舞台对象/关系/连接关系/有向边CR曲线说明 全称为：Cublic Catmull Rom Spline，一般项目里面简称CR曲线。 计算 CR曲线本质是一个参数方程，给定四个控制点p0,p1,p2,p3\boldsymbol{p_0},\boldsymbol{p_1},\boldsymbol{p_2},\boldsymbol{p_3}p0,p1,p2,p3以及α,τ\alpha,\tauα,τ通过如下方法计算： t01=∣p0−p1∣αt12=∣p1−p2∣αt23=∣p2−p3∣αt_{01}=\lvert \boldsymbol{p_0}-\boldsymbol{p_1} \rvert ^\alpha \\ t_{12}=\lvert \boldsymbol{p_1}-\boldsymbol{p_2} \rvert ^\alpha \\ t_{23}=\lvert \boldsymbol{p_2}-\boldsymbol{p_3} \rvert ^\alphat01=∣p0−p1∣αt12=∣p1−p2∣αt23=∣p2−p3∣α m1=(1−τ)(p2−p1+t12(p1−p0t01−p2−p0t01+t12))\boldsymbol{m_1}=(1-\tau) (\boldsymbol{p_2} - \boldsymbol{p_1} + t_{12} ( \frac{\boldsymbol{p_1} - \boldsymbol{p_0}}{t_{01}} - \frac{\boldsymbol{p_2} - \boldsymbol{p_0}}{t_{01} + t_{12}} ))m1=(1−τ)(p2−p1+t12(t01p1−p0−t01+t12p2−p0)) m2=(1−τ)(p2−p1+t12(p3−p2t23−p3−p1t12+t23))\boldsymbol{m_2}=(1-\tau) (\boldsymbol{p_2} - \boldsymbol{p_1} + t_{12} ( \frac{\boldsymbol{p_3} - \boldsymbol{p_2}}{t_{23}} - \frac{\boldsymbol{p_3} - \boldsymbol{p_1}}{t_{12} + t_{23}} ))m2=(1−τ)(p2−p1+t12(t23p3−p2−t12+t23p3−p1)) a=2(p1−p2)+m1+m2b=−3(p1−p2)−m1−m1−m2c=m1d=p1\boldsymbol{a}=2(\boldsymbol{p_1}-\boldsymbol{p_2})+\boldsymbol{m_1}+\boldsymbol{m_2} \\ \boldsymbol{b}=-3(\boldsymbol{p_1}-\boldsymbol{p_2})-\boldsymbol{m_1}-\boldsymbol{m_1}-\boldsymbol{m_2} \\ \boldsymbol{c}=\boldsymbol{m_1} \\ \boldsymbol{d}=\boldsymbol{p_1} \\a=2(p1−p2)+m1+m2b=−3(p1−p2)−m1−m1−m2c=m1d=p1 p(t)=at3+bt2+ct+d\boldsymbol{p}(t)=\boldsymbol{a}t^3+\boldsymbol{b}t^2+\boldsymbol{c}t+\boldsymbol{d}p(t)=at3+bt2+ct+d 弧长 先求导 p′=dp(t)dt=3t2a+2tb+c{\boldsymbol{p}}^{'}=\frac{d\boldsymbol{p}(t)}{dt}=3t^2\boldsymbol{a}+2t\boldsymbol{b}+\boldsymbol{c}p′=dtdp(t)=3t2a+2tb+c 弧长: s=∫01∣p′∣dts=\int_0^1 \lvert {\boldsymbol{p}}^{'} \rvert dts=∫01∣p′∣dt s(t0,t1)=∫t0t1∣p′∣dt=(t1−t0)∫01∣p′(x(t1−t0)+t0)∣dxs(t_0, t_1) = \int_{t_0}^{t_1} \lvert {\boldsymbol{p}}^{'} \rvert dt = (t_1-t_0) \int_0^1 \lvert {\boldsymbol{p}}^{'}(x(t_1-t_0)+t_0)\rvert dxs(t0,t1)=∫t0t1∣p′∣dt=(t1−t0)∫01∣p′(x(t1−t0)+t0)∣dx 其中 x=t−t0t1−t0x = \frac{t-t_0}{t_1-t_0}x=t1−t0t−t0有向边Previous Page连线 LineEdgeNext Page