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Diferencijalna geometrija - zadaci

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Zadatak 2.

Kako glase jednadžbe tangencijalnih ravnina plohe r(u,v)={u,u+v,u2+v2}\quad\vec{r}(u,v)=\left \{ u,u+v,u^2+v^2 \right \} \quad koje prolaze pravcem x22=y33=z110. \quad \frac{x-2}{2}=\frac{y-3}{3}=\frac{z-1}{10}.

r(u,v)=\vec{r}(u,v)= (u,u+v,u2+v2)\displaystyle \left(u,\,u + v,\,u^{2} + v^{2}\right)
ru=\vec{r}_{u}= (1,1,2u)\displaystyle \left(1,\,1,\,2 \, u\right)
rv=\vec{r}_{v}= (0,1,2v)\displaystyle \left(0,\,1,\,2 \, v\right)
ru×rv=\vec{r}_{u}\times \vec{r}_{v}= (2u+2v,2v,1)\displaystyle \left(-2 \, u + 2 \, v,\,-2 \, v,\,1\right)
p=\vec{p}= (4,3,10)\displaystyle \left(4,\,3,\,10\right)
Np=\vec{N}\cdot\vec{p}= 8u+2v+10\displaystyle -8 \, u + 2 \, v + 10 =0=0
v=4u5v=4u-5
r(u)=\vec{r}(u)= (u,5u5,(4u5)2+u2)\displaystyle \left(u,\,5 \, u - 5,\,{\left(4 \, u - 5\right)}^{2} + u^{2}\right)
N(u)=\vec{N}(u)= (6u10,8u+10,1)\displaystyle \left(6 \, u - 10,\,-8 \, u + 10,\,1\right)
A(xx0)+B(yy0)+C(zz0)=0A(x-x_{0})+B(y-y_{0})+C(z-z_{0})=0
T(2,3,10)T(2,3,10)
(6u10)(2u)+(8u+10)(35u+5)+1((4u5)2+u2)=0(6u-10)(2-u)+(-8u+10)(3-5u+5)+1-((4u-5)^2+u^2)=0
[u=(1817)\displaystyle u = \left(\frac{18}{17}\right), u=2\displaystyle u = 2]
2x6y+z+13\displaystyle 2 \, x - 6 \, y + z + 13 =0=0
6217x+2617y+z+2917\displaystyle -\frac{62}{17} \, x + \frac{26}{17} \, y + z + \frac{29}{17} =0=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
n(-0.053+4507228.4978+0.0039*0.000001*4507228.4978+0.0008*pi/(180*3600)*4369598.4978+0.00039*pi/(180*3600)*1067383.4978)
4.50722848134390e6
0.25+1
1.25000000000000
pi

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1013, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'PI' is not defined