Let S:\(\vec{r}=\vec{r}(u,v)\) be a surface.
Then tangent line to the surface is described by
\( \vec{T}=\lambda \vec{r}_1+\mu \vec{r}_2 \)
Here
\( (\lambda, \mu )\) is called direction components
If \( \vec{e} \) is unit vector along this tangent line, then
\( \vec{e}=l \vec{r}_1+m\vec{r}_2\)
Here
\( (l, m )\) is called direction coefficients
in which
\( El^2+2Flm+Gm^2=1\)
Relation between \( (\lambda,\mu)\) and \( (l,m) \)
Let S:\(\vec{r}=\vec{r}(u,v)\) be a surface with direction components \( (\lambda, \mu )\) and direction coefficients \( (l, m )\) then
\( \frac{l}{\lambda},\frac{m}{\mu} =k \), (say)
or \( l=\lambda k, m= \mu k\)
Now,
\( El^2+2Flm+Gm^2=1\)
or\( E(\lambda k)^2+2F(\lambda k)(\mu k)+G(\mu k)^2=1\)
or\( k^2 (E \lambda ^2+2F \lambda \mu +G \mu^2) =1\)
or \( k=\frac{1}{\sqrt{E \lambda ^2+2F \lambda \mu +G \mu^2}}\)
Thus,
\( (l, m )= \left ( \frac{\lambda}{\sqrt{E \lambda ^2+2F \lambda \mu +G \mu^2}},\frac{\mu}{\sqrt{E \lambda ^2+2F \lambda \mu +G \mu^2}} \right )\)
Direction coefficients of parametric curves
Let S:\(\vec{r}=\vec{r}(u,v)\) be a surface with parametric curves
u=constant (v-curve) and v=constant (u-curve).
Then, \(\vec{r}_1\) is tangent to v= constant curve, thus
\(\vec{r}_1=1\cdot \vec{r}_1+0\cdot \vec{r}_2\)
Here, component of \(\vec{r}_1\) is (1,0), so direction coefficient of v= constant curve (u-curve) is
\((l,m)= \left ( \frac{\lambda}{\sqrt{E \lambda ^2+2F \lambda \mu +G \mu^2}},\frac{\mu}{\sqrt{E \lambda ^2+2F \lambda \mu +G \mu^2}} \right )\)
or\((l,m)= \left ( \frac{1}{\sqrt{E}},0 \right )\)
Similarly,
we know that, \(\vec{r}_2\) is tangent to u= constant curve (v-curve), thus
\(\vec{r}_2=0 \cdot \vec{r}_1+1 \cdot \vec{r}_2\)
Here, component of \(\vec{r}_2\) is (0,1), so direction coefficient of u= constant curve (v-curve) is
\((l,m)= \left ( \frac{\lambda}{\sqrt{E \lambda ^2+2F \lambda \mu +G \mu^2}},\frac{\mu}{\sqrt{E \lambda ^2+2F \lambda \mu +G \mu^2}} \right )\)
or\((l,m)= \left ( 0,\frac{1}{\sqrt{G}} \right )\)
Angle between two directions
Let S: \(\vec{r}=\vec{r}(u,v)\) be a surface \( (l,m)\) and \( (l' ,m')\) be two direction at P, then their corresponding unit vectors are
\( \vec{e} = l\vec{r}_1+m \vec{r}_2 \)
\( \vec{e}' = l' \vec{r}_1+m' \vec{r}_2 \)
If θ be the angle between these directions then
\( \cos \theta = \vec{e}. \vec{e}' \)
or
\( \cos \theta = (l\vec{r}_1+m \vec{r}_2) . ( l' \vec{r}_1+m' \vec{r}_2 ) \)
or
\( \cos \theta = Ell'+F(lm'+l'm)+Gmm' \)
Also
\( \sin \theta = | \vec{e} \times \vec{e}' | \)
or
\( \sin \theta = |(l\vec{r}_1+m \vec{r}_2) \times ( l' \vec{r}_1+m' \vec{r}_2 )| \)
or
\( \sin \theta =H(lm'-l'm) \)
Therefore
\( \tan \theta = \frac{H(lm'-l'm)}{Ell'+F(lm'+l'm)+Gmm'} \)
Note
If two directions \( (l,m)\) and \( (l' ,m')\) are orthogonal, then
\( Ell'+F(lm'+l'm)+Gmm'=0 \)
or
\( E \frac{l}{m} \frac{l'}{m'}+F \left (\frac{l}{m} + \frac{l'}{m'} \right )+G'=0 \)
Equivalently, if two directions \( (\lambda,\mu)\) and \( (\lambda' ,\mu')\) are orthogonal, then
\( E \lambda \lambda '+F(\lambda \mu '+\lambda ' \mu)+G \mu \mu'=0 \)
or
\( E \frac{\lambda}{\mu} \frac{\lambda'}{\mu'}+F \left (\frac{\lambda}{\mu} + \frac{\lambda'}{\mu'} \right )+G'=0 \)
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