Ruled Surface
A surface generated by motion of straight line along a given curve is called ruled surface, in which, striaght line is called generator and given curve is called directrix of the ruled surface.For example, a cone is formed by keeping one point of a line fixed and moving the line along a circle, thus, cone is a ruled surface.
Other examples of ruled surface are cylinder, right conoid and helicoid.
Equation of Ruled Surface
Let S:\( \vec{r}=\vec{r}( s ) \) be a directrix of a ruled surface and P be a point on it with
\( \overrightarrow{OP}=\vec{r} \)
If \( \vec{g}( s ) \) be unit vector along generator and \( R \) be arbitrary point T on the generator, then equation of ruled surface S is
\( R=\vec{r}( s )+v\vec{g}( s ) \) (i)where \( v \) is parameter
Types of Ruled Surface
There are two types of ruled surface. One is developable and other is skew.- A ruled surface whose consecutive generators do intersect is called developable surface.
For example, cone is developable surface.
- A ruled surface whose consecutive generators do not intersect is called skew surface.
For example, cylinder is skew surface.
Theorem
Show that necessary and sufficient condition for a ruled surface to be a developable is \([ \vec{t},\vec{g},\vec{g}' ]=0 \).
Proof
Let S be a ruled surface, then its equation is
\(R=\vec{r}( s )+v\vec{g}( s ) \) (i)\(v \) is real parameter
Differentiating (i) w. r. to. s and v respectively, we get
\({R_1} =\frac{\partial R}{\partial s}= \vec{t}+v \vec{g}'\)
\({R_2}=\frac{\partial R}{\partial v}=\vec{g} \)
\({R_{11}}=\vec{t}'+v\vec{g}''\)
\({R_{12}}=\vec{g}'\)
\({R_{22}}=0 \)
Here
\([ {R_1},{R_2},{R_{12}} ]=HM\)
or
\([ \vec{t}+v\vec{g}',\vec{g},\vec{g}' ]= HM \)
or
\( [ \vec{t},\vec{g},\vec{g}' ]=HM\)
Next
\([ {R_1},{R_2},{R_{22}} ]=HN\)
or
\([ \vec{t}+v\vec{g}',\vec{g},0 ]=HN \)
or
\(0=HN \)
or
\(N=0 \)
Now, Gaussian curvature of the surface is
\(K =\frac{LN-M^2}{H^2}\)
or
\( K= \frac{-M^2}{H^2} \)
or
\( K=-\frac{[ \vec{t},\vec{g},\vec{g}' ]^2}{H^4} \)
Since, necessary and sufficient condition for a surface to be developable surface is
\(K=0 \)
So, necessary and sufficient condition for a ruled surface to be a developable surface is
\( \frac{[ \vec{t},\vec{g},\vec{g}' ]^2}{H^4}=0 \)
or
\( [\vec{t},\vec{g},\vec{g}' ]=0 \)
Theorem 2
Show that necessary and sufficient condition for a ruled surface to be skew is \([ \vec{t},\vec{g},\vec{g}' ] \ne 0 \).
Proof
Let S be a ruled surface, then
\( K=-\frac{[ \vec{t},\vec{g},\vec{g}' ]^2}{H^4} \)
Since, necessary and sufficient condition for a surface to be skew surface is
\(K \ne 0 \)
So, necessary and sufficient condition for a ruled surface to be a skew surface is
\( \frac{[ \vec{t},\vec{g},\vec{g}' ]^2}{H^4}\ne 0\)
or
\( [\vec{t},\vec{g},\vec{g}' ]\ne 0 \)
Example
Find a condition that \( x=az+\alpha ,y=bz+\beta \) generates a skew surface where \( a,b,\alpha ,\beta \) are function of s.
Solution
Given the equation of line is
\(x=az+\alpha ,y=bz+\beta \)
or
\( \frac{x-\alpha }{a}=z,\frac{y-\beta }{b}=z\)
or
\(\frac{x-\alpha }{a}=\frac{y-\beta }{b}=\frac{z}1\)
Here
\( \vec{r}=( \alpha ,\beta ,0 )\) and \(\vec{g}=( a,b,1 )\)
Thus
\( \vec{t}=( \alpha',\beta ',0 )\) and \(\vec{g}'=( a',b',0 ) \)
So
\( \vec{g}\times \vec{g}'=( a,b,1 )\times ( a',b',0 ) \)
or
\( \vec{g}\times \vec{g}'=( -{b}',{a}',a{b}'-a'b )\)
Now, condition that the line generates a skew surface is
\([ \vec{t},\vec{g},\vec{g}']\ne 0 \)
or
\(\vec{t}.( \vec{g}\times \vec{g}' )\ne 0\)
or
\(( \alpha',\beta ',0 ).( -b',a',ab'-a'b )\ne 0 \)
or
\(a' \beta '-b'\alpha ' \ne 0 \)
Example
Show that a line \( x=3t^2z+2t( 1-3t^2 ),y=-2tz+t^2( 3+4t^2)\) generates a skew surface.
Solution
Given equation of line is
\(x=3t^2z+2t( 1-3t^2 ),y=-2tz+t^2( 3+4t^2 )\)
or
\(x-2t( 1-3t^2 )=3t^2z,y-t^2( 3+4t^2 )=-2tz\)
or
\(\frac{x-2t( 1-3t^2 )}{3t^2}=z,\frac{y-t^2( 3+4t^2 )}{-2t}=z\)
or
\( \frac{x-2t( 1-3t^2 )}{3t^2}=\frac{y-t^2( 3+4t^2 )}{-2t}=\frac{z}{1} \)
Here
\(\vec{r}=\{ 2t( 1-3t^2 ),t^2( 3+4t^2 ),0 \}\) and \( \vec{g}=( 3t^2,-2t,1 )\)
Thus
\(\vec {t}=( 2-18t^2,6t+16t^3,0 )\) and \( \vec{g}'=( 6t,-2,0 )\)
So
\( \vec{g}\times \vec{g}'=( 3t^2,-2t,1 )\times ( 6t,-2,0 )\)
or
\( \vec{g}\times \vec{g}'=( -2,6t,18t^2 )\)
Now, we have
\([ \vec {t},\vec{g},\vec{g}' ]=\vec{t}.( \vec{g}\times \vec{g}' )\)
or
\([ \vec {t},\vec{g},\vec{g}' ] =( 2-18t^2,6t+16t^3,0 ).( -2,6t,18t^2 )\)
or
\([ \vec {t},\vec{g},\vec{g}' ]\ne 0 \)
Hence, given line generates a skew surface.
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