Some Common Example of Surface
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Plane surface
Plane is a surface traced by a straight line whose parameters are of degree 1. One example of plane surface is given by
\( \vec{r}=(u, v,u+v) \)
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Cylinder
Cylinder is a surface traced by a straight line being parallel to a fixed vector. It is given by an equation
\( \vec{r}=(r \cos u, r \sin u,v) \)
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Cone
Cone is a surface traced by a straight line being fixed to a fixed point. It is given by an equation
\( \vec{r}=(v \cos u, v \sin u,v) \)
Paraboloid
\(\vec{r}=( u,v,u^2+v^2 ) \)Hyperboloid
\( \vec{r}=(x,y,x^2-y^2) \)Minimal surface
\( \vec{r}=(x,y,\log \cos y-\log \cos x) \)The helicoid
\( \vec{r}=(u \cos v,u \sin v ,v) \)Pseduo-sphere
\(\vec{r}=(\sec h u \cos v, \sec h u \sin v, u-\tan h )\)Monge’s form
\( \vec{r}=( u,v,f( u,v ) )\)Surface of revolution
\(\vec{r}=(u \cos v,u \sin v, f( u ))\)Conoidal surface
\( \vec{r}=( u \cos v,u \sin v,f( v ) ) \)Saddle surface
\( \vec{r}=( u,v,uv) \)
Compute Fundamental Cofficients of surface
Compute fundamental coefficients for a saddle surface \( \vec{r}=(u,v,uv) \)Solution
The saddle surface is
\( \vec{r}=(u,v,uv) \)(i)
Differentiation of (i) w. r. to. u and v, we get
\( \vec{r}_1=(1,0,v) \)
\( \vec{r}_2=(0,1,u) \)
\( \vec{r}_{11}=(0,0,0) \)
\( \vec{r}_{12}=(0,0,1) \)
\( \vec{r}_{22}=(0,0,0) \)
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
\( E=\vec{r}_1^2=(1,0,v)^2=1+v^2 \)
\( F=\vec{r}_1. \vec{r}_2=(1,0,v).(0,1,u)=uv \)
\( G=\vec{r}_2^2=(1,0,u)^2=1+u^2 \)
Next,we have to compute second fundamental cofficients,for this
\( H\vec{N}=\vec{r}_1\times \vec{r}_2 \)
or \( H\vec{N}=(1,0,v)\times (0,1,u) \)
or \( H\vec{N}=(-v,-u,1) \) (A)
Taking magnitude, we get
\( H=\sqrt{1+u^2+v^2}\)
And substituting H in (A) we get
\( \vec{N}=\frac{(-v,-u,1)}{\sqrt{1+u^2+v^2}}\)
Hence, the second order fundamental coefficients are
\(L= \vec{r}_{11}.\vec{N}=(0,0,0).\frac{(-v,-u,1)}{\sqrt{1+u^2+v^2}}=0\)
\(M= \vec{r}_{12}.\vec{N}=(0,0,1).\frac{(-v,-u,1)}{\sqrt{1+u^2+v^2}}=\frac{1}{\sqrt{1+u^2+v^2}}\)
\(N= \vec{r}_{22}.\vec{N}=(0,0,0).\frac{(-v,-u,1)}{\sqrt{1+u^2+v^2}}=0\)
This completes the solution
Find fundamental coefficients of following surface
- Monge’s form: \( \vec{r}=( x,y,f( x,y ) )\)
Answer:
\( \vec{N}=\left\{-\frac{f_1}{\sqrt{1 + f_1^2 + f_2^2}}, -\frac{f_2}{\sqrt{1 + f_1^2 + f_2^2}}, \frac{1}{\sqrt{1 + f_1^2 + f_2^2}}\right\} \)
Cofficients: \( \left\{1 + f_1^2, f_1 f_2, 1 + f_2^2, \frac{f_{11}}{\sqrt{1 + f_1^2 + f_2^2}}, \frac{f_{12}}{\sqrt{1 + f_1^2 + f_2^2}}, \frac{f_{22}}{\sqrt{1 + f_1^2 + f_2^2}}\right\} \) - Surface of revolution: \(\vec{r}=( u \cos v,u \sin v,f( u ) )\)
Answer:
\( \vec{N}=\left\{-\frac{f_1 u \cos v}{H}, -\frac{f_1 u \sin v}{H}, \frac{u}{H}\right\} \)
Cofficients: \( \left\{1 + f_1^2, 0, u^2, \frac{f_{11} u}{H}, 0, \frac{f_1 u^2}{H}\right\} \) - Conoidal surface:\( \vec{r}=( u \cos v,u \sin v,f( v ) ) \)
Answer:
\( \vec{N}=\left\{\frac{f_2 \sin v}{H}, -\frac{f_2 \cos v}{H}, \frac{u}{H}\right\}\)
Cofficients: \( \left\{1, 0, f_2^2 + u^2, 0, -\frac{f_2}{H}, \frac{f_{22} u}{H}\right\} \) - Right helicoid: \( \vec{r}=( u \cos v,u\sin v,cv ) \)
Answer:
\( \vec{N}=\left\{\frac{c \sin v}{H}, -\frac{c \cos v}{H}, \frac{u}{H}\right\}\)
Cofficients: \( \left\{1, 0, c^2 + u^2, 0, -\frac{c}{H}, 0\right\} \) - Plane surface: \( \vec{r}=( u,v,u+v) \)
Answer:
\( \vec{N}=\left\{\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}, 0\right\}\)
Cofficients: \( \{3, 1, 1, 0, 0, 0\} \) - Saddle surface: \( \vec{r}=( u,v,uv) \)
Answer:
\( \vec{N}=\left\{-\frac{v}{\sqrt{1 + u^2 + v^2}}, -\frac{u}{\sqrt{1 + u^2 + v^2}}, \frac{1}{\sqrt{1 + u^2 + v^2}}\right\}\)
Cofficients: \( \left\{1 + v^2, uv, 1 + u^2, 0, \frac{1}{\sqrt{1 + u^2 + v^2}}, 0\right\}\) - Saddle surface: \( \vec{r}=( u+v,u-v,uv ) \)
Answer:
\( \vec{N}=\left\{\frac{u + v}{\sqrt{2} \sqrt{2 + u^2 + v^2}}, \frac{-u + v}{\sqrt{2} \sqrt{2 + u^2 + v^2}}, -\frac{\sqrt{2}}{\sqrt{2 + u^2 + v^2}}\right\} \)
Cofficients: \( \left\{2 + v^2, uv, 2 + u^2, 0, -\frac{\sqrt{2}}{\sqrt{2 + u^2 + v^2}}, 0\right\}\) - Paraboloid:\(\vec{r}=( u,v,u^2+v^2 ) \)
Answer:
\( \vec{N}=\left\{\frac{v}{\sqrt{2}}, -\frac{v}{\sqrt{2}}, 0\right\}\)
Cofficients: \( \{a^2, 0, 1, -a, 0, 0\} \) - Cylinder: \( \vec{r}=(a\cos u, a\sin u,v ) \)
Answer:
\( \vec{N}=\{\cos u, \sin u, 0\}\)
Cofficients: \( \{2 + 4 u^2, 4 u v, 4 v^2, 0, 0, 0\} \) - Cone: \( \vec{r}=(v\cos u, v\sin u,v ) \)
Answer:
\( \vec{N}=\left\{\frac{v \cos u}{\sqrt{2}}, \frac{v \sin u}{\sqrt{2}}, -\frac{v}{\sqrt{2}}\right\} \)
Cofficients: \( \left\{v^2, 0, 2, -\frac{v^2}{\sqrt{2}}, 0, 0\right\} \) - Sphere: \( \vec{r}=(\sin u \cos v,\sin u \sin v, \cos u) \)
Answer:
\( \vec{N}=\left\{\frac{\cos v \sin^2 u}{H}, \frac{\sin^2 u \sin v}{H}, \frac{\cos u \sin u}{H}\right\} \)
Cofficients: \( \left\{1, 0, \sin^2 u, -\frac{\sin u}{H}, 0, -\frac{\sin^3 u}{H}\right\} \) - Hyperboloid: \( 2z=7x^2+6xy-y^2 \)at origin
Answer:
\( \vec{N}= \{0,0,1\}\)
Cofficients: \( \left\{ 1,0,1,7,3,-1\right\} \) - Minimal surface: \( e^z \cos x=\cos y \)
Answer:
\( \vec{N}= \left\{-\frac{\tan x}{H}, \frac{\tan y}{H}, \frac{1}{H}\right\} \)
Cofficients: \( \left\{\sec^2 x, -\tan x \tan y, \sec^2 y, \frac{\sec^2 x}{H}, 0, -\frac{\sec^2 y}{H}\right\} \)
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