b + sin(\\theta)r\\\\
\n\\end{aligned}
\n\\right.
\n\\end{equation}$$<\/p>\n

\u53c2\u6570\u66f2\u7ebf\u5efa\u7acb\u4e86\\(t\\rightarrow (x,y,z)\\)\u7684\u6620\u5c04\uff0c\u5373\u5bf9\u5e94\u5b9a\u4e49\u57df\u5185\u4efb\u610f\u4e00\u70b9t,\u5747\u6709\u7a7a\u95f4\u4f4d\u7f6e\\((x,y,z)\\)\u4e0e\u4e4b\u5bf9\u5e94\u3002\u53c2\u6570\u7684\u53d6\u503c\uff0c\u6709\u65f6\u5019\u5177\u6709\u5176\u7269\u7406\u610f\u4e49\u3002\u6bd4\u5982\u5e38\u89c1\u7684\u4ee5\u65f6\u95f4t\u4e3a\u53c2\u6570\u8868\u8fbe\u7684\u91cc\u7a0b\u4e0e\u65f6\u95f4\u7684\u5173\u7cfb\u66f2\u7ebf\u3002NURBS\u66f2\u7ebf\u5c31\u662f\u4ee5\u53c2\u6570\u66f2\u7ebf\u7684\u5f62\u5f0f\u5b9a\u4e49\u7684\u3002<\/p>\n

\u6b63\u5207\u5411\u91cf\u4e0e\u5207\u7ebf<\/h3>\n

\u5728\u53c2\u6570\u66f2\u7ebf\u4e0a\u5bf9\u5e94\u53c2\u6570t\u5904\u7684\u6b63\u5207\u5411\u91cf,\u4e3a\u5404\u4e2a\u7ef4\u5ea6\u5bf9t\u6c42\u5bfc\u540e\u7684\u6570\u5b57\u6784\u6210\u7684\u5411\u91cf\uff0c\u5982\u679c\u53c2\u6570\u66f2\u7ebf\u5f62\u5f0f\u4e3a \\(F(t)=(f(t),h(t),g(t))\\)\uff0c\u90a3\u4e48\u5176\u6b63\u5207\u5411\u91cf\u53ef\u4ee5\u8868\u8fbe\u4e3a \\(F'(t)=[f'(t),h'(t),g'(t)]^T\\)\u3002\u901a\u5e38\uff0c\u8fd9\u6837\u6c42\u51fa\u6b63\u5207\u5411\u91cf\u7684\u6a21\u957f\u4e0d\u552f\u4e00\uff0c\u9700\u8981\u5c06\u5176\u5f52\u4e00\u5316\u3002<\/p>\n

\u53c2\u6570\u66f2\u7ebf\u5728t\u5904\u7684\u5207\u7ebf\u4e3a \\(l = [f(t),h(t),g(t)]^T + [f'(t),h'(t),g'(t)]^Tu\\)\u3002\u6211\u7ed9\u51fa\u7684\u5207\u7ebf\u7684\u5b9a\u4e49\u4e5f\u662f\u53c2\u6570\u66f2\u7ebf\u3002<\/p>\n

\u66f2\u7387<\/h3>\n

\u66f2\u7387\u662f\u66f2\u7387\u534a\u5f84\u7684\u5012\u6570\uff0c\u662f\u7528\u6765\u63cf\u8ff0\u66f2\u7ebf\u5728\u67d0\u70b9\u5904\u5f2f\u66f2\u7a0b\u5ea6\u7684\u6307\u6807\u3002\u4ece\u9ad8\u4e2d\u8bfe\u672c\u4e0a\uff0c\u6211\u4eec\u5df2\u7ecf\u77e5\u9053\\(y=f(x)\\)\u7684\u66f2\u7387\u4e3a\uff1a$$\\kappa =\\frac{|y”|}{(1+y’^2)^{3\/2}}$$\u53c2\u6570\u66f2\u7ebf\u7684\u66f2\u7387\u662f\u8fd9\u6837\u7684\uff1a$$\\kappa = \\frac{|f'(t)\\times f”(t)|}{|f'(t)^3|}$$<\/p>\n

\u6211\u4eec\u53ef\u4ee5\u8bc1\u660e\u4e00\u4e0b\\(y=f(x)\\)\u7684\u66f2\u7387\u53ea\u662f\u53c2\u6570\u66f2\u7ebf\u66f2\u7387\u7684\u7279\u6b8a\u5f62\u5f0f\u3002\\(y=f(x)\\)\u53ef\u4ee5\u7528 \\(y=f(t),x=t\\)\u6765\u8868\u793a\uff0c\u90a3\u4e48 \\(F'(t) = [f'(x),1]^T,F”(t)=[f”(x),0]\\)\u3002$$\\kappa = \\frac{|f'(x)\\times 0 – f”(x)\\times 1|}{|1*1+f'(x)^2|^{3\/2}} = \\frac{|y”|}{(1+y’^2)^{3\/2}}$$<\/p>\n

\u6240\u4ee5\uff0c\u4ee5\u540e\u5927\u5bb6\u76f4\u63a5\u8bb0\u4f4f\u53c2\u6570\u66f2\u7ebf\u7684\u6c42\u5bfc\u516c\u5f0f\u5c31\u597d\u4e86\u3002<\/p>\n

\u8fde\u7eed\u6027<\/h3>\n

\u5bf9\u4e8e\u4e24\u6bb5\u8fde\u7eed\u7684\u66f2\u7ebf\uff0c\u8003\u8651\u4ece\u4e00\u4e2a\u66f2\u7ebf\u8fc7\u6e21\u5230\u53e6\u4e00\u4e2a\u66f2\u7ebf\u7684\u201c\u5e73\u6ed1\u201d\u7a0b\u5ea6\uff0c\u6210\u4e3a\u53c2\u6570\u66f2\u7ebf\u7684\u8fde\u7eed\u6027\u95ee\u9898\u3002\u5f88\u591a\u4e1a\u52a1\u90fd\u5bf9\u66f2\u7ebf\u63d0\u51fa\u4e86\u8fde\u7eed\u6027\u8981\u6c42\u3002\u5982\u516c\u8def\u8bbe\u8ba1\u3002\u5047\u8bbe\u4f60\u5728\u516c\u8def\u4e0a\u884c\u9a76\uff0c\u5f53\u4ece\u4e00\u6bb5\u516c\u8def\u884c\u9a76\u5230\u53e6\u4e00\u6bb5\u516c\u8def\u65f6\uff0c\u5982\u679c\u8fde\u63a5\u5904\u51e0\u4f55\u5750\u6807\u76f8\u540c\uff08\u5f53\u7136\uff0c\u5426\u5219\u4f60\u5c31\u6389\u6c9f\u91cc\u53bb\u4e86\uff09\uff0c\u90a3\u4e48\u6211\u4eec\u79f0\u53c2\u6570\u66f2\u7ebf\\(C^0\\)\u8fde\u7eed\u3002\u5982\u679c\u5728\u5207\u6362\u65f6\u901f\u5ea6\u5927\u5c0f\u65b9\u5411\u4e0d\u53d8\uff08\u5373\u5207\u5411\u91cf\u4e0d\u53d8\uff09\uff0c\u90a3\u4e48\u79f0\u4e4b\u4e3a\\(C^1\\)\u8fde\u7eed\u3002\u5982\u679c\u52a0\u901f\u5ea6\u6ca1\u6709\u53d8\u5316\uff0c\u5219\u6210\u4e3a\\(C^2\\)\u8fde\u7eed\uff0c\u4ee5\u6b64\u7c7b\u63a8\u3002\u5c31\u662f\u8bf4\uff0c\u5982\u679c\u4e24\u4e2a\u66f2\u7ebf\u80fd\u591f\u4fdd\u8bc10\u5230k\u9636\u7684\u5bfc\u6570\u5411\u91cf\u5747\u76f8\u7b49\uff0c\u90a3\u4e48\u79f0\u4e4b\u4e3a\\(C^k\\)\u8fde\u7eed\u3002\u5982\u679c\u66f2\u7387\u6ca1\u6709\u53d1\u751f\u53d8\u5316\uff0c\u5219\u79f0\u4e3a\u66f2\u7387\u8fde\u7eed\u3002\u6ce8\u610f\uff0c\u5982\u679c\\(C^2\\)\u8fde\u7eed\uff0c\u90a3\u4e48\u66f2\u7387\u4e00\u5b9a\u8fde\u7eed\uff0c\u5982\u679c\u66f2\u7387\u8fde\u7eed\uff0c\u4e0d\u4e00\u5b9a\\(C^2\\)\u8fde\u7eed\u3002\u66f2\u7387\u662f\u4e00\u4e2a\u6807\u91cf\u3002<\/p>\n

\u7279\u522b\u9700\u8981\u63d0\u5230\u7684\u662f\uff0c\u53c2\u6570\u7684\u9009\u53d6\u4e0e\u8fde\u7eed\u6027\u6709\u76f4\u63a5\u5173\u7cfb\u3002\u8fd8\u4ee5\u516c\u91cc\u884c\u9a76\u4e3a\u4f8b\uff0c\u5982\u679c\u4ee5\u65f6\u95f4t\u4e3a\u53c2\u6570\u8868\u793a\u91cc\u7a0b\uff0c\u90a3\u4e48\u5207\u5411\u91cf\u5c31\u662f\u901f\u5ea6v\u3002\u5982\u679c\u5728\u4e00\u6bb5\u8def\u4e0a\uff0c\u4f60\u7684\u901f\u5ea6\u8868\u662f\u4ee5\u00a0\u516c\u91cc\/\u5c0f\u65f6\u8bb0\u5f55\u7684\uff0c\u53e6\u4e00\u6bb5\u8def\u4e0a\uff0c\u4f60\u7684\u901f\u5ea6\u8ba1\u6539\u4e3a mile\/h\u3002\u5373\u4f7f\u4f60\u5b9e\u9645\u4e0a\u5e76\u6ca1\u6709\u6539\u53d8\u8f66\u901f\uff0c\u4f46\u662f\u56e0\u4e3a\u4e24\u6bb5\u8def\u53c2\u6570\u5316\u65b9\u5f0f\u7684\u4e0d\u4e00\u6837\uff0c\u4ecd\u7136\u8fbe\u4e0d\u5230\\(C^1\\)\u8fde\u7eed\u3002<\/p>\n

\u56e0\u6b64\uff0c\u6211\u4eec\u91c7\u7528\u201cGeometric Continuity\u201d\uff0c\u51e0\u4f55\u8fde\u7eed\u6765\u8868\u8fbe\u8fd9\u6837\u7684\u8fde\u7eed\u95ee\u9898\uff0c\u5373\u5982\u679c\u5b58\u5728\u4e24\u79cd\u53c2\u6570\u5316\u65b9\u5f0f\uff0c\u80fd\u591f\u4f7f\u5f97\u4e24\u4e2a\u53c2\u6570\u66f2\u7ebf\\(C^k\\)\u8fde\u7eed\uff0c\u90a3\u4e48\u5219\u8ba4\u4e3a\u4e24\u4e2a\u66f2\u7ebf\\(G^k\\)\u8fde\u7eed\u3002\u5bf9\u4e8e\u6211\u4eec\u5e38\u7528\u7684\\(G^1,G^2\\)\uff0c\u53ef\u4ee5\u7528\u66f4\u7279\u5b9a\u7684\u65b9\u5f0f\u63cf\u8ff0\uff1a<\/p>\n

    \n
  1. \u4e24\u4e2a\\(C^0\\)\u66f2\u7ebf\u5f53\u4e14\u4ec5\u5f53\u5728\u8fde\u63a5\u5904\u4e00\u9636\u5bfc\u6570\u65b9\u5411\u76f8\u540c\u65f6\uff08\u6ce8\u610f\uff0c\u4e0d\u80fd\u662f\u76f8\u53cd\uff0c\u5373\u4e24\u4e2a\u5207\u5411\u91cf\u70b9\u79ef\u5927\u4e8e0\uff09\\(G^1\\)\u8fde\u7eed\u3002<\/li>\n
  2. \u4e24\u4e2a\\(C^1\\)\u66f2\u7ebf\u5f53\u4e14\u4ec5\u5f53\u5728\u8fde\u63a5\u5904\u7684\u00a0\u4e8c\u9636\u5bfc\u6570\u5dee\u503c\\(f”(u)-g”(u)\\)\u4e0e\u4e00\u9636\u5bfc\u6570\u5e73\u884c\u65f6\\(G^2\\)\u8fde\u7eed\u3002<\/li>\n<\/ol>\n

    \u5728\u4e00\u4e9b\u5546\u4e1a\u8f6f\u4ef6\u4e2d\uff0c\u6709\u5bf9\u5e94\u7684\u5b9e\u73b0\u3002\u6211\u539f\u6765\u505a\u5730\u56fe\u65f6\u7528\u5230\u7684coreldraw ,\u5373\u6709\u975e\u5e38\u65b9\u4fbf\u7684\u8fde\u63a5\u5de5\u5177\uff0c\u4f7f\u5f97bezier\u66f2\u7ebf\u8fde\u63a5\u65f6\u4fdd\u8bc1\u65b9\u5411\u4e0d\u53d1\u751f\u53d8\u5316\u3002<\/p>\n

    \n

    \u672c\u6587\u53c2\u8003\u4e86\u00a0Dr. Ching-Kuang Shene<\/a>\u00a0\u7684\u00a0 Introduction to Computing with Geometry<\/a>\u00a0\u8bfe\u7a0b\uff0c\u975e\u5e38\u68d2\u7684\u6750\u6599\uff0c\u770b\u5230\u6d59\u5927CAD\u5b9e\u9a8c\u5ba4\u7684\u6709\u5173nurbs\u7684\u6559\u7a0b\u4e5f\u53c2\u8003\u4e86\u3002\u63a8\u8350\u9605\u8bfb\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

    \u4ece\u672c\u7bc7\u5f00\u59cb\uff0c\u6211\u5c06\u9646\u7eed\u4ecb\u7ecdNURBS(Non-Uniform Relational … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[23],"tags":[],"_links":{"self":[{"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/posts\/349"}],"collection":[{"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=349"}],"version-history":[{"count":17,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/posts\/349\/revisions"}],"predecessor-version":[{"id":373,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/posts\/349\/revisions\/373"}],"wp:attachment":[{"href":"http:\/\/www.whudj.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=349"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=349"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=349"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}