Fundamental Cofficients of Surface

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Some Common Example of Surface

1. Plane surface

   0,0

   az = 1.00

   el = 0.30
   
   Plane is a surface traced by a straight line whose parameters are of degree 1. One example of plane surface is given by
   $\overrightarrow{r}=(u,v,u+v)$
   

2. Cylinder

   0,0

   az = 1.00

   el = 0.30
   
   Cylinder is a surface traced by a straight line being parallel to a fixed vector. It is given by an equation
   $\overrightarrow{r}=(r\cos u,r\sin u,v)$
   

3. Cone

   0,0

   az = 1.00

   el = 0.30
   
   Cone is a surface traced by a straight line being fixed to a fixed point. It is given by an equation
   $\overrightarrow{r}=(v\cos u,v\sin u,v)$
   

4. Paraboloid

   $\overrightarrow{r}=(u,v,u^2+v^2)$

5. Hyperboloid

   $\overrightarrow{r}=(x,y,x^2-y^2)$

6. Minimal surface

   $\overrightarrow{r}=(x,y,\log \cos y-\log \cos x)$

7. The helicoid

   $\overrightarrow{r}=(u\cos v,u\sin v,v)$

8. Pseduo-sphere

   $\overrightarrow{r}=(\sec hu\cos v,\sec hu\sin v,u-\tan h)$

9. Monge’s form

   $\overrightarrow{r}=(u,v,f(u,v))$

10. Surface of revolution

    $\overrightarrow{r}=(u\cos v,u\sin v,f(u))$

11. Conoidal surface

    $\overrightarrow{r}=(u\cos v,u\sin v,f(v))$

12. Saddle surface

    $\overrightarrow{r}=(u,v,uv)$

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Compute Fundamental Cofficients of surface

Compute fundamental coefficients for a saddle surface $\overrightarrow{r}=(u,v,uv)$
Solution
The saddle surface is
$\overrightarrow{r}=(u,v,uv)$(i)
Differentiation of (i) w. r. to. u and v, we get
${\overrightarrow{r}}_1=(1,0,v)$
${\overrightarrow{r}}_2=(0,1,u)$
${\overrightarrow{r}}_{11}=(0,0,0)$
${\overrightarrow{r}}_{12}=(0,0,1)$
${\overrightarrow{r}}_{22}=(0,0,0)$
Here, we used the suffix 1 and 2 for derivatives with respect to u and v and respectively, and similarly for higher derivatives.
Now, first order fundamental coefficients are
$E={\overrightarrow{r}}_1^2=(1,0,v{)}^2=1+v^2$
$F={\overrightarrow{r}}_1.{\overrightarrow{r}}_2=(1,0,v).(0,1,u)=uv$
$G={\overrightarrow{r}}_2^2=(1,0,u{)}^2=1+u^2$
Next,we have to compute second fundamental cofficients,for this
$H\overrightarrow{N}={\overrightarrow{r}}_1\times {\overrightarrow{r}}_2$
or $H\overrightarrow{N}=(1,0,v)\times (0,1,u)$
or $H\overrightarrow{N}=(-v,-u,1)$ (A)
Taking magnitude, we get
$H=\sqrt{1+u^2+v^2}$
And substituting H in (A) we get
$\overrightarrow{N}=\frac{(-v,-u,1)}{\sqrt{1+u^2+v^2}}$
Hence, the second order fundamental coefficients are
$L={\overrightarrow{r}}_{11}.\overrightarrow{N}=(0,0,0).\frac{(-v,-u,1)}{\sqrt{1+u^2+v^2}}=0$
$M={\overrightarrow{r}}_{12}.\overrightarrow{N}=(0,0,1).\frac{(-v,-u,1)}{\sqrt{1+u^2+v^2}}=\frac{1}{\sqrt{1+u^2+v^2}}$
$N={\overrightarrow{r}}_{22}.\overrightarrow{N}=(0,0,0).\frac{(-v,-u,1)}{\sqrt{1+u^2+v^2}}=0$
This completes the solution

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Find fundamental coefficients of following surface

1. Monge’s form: $\overrightarrow{r}=(x,y,f(x,y))$
   

   Answer:
   $\overrightarrow{N}=\{-\frac{f_1}{\sqrt{1+f_1^2+f_2^2}},-\frac{f_2}{\sqrt{1+f_1^2+f_2^2}},\frac{1}{\sqrt{1+f_1^2+f_2^2}}\}$
   Cofficients: $\{1+f_1^2,f_1{f}_2,1+f_2^2,\frac{f_{11}}{\sqrt{1+f_1^2+f_2^2}},\frac{f_{12}}{\sqrt{1+f_1^2+f_2^2}},\frac{f_{22}}{\sqrt{1+f_1^2+f_2^2}}\}$

2. Surface of revolution: $\overrightarrow{r}=(u\cos v,u\sin v,f(u))$
   

   Answer:
   $\overrightarrow{N}=\{-\frac{f_{1}u\cos v}{H},-\frac{f_{1}u\sin v}{H},\frac{u}{H}\}$
   Cofficients: $\{1+f_1^2,0,u^2,\frac{f_{11}u}{H},0,\frac{f_1{u}^2}{H}\}$

3. Conoidal surface:$\overrightarrow{r}=(u\cos v,u\sin v,f(v))$
   

   Answer:
   $\overrightarrow{N}=\{\frac{f_2\sin v}{H},-\frac{f_2\cos v}{H},\frac{u}{H}\}$
   Cofficients: $\{1,0,f_2^2+u^2,0,-\frac{f_2}{H},\frac{f_{22}u}{H}\}$

4. Right helicoid: $\overrightarrow{r}=(u\cos v,u\sin v,cv)$
   

   Answer:
   $\overrightarrow{N}=\{\frac{c\sin v}{H},-\frac{c\cos v}{H},\frac{u}{H}\}$
   Cofficients: $\{1,0,c^2+u^2,0,-\frac{c}{H},0\}$

5. Plane surface: $\overrightarrow{r}=(u,v,u+v)$
   

   Answer:
   $\overrightarrow{N}=\{\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0\}$
   Cofficients: $\{3,1,1,0,0,0\}$

6. Saddle surface: $\overrightarrow{r}=(u,v,uv)$
   

   Answer:
   $\overrightarrow{N}=\{-\frac{v}{\sqrt{1+u^2+v^2}},-\frac{u}{\sqrt{1+u^2+v^2}},\frac{1}{\sqrt{1+u^2+v^2}}\}$
   Cofficients: $\{1+v^2,uv,1+u^2,0,\frac{1}{\sqrt{1+u^2+v^2}},0\}$

7. Saddle surface: $\overrightarrow{r}=(u+v,u-v,uv)$
   

   Answer:
   $\overrightarrow{N}=\{\frac{u+v}{\sqrt{2}\sqrt{2+u^2+v^2}},\frac{-u+v}{\sqrt{2}\sqrt{2+u^2+v^2}},-\frac{\sqrt{2}}{\sqrt{2+u^2+v^2}}\}$
   Cofficients: $\{2+v^2,uv,2+u^2,0,-\frac{\sqrt{2}}{\sqrt{2+u^2+v^2}},0\}$

8. Paraboloid:$\overrightarrow{r}=(u,v,u^2+v^2)$
   

   Answer:
   $\overrightarrow{N}=\{\frac{v}{\sqrt{2}},-\frac{v}{\sqrt{2}},0\}$
   Cofficients: $\{a^2,0,1,-a,0,0\}$

9. Cylinder: $\overrightarrow{r}=(a\cos u,a\sin u,v)$
   

   Answer:
   $\overrightarrow{N}=\{\cos u,\sin u,0\}$
   Cofficients: $\{2+4u^2,4uv,4v^2,0,0,0\}$

10. Cone: $\overrightarrow{r}=(v\cos u,v\sin u,v)$
    

    Answer:
    $\overrightarrow{N}=\{\frac{v\cos u}{\sqrt{2}},\frac{v\sin u}{\sqrt{2}},-\frac{v}{\sqrt{2}}\}$
    Cofficients: $\{v^2,0,2,-\frac{v^2}{\sqrt{2}},0,0\}$

11. Sphere: $\overrightarrow{r}=(\sin u\cos v,\sin u\sin v,\cos u)$
    

    Answer:
    $\overrightarrow{N}=\{\frac{\cos v{\sin }^{2}u}{H},\frac{{\sin }^{2}u\sin v}{H},\frac{\cos u\sin u}{H}\}$
    Cofficients: $\{1,0,{\sin }^{2}u,-\frac{\sin u}{H},0,-\frac{{\sin }^{3}u}{H}\}$

12. Hyperboloid: $2z=7x^2+6xy-y^2$at origin
    

    Answer:
    $\overrightarrow{N}=\{0,0,1\}$
    Cofficients: $\{1,0,1,7,3,-1\}$

13. Minimal surface: $e^z\cos x=\cos y$
    

    Answer:
    $\overrightarrow{N}=\{-\frac{\tan x}{H},\frac{\tan y}{H},\frac{1}{H}\}$
    Cofficients: $\{{\sec }^{2}x,-\tan x\tan y,{\sec }^{2}y,\frac{{\sec }^{2}x}{H},0,-\frac{{\sec }^{2}y}{H}\}$

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