CR görbe kifejtve

gistfile1.tex

r_i(t)=r_i+(t-t_i)\frac{\frac{r_{i+1}-r_i}{t_{i+1}-t_{i}}+\frac{r_i-r_{i-1}}{t_i-t_{i-1}}}{2}+(t-t_i)^2\left(\frac{3(r_{i+1}-r_i)}{(t_{i+1}-t_i)^2}-\frac{\frac{\frac{r_{i+2}-r_{i+1}}{t_{i+2}-t_{i+1}}+\frac{r_{i+1}-r_i}{t_{i+1}-t_i}}{2}+2\frac{\frac{r_{i+1}-r_i}{t_{i+1}-t_i}+\frac{r_i-r_{i-1}}{t_i-t_{i-1}}}{2}}{t_{i+1}-t_i} \right )+(t-t_i)^3\left( 2\frac{(r_i-r_{i+1})}{(t_{i+1}-t_i)^3}+\frac{\frac{\frac{r_{i+2}-r_{i+1}}{t_{i+2}-t_{i+1}}+\frac{r_{i+1}-r_i}{t_{i+1}-t_i}}{2}+\frac{\frac{r_{i+1}-r_i}{t_{i+1}-t_i}+\frac{r_i-r_{i-1}}{t_i-t_{i-1}}}{2}}{(t_{i+1}-t_i)^2} \right )

r_i(t)=r_i+(t-t_i)\frac{\frac{-r_{i}}{t_{i+1}-t_i}+\frac{r_i}{t_i-t_{i-1}}}{4}+(t-t_i)^2\left(\frac{3(-r_i)}{(t_{i+1}-t_i)^2}-\frac{\frac{\frac{-r_i}{t_{i+1}-t_i}}{4}+2\frac{\frac{-r_i}{t_{i+1}-t_i}+\frac{r_i}{t_i-t_{i-1}}}{4}}{t_{i+1}-t_i} \right )+(t-t_i)^3\left( 2\frac{(r_i)}{(t_{i+1}-t_i)^3}+\frac{\frac{\frac{-r_i}{t_{i+1}-t_i}}{4}+\frac{\frac{-r_i}{t_{i+1}-t_i}+\frac{r_i}{t_i-t_{i-1}}}{4}}{(t_{i+1}-t_i)^2} \right )

r_i+(t-t_i)\frac{\frac{-r_{i}}{t_{i+1}-t_i}+\frac{r_i}{t_i-t_{i-1}}}{2}+(t-t_i)^2\left(\frac{3(-r_i)}{(t_{i+1}-t_i)^2}-\frac{\frac{\frac{-r_i}{t_{i+1}-t_i}}{2}+\frac{-r_i}{t_{i+1}-t_i}+\frac{r_i}{t_i-t_{i-1}}}{t_{i+1}-t_i} \right )+(t-t_i)^3\left( 2\frac{(r_i)}{(t_{i+1}-t_i)^3}+\frac{\frac{\frac{-r_i}{t_{i+1}-t_i}}{2}+\frac{\frac{-r_i}{t_{i+1}-t_i}+\frac{r_i}{t_i-t_{i-1}}}{2}}{(t_{i+1}-t_i)^2} \right )

r_i ~ 1+(t-t_i)\left( \frac{-0.5}{t_{i+1}-t_i}+\frac{0.5}{t_i-t_{i-1}}\right )+(t-t_i)^2\left(\frac{-4.5}{(t_{i+1}-t_i)^2}-\frac{1}{(t_i-t_{i-1})(t_{i+1}-t_i)} \right )+(t-t_i)^3\left( \frac{1}{(t_{i+1}-t_i)^3}+\frac{0.5}{(t_i-t_{i-1})(t_{i+1}-t_i)^2} \right )

r_{i-1} ~ (t-t_i)\frac{-0.5}{t_i-t_{i-1}}+(t-t_i)^2\left(\frac{1}{(t_i-t_{i-1})(t_{i+1}-t_i)} \right )+(t-t_i)^3\left( \frac{-0.5}{(t_i-t_{i-1})(t_{i+1}-t_i)^2} \right )

r_{i+2} ~ (t-t_i)^2\frac{-0.5}{(t_{i+2}-t_{i+1})(t_{i+1}-t_i)}+(t-t_i)^3\frac{0.5}{(t_{i+2}-t_{i+1})(t_{i+1}-t_i)^2}

p_{i+1} \left(\frac{(1-\tau ) \left(t-t_i\right){}^3}{2 \left(t_{i+1}-t_i\right){}^3}+\frac{(\tau -1) \left(t-t_i\right){}^2}{\left(t_{i+1}-t_i\right){}^2}+\frac{(1-\tau )
   \left(t-t_i\right)}{2 \left(t_{i+1}-t_i\right)}-\frac{2 \left(t-t_i\right){}^3}{\left(t_{i+1}-t_i\right){}^3}+\frac{3
   \left(t-t_i\right){}^2}{\left(t_{i+1}-t_i\right){}^2}\right)+p_{i-1} \left(\frac{(\tau -1) \left(t-t_i\right){}^3}{2 \left(t_i-t_{i-1}\right)
   \left(t_{i+1}-t_i\right){}^2}+\frac{(1-\tau ) \left(t-t_i\right){}^2}{\left(t_i-t_{i-1}\right) \left(t_{i+1}-t_i\right)}+\frac{(\tau -1) \left(t-t_i\right)}{2
   \left(t_i-t_{i-1}\right)}\right)+p_i \left(\frac{(1-\tau ) \left(t-t_i\right){}^3}{2 \left(t_i-t_{i-1}\right) \left(t_{i+1}-t_i\right){}^2}+\frac{(\tau -1)
   \left(t-t_i\right){}^3}{2 \left(t_{i+1}-t_i\right){}^3}+\frac{(\tau -1) \left(t-t_i\right){}^2}{\left(t_i-t_{i-1}\right) \left(t_{i+1}-t_i\right)}+\frac{(1-\tau )
   \left(t-t_i\right){}^2}{\left(t_{i+1}-t_i\right){}^2}+\frac{(1-\tau ) \left(t-t_i\right)}{2 \left(t_i-t_{i-1}\right)}+\frac{(\tau -1) \left(t-t_i\right)}{2
   \left(t_{i+1}-t_i\right)}+\frac{2 \left(t-t_i\right){}^3}{\left(t_{i+1}-t_i\right){}^3}-\frac{3 \left(t-t_i\right){}^2}{\left(t_{i+1}-t_i\right){}^2}+1\right)

\left( 2\frac{(p_i-p_{i+1})}{(t_{i+1}-t_i)^3}+\frac{\frac{\frac{p_{i+1}-p_i}{t_{i+1}-t_i}+\frac{p_i-p_{i-1}}{t_i-t_{i-1}}}{2}(1-\tau)}{(t_{i+1}-t_i)^2} \right )(t-t_i)^3+\left(\frac{3(p_{i+1}-p_i)}{(t_{i+1}-t_i)^2}-\frac{(1-\tau)2\frac{\frac{p_{i+1}-p_i}{t_{i+1}-t_i}+\frac{p_i-p_{i-1}}{t_i-t_{i-1}}}{2}}{t_{i+1}-t_i} \right )(t-t_i)^2+\frac{\frac{p_{i+1}-p_i}{t_{i+1}-t_{i}}+\frac{p_i-p_{i-1}}{t_i-t_{i-1}}}{2}(t-t_i)(1-\tau)+p_i

i+2 nélkül:
p_{i+1} \left(\frac{(1-\tau ) \left(t-t_i\right){}^3}{2 \left(t_{i+1}-t_i\right){}^3}+\frac{(\tau -1) \left(t-t_i\right){}^2}{\left(t_{i+1}-t_i\right){}^2}+\frac{(1-\tau )
   \left(t-t_i\right)}{2 \left(t_{i+1}-t_i\right)}-\frac{2 \left(t-t_i\right){}^3}{\left(t_{i+1}-t_i\right){}^3}+\frac{3
   \left(t-t_i\right){}^2}{\left(t_{i+1}-t_i\right){}^2}\right)+p_{i-1} \left(\frac{(\tau -1) \left(t-t_i\right){}^3}{2 \left(t_i-t_{i-1}\right)
   \left(t_{i+1}-t_i\right){}^2}+\frac{(1-\tau ) \left(t-t_i\right){}^2}{\left(t_i-t_{i-1}\right) \left(t_{i+1}-t_i\right)}+\frac{(\tau -1) \left(t-t_i\right)}{2
   \left(t_i-t_{i-1}\right)}\right)+p_i \left(\frac{(1-\tau ) \left(t-t_i\right){}^3}{2 \left(t_i-t_{i-1}\right) \left(t_{i+1}-t_i\right){}^2}+\frac{(\tau -1)
   \left(t-t_i\right){}^3}{2 \left(t_{i+1}-t_i\right){}^3}+\frac{(\tau -1) \left(t-t_i\right){}^2}{\left(t_i-t_{i-1}\right) \left(t_{i+1}-t_i\right)}+\frac{(1-\tau )
   \left(t-t_i\right){}^2}{\left(t_{i+1}-t_i\right){}^2}+\frac{(1-\tau ) \left(t-t_i\right)}{2 \left(t_i-t_{i-1}\right)}+\frac{(\tau -1) \left(t-t_i\right)}{2
   \left(t_{i+1}-t_i\right)}+\frac{2 \left(t-t_i\right){}^3}{\left(t_{i+1}-t_i\right){}^3}-\frac{3 \left(t-t_i\right){}^2}{\left(t_{i+1}-t_i\right){}^2}+1\right)

i-1 nélkül:
\left( 2\frac{(p_i-p_{i+1})}{(t_{i+1}-t_i)^3}+\frac{\frac{\frac{p_{i+2}-p_{i+1}}{t_{i+2}-t_{i+1}}+\frac{p_{i+1}-p_i}{t_{i+1}-t_i}}{2}(1-\tau)}{(t_{i+1}-t_i)^2} \right )(t-t_i)^3+\left(\frac{3(p_{i+1}-p_i)}{(t_{i+1}-t_i)^2}-\frac{\frac{\frac{p_{i+2}-p_{i+1}}{t_{i+2}-t_{i+1}}+\frac{p_{i+1}-p_i}{t_{i+1}-t_i}}{2}(1-\tau)}{t_{i+1}-t_i} \right )(t-t_i)^2+p_i

átrendezve:
p_i \left(\frac{(\tau -1) \left(t-t_i\right){}^3}{2 \left(t_{i+1}-t_i\right){}^3}+\frac{(1-\tau ) \left(t-t_i\right){}^2}{2 \left(t_{i+1}-t_i\right){}^2}+\frac{2
   \left(t-t_i\right){}^3}{\left(t_{i+1}-t_i\right){}^3}-\frac{3 \left(t-t_i\right){}^2}{\left(t_{i+1}-t_i\right){}^2}+1\right)+p_{i+1} \left(\frac{(\tau -1)
   \left(t-t_i\right){}^3}{2 \left(t_{i+1}-t_i\right){}^2 \left(t_{i+2}-t_{i+1}\right)}+\frac{(1-\tau ) \left(t-t_i\right){}^3}{2 \left(t_{i+1}-t_i\right){}^3}+\frac{(1-\tau )
   \left(t-t_i\right){}^2}{2 \left(t_{i+1}-t_i\right) \left(t_{i+2}-t_{i+1}\right)}+\frac{(\tau -1) \left(t-t_i\right){}^2}{2 \left(t_{i+1}-t_i\right){}^2}-\frac{2
   \left(t-t_i\right){}^3}{\left(t_{i+1}-t_i\right){}^3}+\frac{3 \left(t-t_i\right){}^2}{\left(t_{i+1}-t_i\right){}^2}\right)+p_{i+2} \left(\frac{(1-\tau )
   \left(t-t_i\right){}^3}{2 \left(t_{i+1}-t_i\right){}^2 \left(t_{i+2}-t_{i+1}\right)}+\frac{(\tau -1) \left(t-t_i\right){}^2}{2 \left(t_{i+1}-t_i\right)
   \left(t_{i+2}-t_{i+1}\right)}\right)

minden:
p_{-1+i} \left(\frac{(-1+\tau ) \left(t-t_i\right)}{2 \left(-t_{-1+i}+t_i\right)}+\frac{(-1+\tau ) \left(t-t_i\right){}^3}{2 \left(-t_{-1+i}+t_i\right) \left(-t_i+t_{1+i}\right){}^2}+\frac{(1-\tau ) \left(t-t_i\right){}^2}{\left(-t_{-1+i}+t_i\right) \left(-t_i+t_{1+i}\right)}\right)+p_i \left(1+\frac{(1-\tau ) \left(t-t_i\right)}{2 \left(-t_{-1+i}+t_i\right)}+\frac{2 \left(t-t_i\right){}^3}{\left(-t_i+t_{1+i}\right){}^3}+\frac{(-1+\tau ) \left(t-t_i\right){}^3}{\left(-t_i+t_{1+i}\right){}^3}-\frac{3 \left(t-t_i\right){}^2}{\left(-t_i+t_{1+i}\right){}^2}+\frac{3 (1-\tau ) \left(t-t_i\right){}^2}{2 \left(-t_i+t_{1+i}\right){}^2}+\frac{(1-\tau ) \left(t-t_i\right){}^3}{2 \left(-t_{-1+i}+t_i\right) \left(-t_i+t_{1+i}\right){}^2}+\frac{(-1+\tau ) \left(t-t_i\right)}{2 \left(-t_i+t_{1+i}\right)}+\frac{(-1+\tau ) \left(t-t_i\right){}^2}{\left(-t_{-1+i}+t_i\right) \left(-t_i+t_{1+i}\right)}\right)+p_{1+i} \left(-\frac{2 \left(t-t_i\right){}^3}{\left(-t_i+t_{1+i}\right){}^3}+\frac{(1-\tau ) \left(t-t_i\right){}^3}{\left(-t_i+t_{1+i}\right){}^3}+\frac{3 \left(t-t_i\right){}^2}{\left(-t_i+t_{1+i}\right){}^2}-\frac{3 (1-\tau ) \left(t-t_i\right){}^2}{2 \left(-t_i+t_{1+i}\right){}^2}+\frac{(1-\tau ) \left(t-t_i\right)}{2 \left(-t_i+t_{1+i}\right)}+\frac{(-1+\tau ) \left(t-t_i\right){}^3}{2 \left(-t_i+t_{1+i}\right){}^2 \left(-t_{1+i}+t_{2+i}\right)}+\frac{(1-\tau ) \left(t-t_i\right){}^2}{2 \left(-t_i+t_{1+i}\right) \left(-t_{1+i}+t_{2+i}\right)}\right)+p_{2+i} \left(\frac{(1-\tau ) \left(t-t_i\right){}^3}{2 \left(-t_i+t_{1+i}\right){}^2 \left(-t_{1+i}+t_{2+i}\right)}+\frac{(-1+\tau ) \left(t-t_i\right){}^2}{2 \left(-t_i+t_{1+i}\right) \left(-t_{1+i}+t_{2+i}\right)}\right)