\u5728\u66f2\u7ebf\u62df\u5408\u95ee\u9898\u4e2d\uff0c\u901a\u5e38\u9700\u8981\u6839\u636e\u5df2\u77e5\u66f2\u7ebf\u4e0a\u7684\u79bb\u6563\u70b9\uff0c\u4f30\u7b97\u51fa\u66f2\u7ebf\u5728\u7aef\u70b9\u5904\u7684\u5bfc\u6570\uff08\u4e25\u683c\u6765\u8bf4\u662f\u5bfc\u77e2\uff09\uff0c\u5e38\u7528\u7684\u4e00\u79cd\u5bfc\u6570\u4f30\u8ba1\u65b9\u6cd5\u79f0\u4e3a\u201cBessel Tangent\u201d\u65b9\u6cd5\u3002<\/p>\n
\u5df2\u77e5\u70b9\\(P_{i-1},P_i,P_{i+1}\\)\uff0c\u6c42\u4e00\u79cd\u5bfc\u6570\uff08\u77e2\uff09\u4f30\u7b97\u7b97\u6cd5\uff0c\u8ba1\u7b97\\(P_{i-1}^\\prime,P_i^\\prime,P_{i+1}^\\prime\\)\u3002Bessel tangent \u7b97\u6cd5\u7684\u4e3b\u8981\u601d\u8def\u662f\uff0c\u63d2\u503c\u4e00\u6761\u901a\u8fc7\u4e09\u70b9\u7684\u4e8c\u6b21\u66f2\u7ebf\uff0c\u7136\u540e\u901a\u8fc7\u8ba1\u7b97\u66f2\u7ebf\u4e0a\u4e09\u70b9\u7684\u5bfc\u6570\u7684\u65b9\u5f0f\u7ed9\u51fa\u4e09\u70b9\u7684\u4f30\u8ba1\u5bfc\u6570\u3002<\/p>\n
\u4ee4 \\(P_{i-1},P_{i},P_{i+1}\\) \u7684\u53c2\u6570\u5206\u522b\u4e3a \\(t_{i-1},t_{i},t_{i+1}\\)<\/p>\n
\u63d2\u503c\u7684\u66f2\u7ebf\u7684\u8868\u8fbe\u5f0f\u5982\u4e0b\uff1a<\/p>\n
$$P(t)=\\frac{(t-t_i)(t-t_{i+1})}{(t_{i-1}-t_i)(t_{i-1}-t_{i+1})}(P_{i-1}-P_{i})+\\frac{(t-t_{i-1})(t-t_{i})}{(t_{i+1}-t_{i-1})(t_{i+1}-t_{i})}(P_{i+1}-P_{i})+P_{i}\\tag1$$<\/p>\n
1) \\(P_i\\)\u5bfc\u6570<\/strong><\/p>\n $$P(t_i)^\\prime=\\frac{t_i-t_{i+1}}{(t_{i-1}-t_i)(t_{i-1}-t_{i+1})}(P_{i-1}-P_{i})+\\frac{t_i-t_{i-1}}{(t_{i+1}-t_{i-1})(t_{i+1}-t_{i})}(P_{i+1}-P_{i})$$<\/p>\n \u4ee4 \\(\\triangle t_i = t_{i+1}-t_i;\\triangle P_i=\\frac{P_{i+1}-P_i}{\\triangle t_i}\\)<\/p>\n $$\\begin{align} 2)\\(P_{i-1},P_{i+1}\\)\u5bfc\u6570<\/strong><\/p>\n \u6839\u636e\u516c\u5f0f(1)<\/p>\n $$\\begin{align} $$\\begin{align} \u53c2\u6570\u5316\u7684\u9009\u62e9<\/strong><\/p>\n \u5bf9\u4e8e\u4e09\u70b9\u7684\u53c2\u6570\u53ef\u4ee5\u6309\u7167\u7d2f\u8ba1\u5f26\u957f\u5316\u786e\u5b9a\u53c2\u6570$$ \\left\\{ \u6211\u5bf9\u7b97\u6cd5\u7684C++\u5b9e\u73b0\u662f\uff1a<\/p>\n \u53c2\u8003\u8d44\u6599\uff1a<\/p>\n The Essentials of CAGD : Chapter 11 Working with B Spline Curves\u00a0<\/a><\/p>\n \u4e8e\u8c26. \u57fa\u4e8e\u5207\u5411\u7ea6\u675f\u7684\u4e8c\u6b21B\u6837\u6761\u63d2\u503c\u66f2\u7ebf\u7684\u7814\u7a76[D]. \u5c71\u4e1c\u5927\u5b66, 2009.<\/p>\n","protected":false},"excerpt":{"rendered":" \u5728\u66f2\u7ebf\u62df\u5408\u95ee\u9898\u4e2d\uff0c\u901a\u5e38\u9700\u8981\u6839\u636e\u5df2\u77e5\u66f2\u7ebf\u4e0a\u7684\u79bb\u6563\u70b9\uff0c\u4f30\u7b97\u51fa\u66f2\u7ebf\u5728\u7aef\u70b9\u5904\u7684\u5bfc\u6570\uff08\u4e25\u683c … Continue reading
\nP(t_i)^\\prime&=\\frac{t_i-t_{i+1}}{t_{i-1}-t_{i+1}}\\triangle P_{i-1}+\\frac{t_i-t_{i-1}}{t_{i+1}-t_{i-1}}\\triangle P_{i}\\\\
\n&=\u00a0\\frac{t_i-t_{i+1}}{t_{i-1}-t_{i+1}}\\triangle P_{i-1}+\\frac{t_i-t_{i-1}}{t_{i+1}-t_{i-1}}\\triangle P_{i} \\\\
\n&= \\frac{\\triangle t_{i}}{t_{i+1}-t_{i}+t_{i}-t_{i-1}}\\triangle P_{i-1} + \\frac{\\triangle t_{i-1}}{t_{i+1}-t_{i}+t_{i}-t_{i-1}}\\triangle P_{i}\\\\
\n&=\\frac{\\triangle t_{i}}{\\triangle t_{i}+\\triangle t_{i-1}}\\triangle P_{i-1}+\\frac{\\triangle t_{i-1}}{\\triangle t_{i}+\\triangle t_{i-1}}\\triangle P_{i} \\tag2
\n\\end{align}$$\u5373\\(P_i^\\prime\\)\u662f\\(\\triangle P_{i-1}\\)\uff0c\\(\\triangle P_{i}\\)\u7684\u52a0\u6743\u5e73\u5747\u3002<\/p>\n
\nP(t_{i-1})^\\prime&=\\frac{2t_{i-1}-2t_{i+1}+t_{i+1}-t_{i}}{t_{i-1}-t_{i+1}}\\triangle P_{i-1}+\\frac{t_{i-1}-t_{i}}{t_{i+1}-t_{i-1}}\\triangle P_{i}\\\\
\n&=2\\triangle P_{i-1}-(\\frac{\\triangle t_{i}}{\\triangle t_{i}+\\triangle t_{i-1}}\\triangle P_{i-1}+\\frac{\\triangle t_{i-1}}{\\triangle t_{i}+\\triangle t_{i-1}}\\triangle P_{i})\\\\
\n&=2\\triangle P_{i-1}-P(t_i)^\\prime \\tag3
\n\\end{align}$$<\/p>\n
\nP(t_{i+1})^\\prime&=-(\\frac{\\triangle t_{i}}{\\triangle t_{i}+\\triangle t_{i-1}}\\triangle P_{i-1}+\\frac{\\triangle t_{i-1}}{\\triangle t_{i}+\\triangle t_{i-1}}\\triangle P_{i} )+2\\triangle P_i \\\\
\n&=2\\triangle P_i-P(t_{i})^\\prime \\tag4
\n\\end{align}$$<\/p>\n
\n\\begin{aligned}
\nt_{i-1} & = 0 \\\\
\nt_{i} & = t_0+|P_{i-1}P_i| \\\\
\nt_{i+1} & = t_1+|P_iP_{i+1}|
\n\\end{aligned}
\n\\right.
\n$$<\/p>\n \/*!\r\n *\\brief Bessel Tangent \u65b9\u6cd5\u4f30\u7b97\u5bfc\u6570\r\n *\\ param const Point & p0,p1,p2 \u5f85\u6c42\u70b9\r\n *\\ param Vec3d & p0deriv,p1deriv,p2deriv \u4e09\u70b9\u4e0a\u5bfc\u6570\u7684\u4f30\u8ba1\u503c\r\n *\\ Returns: \r\n *\/\r\nvoid BesselTanget(const Point& p0,const Point& p1,const Point& p2, \r\n\t\t\t\t Vec3d& p0deriv,Vec3d& p1deriv,Vec3d& p2deriv)\r\n{\r\n\tdouble delta_t1 =PointDistance(p2,p1);\r\n\tdouble delta_t0 = PointDistance(p1,p0);\r\n\tVec3d delta_p0 = (p1-p0)\/delta_t0;\r\n\tVec3d delta_p1 = (p2-p1)\/delta_t1;\r\n\tdouble sum = delta_t0+delta_t1;\r\n\tp1deriv = delta_t1\/sum * delta_p0 + delta_t0\/sum * delta_p1;\r\n\tp0deriv = 2*delta_p0 - p1deriv;\r\n\tp2deriv = 2*delta_p1 - p1deriv;\r\n}<\/pre>\n