Let \( S:\vec{r}=\vec{r}( u,v )\) be a surface. Also let \( \Gamma \) and \( \Gamma' \) be tangent plane at \( P\) and \(Q\) respectively in which \( \Gamma \cup \Gamma' =l \). Now, limiting form of directions \(PQ\) and \(l\) as\(Q \to P\), are called conjugate directions at P. Conjugate direction is a pair of directions, one of which is direction of curve and other is direction of characteristic line of tangent planes. Conjugate directions are direction of a pair of curves on a surface whose principal normal are same.

#### Equation of Conjugate Directions

Let \( S:\vec{r}=\vec{r}( u,v )\) be a surface in which let \(P( u,v )\) and \( Q( u+du,v+dv )\) are two neighboring points such that\(\overrightarrow{OP}=\vec{r}, \overrightarrow{OQ}=\vec{r}+d\vec{r}\)

Also, consider tangent plane \( \Gamma \) and \( \Gamma' \) respectively with \( \Gamma \cup \Gamma' =l \). Let R be a point on \(l\) such that

\(\overrightarrow{OR}=\vec{r}+\delta \vec{r}\)

Then

\( \overrightarrow{PR}=\overrightarrow{OR}-\overrightarrow{OP}\)

\( \overrightarrow{PR}=\delta \vec{r}\)

\( \overrightarrow{PR}=\vec{r}_1 \delta u+\vec{r}_2 \delta v \)

Assume \( \vec{N}\) and \( \vec{N}+d\vec{N}\) be surface normal at P and Q respectively.Then,

\( \vec{N} \cdot \overrightarrow{PR}=0\) and\(( \vec{N}+d \vec{N} ) \cdot \overrightarrow{PR}=0\)

because \( \overrightarrow{PR}\) lies on tangent planes

or\( d \vec{N} \cdot \overrightarrow{PR}=0\)

or\(( \vec{N}_1 du+ \vec{N}_2 dv ) \cdot ( \vec{r}_1 \delta u+\vec{r}_2 \delta v )=0\)

or\(L du\delta u+M( du\delta v+\delta udv )+Ndv\delta v=0\)

This is equation of conjugate directions.

Note

Let \(( l,m )\) and \(( l',m' )\) be two directions on a surface, then \(( l,m )\) and \(( l',m' )\) will be conjugate if and only if

\( Lll'+M( lm'+l'm )+Nmm'=0\)

#### Condition to be Conjugate Direction

Let \( S:\vec{r}=\vec{r}( u,v )\) be a surface and \( Pdu^2+2Qdudv+Rdv^2=0\) be double family of curves, then\(P( \frac{du}{dv} )^2+2Q \frac{du}{dv}+R=0\) (i)

If \( \frac{l}{m}\) and \( \frac{l'}{m'}\) be roots given by (i), then

\( \frac{l}{m}+\frac{l'}{m'}=-\frac{2Q}{P}\) and \(\frac{l}{m}\frac{l'}{m'}=\frac{R}{P}\)

Now, two directions given by (i) will be conjugate if

\( L \frac{l}{m}\frac{l'}{m'}+M( \frac{l}{m}+\frac{l'}{m'} )+N=0\)

or\(L\frac{R}{P}+M( -\frac{2Q}{P} )+N=0\)

or\(LR-2MQ+NP=0\)

#### An important result:

The necessary and sufficient condition for parametric curves to have conjugate direction is \(M=0\)Proof

Let \( \vec{r}=\vec{r}( u,v )\) be a surface. Now, equation of parametric curves is

\(dudv=0\) (i)

Also, equation of double family of curves is

\(Pdu^2+2Qdudv+Rdv^2=0\) (ii)

Now, comparing (i) and (ii), we get

\(P=R=0,Q \ne 0\)

Now, necessary and sufficient condition for parametric curves to have conjugate direction is

\(LR-2MQ+NP=0\)

or\(M=0\)

This completes the proof.

#### Show that parametric curves of general surface of revolution have conjugate directions.

SolutionThe equation of general surface of revolution is

\(\vec{r}=( u \cos v,u \sin v,f( u ) )\) (i)

The fundamental coefficients are

\(F=0,M=0\)

Thus, parametric curves are conjugate.

#### Principal directions are orthogonal conjugate

ProofLet \( \vec{r}=\vec{r}( u,v )\) be a surface and \(( du,dv )\) be principal direction. Then differential equation of principal direction is

\(( EM-FL )du^2+( EN-GL )dudv+( FN-GM )dv^2=0\) (i)

Also, equation of double family of curves is

\(Pdu^2+2Qdudv+Rdv^2=0\) (ii)

Now, comparing (i) and (ii), we get

\(( EM-FL )=P,( EN-GL )=2Q,( FN-GM )=R\)

Here, condition for orthogonal is

\(ER-2FQ+GP\)

= \(E( FN-GM )-F( EN-GL )+G( EM-FL )\)

=\(0\)

Here, condition for conjugate is

\(LR-2MQ+NP=0\)

Hence, principal directions are orthogonal conjugate

This completes the proof.

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