SharedDG-zadaci.sagewsOpen in CoCalc

Diferencijalna geometrija - zadaci

Krivulje

Zadatak 2.

Kako glase jednadžbe tangencijalnih ravnina plohe \quad\vec{r}(u,v)=\left \{ u,u+v,u^2+v^2 \right \} \quad koje prolaze pravcem \quad \frac{x-2}{2}=\frac{y-3}{3}=\frac{z-1}{10}.

u,v,y,z=var('u,v,y,z')
r=vector((u,u+v,u^2+v^2))
ru=diff(r,u)
rv=diff(r,v)
show('$\\vec{r}(u,v)=$',r)
show('$\\vec{r}_{u}=$',ru)
show('$\\vec{r}_{v}=$',rv)
rurv=ru.cross_product(rv)
show('$\\vec{r}_{u}\\times \\vec{r}_{v}=$',rurv)
p=vector((4,3,10))
show('$\\vec{p}=$',p)
np=rurv.dot_product(p)
show('$\\vec{N}\\cdot\\vec{p}=$',np,'$=0$')
show('$v=4u-5$')
ruu=r(v=4*u-5)
show('$\\vec{r}(u)=$',ruu)
Nu=rurv(v=4*u-5)
show('$\\vec{N}(u)=$',Nu)
show('$A(x-x_{0})+B(y-y_{0})+C(z-z_{0})=0$')
show('$T(2,3,10)$')
show('$(6u-10)(2-u)+(-8u+10)(3-5u+5)+1-((4u-5)^2+u^2)=0$')
show(solve((6*u-10)*(2-u)+(-8*u+10)*(3-5*u+5)+1-((4*u-5)^2+u^2),u))
show(((6*u-10)*(x-u)+(-8*u+10)*(y-5*u+5)+z-((4*u-5)^2+u^2))(u=2),'$=0$')
show(((6*u-10)*(x-u)+(-8*u+10)*(y-5*u+5)+z-((4*u-5)^2+u^2))(u=18/17),'$=0$')

\vec{r}(u,v)= \displaystyle \left(u,\,u + v,\,u^{2} + v^{2}\right)
\vec{r}_{u}= \displaystyle \left(1,\,1,\,2 \, u\right)
\vec{r}_{v}= \displaystyle \left(0,\,1,\,2 \, v\right)
\vec{r}_{u}\times \vec{r}_{v}= \displaystyle \left(-2 \, u + 2 \, v,\,-2 \, v,\,1\right)
\vec{p}= \displaystyle \left(4,\,3,\,10\right)
\vec{N}\cdot\vec{p}= \displaystyle -8 \, u + 2 \, v + 10 =0
v=4u-5
\vec{r}(u)= \displaystyle \left(u,\,5 \, u - 5,\,{\left(4 \, u - 5\right)}^{2} + u^{2}\right)
\vec{N}(u)= \displaystyle \left(6 \, u - 10,\,-8 \, u + 10,\,1\right)
A(x-x_{0})+B(y-y_{0})+C(z-z_{0})=0
T(2,3,10)
(6u-10)(2-u)+(-8u+10)(3-5u+5)+1-((4u-5)^2+u^2)=0
[ \displaystyle u = \left(\frac{18}{17}\right) , \displaystyle u = 2 ]
\displaystyle 2 \, x - 6 \, y + z + 13 =0
\displaystyle -\frac{62}{17} \, x + \frac{26}{17} \, y + z + \frac{29}{17} =0
u,v=var('u,v')
parametric_plot3d((u,u+v,u^2+v^2),(u,-10,10),(v,-10,10),color='green',frame=false)+parametric_plot3d((4*u+2,3*u+3,10*u+1),(u,-10,10),color='red',frame=false)+implicit_plot3d(2*x-6*y+z+13==0,(x,-50,50),(y,-50,50),(z,-50,50),opacity=.40)+implicit_plot3d(-62*x+26*y+17*z+29==0,(x,-50,50),(y,-50,50),(z,-50,50),opacity=.40)
3D rendering not yet implemented
u,v,y,z=var('u,v,y,z')
r=vector((v,(3*u+2*v)/(u+v),2*u))
ru=simplify(diff(r,u))
show(ru(u=1,v=-2))
ruv=simplify(diff(ru,v))
show(ruv(u=1,v=-2))
rv=simplify(diff(r,v))
show(rv(u=1,v=-2))
rvu=simplify(diff(rv,u))
show(rvu)
\displaystyle \left(0,\,-2,\,2\right)
\displaystyle \left(0,\,-3,\,0\right)
\displaystyle \left(1,\,-1,\,0\right)
\displaystyle \left(0,\,\frac{2 \, {\left(3 \, u + 2 \, v\right)}}{{\left(u + v\right)}^{3}} - \frac{5}{{\left(u + v\right)}^{2}},\,0\right)