Let \( S:\vec{r}=\vec{r}( u,v )\) be a surface. Then curves on the surface whose directions are self-conjugate, is called asymptotic lines. The direction of asymptotic lines are called asymptotic direction.

#### Equation of Asymptotic lines

Let \( S:\vec{r}=\vec{r}( u,v )\) be a surface. If \(( du,dv )\) and \(( \delta u,\delta v )\) be conjugate direction, then\(Ldu\delta u+M( du\delta v+\delta udv )+N\delta udv=0\)

In asymptotic line, the directions \(( du,dv )\) and \(( \delta u,\delta v )\) are self-conjugate.

or\(( du,dv )=( \delta u,\delta v )\)

Hence, differential equation of asymptotic line is

\(Ldu^2+2Mdudv+Ndv^2=0\)

This completes the proof.

#### Prove that normal curvature in a direction perpendicular to an asymptotic line is twice the mean curvature.

SolutionLet \( S:\vec{r}=\vec{r}( u,v )\) be a surface and C be an asymptotic line on it. Then, by Euler’s theorem, normal curvature for asymptotic line is

\(\kappa _a \cos ^2 \psi + \kappa _b \sin ^2 \psi =0\) [In asymptotic line , \(\kappa_n =0\) (i)

Let, \(C_1\) be a normal section perpendicular to asymptotic line, then

\(\kappa _a \cos ^2 (90+\psi) + \kappa _b \sin ^2 (90+\psi)=\kappa_n\)

or\(\kappa _a \sin ^2\psi +\kappa_b \cos ^2 \psi =\kappa_n\) (ii)

Adding (i) and (ii), we get

\(\kappa_n=\kappa _a+\kappa _b\)

or\(\kappa_n =2( \frac{\kappa _a+\kappa _b}{2} )\)

or\(\kappa_n=2 \mu \)

This completes the solution.

#### Show that asymptotic lines of general surface of revolution is \(f_{11}du^2+uf_1dv^2=0\)

SolutionLet \( \vec{r}=\vec{r}( u,v )\) be general surface of revolution, then position of arbitrary point on the surface is

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

Also, fundamental coefficients of the surface are

\(L=\frac{u f_{11} }{H},M=0,N=\frac{u^2f_1}{H},H^2=u^2( 1+f_1^2 )\)

Now, the differential equation of asymptotic lines is

\(Ldu^2+2Mdudv+Ndv^2=0\)

or\(\frac{uf_{11}}{H}du^2+\frac{u^2f_1}{H}dv^2=0\)

or\(f_{11}du^2+uf_1dv^2=0\)

This completes the proof.

#### Necessary and sufficient condition for \( \vec{r}=\vec{r}( s )\) on \( \vec{r}=\vec{r}( u,v )\) be asymptotic line is \( d\vec{N} \cdot d\vec{r}=0\)

ProofLet \( \vec{r}=\vec{r}( u,v )\) be a surface then

\(d\vec{N}=\vec{N}_1du+\vec{N}_2dv\)

\(d\vec{r}=\vec{r}_1du+\vec{r}_2dv\)

Now, necessary and sufficient condition for the curve to be asymptotic is

\(Ldu^2+2Mdudv+Ndv^2=0\)

\(( \vec{N}_1du+\vec{N}_2dv ) \cdot (\vec{r}_1du+\vec{r}_2dv ) =0\)

\(d \vec{N} \cdot d\vec{r}=0\)

This completes the proof.

#### Condition for Asymptotic Lines to be Orthogonal

Let \( \vec{r}=\vec{r}( u,v )\) be a surface and C be an asymptotic line on it. Then equation of asymptotic line is\(Ldu^2+2Mdudv+Ndv^2=0\) (i)

Also, equation of double family of curves on the surface is

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

Comparing (i) with (ii), we get

\(P=L,Q=M,R=N\)

Now, condition for the asymptotic line to be orthogonal is

\(ER-2FQ+GP=0\)

or\(EN-2FM+GL=0\)

This completes the proof.

#### Show that asymptotic directions are orthogonal if and only if the surface is minimal.

SolutionLet \( \vec{r}=\vec{r}( u,v )\) be a surface. Assume that the surface is minimal. Then,

\(\mu =0\)

or\(EN-2FM+GL=0\)

This shows that, asymptotic directions are orthogonal.

#### The necessary and sufficient condition for parametric curves to be asymptotic lines is \( L=N=0,M \ne 0\)

ProofLet \(\vec{r}=\vec{r}( u,v )\) be a surface and C be an asymptotic line on it. Then differential equation of asymptotic line is

\(Ldu^2+2Mdudv+Ndv^2=0\) (i)

Also, equation of parametric curves on the surface is

\(dudv=0\) (ii)

Now, necessary and sufficient condition for parametric curves to be asymptotic lines is, (i) and (ii) must be identical

\(L=0,M \ne 0,N=0\)

This completes the proof.

#### Show that parametric curves of right helicoids are asymptotic lines.

SolutionThe position of arbitrary point on the right helicoids is

\(\vec{r}=( ucosv,usinv,cv )\) (i)

Then, the fundamental coefficients of the surface is

\(E=1,F=0,G=u^2+c^2,L=0,M=\frac{-c}{\sqrt{u^2+c^2}},N=0\)

In this case, we have

\(L=0,M=-\frac{c}{H},N=0\)

Hence, on right helicoids, the parametric curves are asymptotic lines.

#### Show that osculating plane on asymptotic line is tangent plane to the surface.

ProofLet \( \vec{r}=\vec{r}( u,v )\)be a surface and C be an asymptotic line on it.

Then

equation of osculating plane on asymptotic line is

\( ( R-\vec{r} )\vec{b}=0\) (i)

Also, equation of tangent plane to the surface is

\( ( R-\vec{r} )\vec{N}=0\) (ii)

Here

\( \vec{N}.\vec{t}=0\) (iii)

Differentiating of (i) w. r. to. s, we get

\(\frac{d\vec{N}}{ds}.\vec{t} +\vec{N}.\kappa \vec{N}=0\)

or \(\frac{d\vec{N}}{ds}.\frac{d\vec{r}}{ds} +\vec{N}.\kappa \vec{N}=0\)

In asymptotic line

\( \frac{d\vec{N}}{ds}.\frac{d\vec{r}}{ds}=0 \)

Thus, we have

\(\frac{d\vec{N}}{ds}.\frac{d\vec{r}}{ds} +\vec{N}.\kappa \vec{N}=0\)

or \(\vec{N}.\kappa \vec{N}=0\)

or \( \vec{N}.\vec{N} =0 \) (iv)

From (iii) and (iv), we can write

\( \vec{N}=\vec{b} \)

Thus, (i) and (ii), are same.

This completes the proof.

#### Show that all straight lines on a surface are asymptotic lines.

SolutionLet \( \vec{r}=\vec{r}( u,v )\) be a surface and C be a curve on it.

Then, C is trraight line if and only if

\( \kappa =0 \) (i)

Also

\( \vec{N} \vec{t}=0 \) (ii)

Differentiating of (ii) w. r. to. s, we get

\(\frac{d\vec{N}}{ds}.\vec{t} +\vec{N}.\kappa \vec{N}=0\)

or \(\frac{d\vec{N}}{ds}.\frac{d\vec{r}}{ds} +\vec{N}.\kappa \vec{N}=0\)

In straight line

\( \kappa =0 \)

Thus, we have

\(\frac{d\vec{N}}{ds}.\frac{d\vec{r}}{ds} +\vec{N}.\kappa \vec{N}=0\)

or \(\frac{d\vec{N}}{ds}.\frac{d\vec{r}}{ds} =0\)

or \(d\vec{N}. d\vec{r} =0\)

or \((\vec{N}_1 du+\vec{N}_2 dv). (\vec{r}_1 du+\vec{r}_2 dv)=0\)

or \(Ldu^2+2Mdudv+Ndv^2=0\)

This is equation of asymptotic line.

Thus, all straight lines on a surface are asymptotic lines.

#### Curvature and Torsions of Asymptotic Lines

Let \( \vec{r}=\vec{r}( u,v )\) be a surface and \(\vec{r}=\vec{r}( s )\) be asymptotic line on it.Then expression of curvature for asymptotic line C is

\( \vec{t}'=\kappa \vec{N}\)

Operating dot product on both side by \(\vec{N} \), we get

\(\kappa =\vec{t}'.\vec{N}\)

or \(\kappa =\vec{t}'.(\vec{b}\times \vec{t} )\)

or \(\kappa =[ \vec{t}',\vec{b},\vec{t} ]\)

or \(\kappa =[ \vec{t},\vec{t}',\vec{b} ]\)

For asymptotic line, we have

\( \vec{N}=\vec{b}\).v Thus, the curvature of asymptotic line is,

\( \kappa =[ \vec{t},\vec{t}',\vec{N} ]\)

Next, the torsion of asymptotic line C is

\( \vec{b}'=-\tau \vec{N} \)

Operating dot product on both side by \(\vec{N}\), we get

\( \tau =-\vec{b}'.\vec{N}\)

or \( \tau =-\vec{b}'.( \vec{b}\times \vec{t} )\)

or \( \tau =-[ \vec{b}',\vec{b},\vec{t} ]\)

or \( \tau =[ \vec{b},\vec{b}',\vec{t} ]\)

For asymptotic line, we have

\( \vec{N}=\vec{b}\).

Thus, torsion of asymptotic line is,

\(\tau =[ \vec{N},\vec{N}',\vec{t}]\)

Hence, curvature and torsion of asymptotic lines are

\( \kappa =[ \vec{t},\vec{t}',\vec{N} ]\) and \(\tau =[ \vec{N},\vec{N}',\vec{t}]\)

#### Theorems of Beltrami and Ennper

Torsion of asymptotic line is \( \tau =\pm \sqrt{-K}\), where K is the Gaussian Curvature.Proof

Let \( \vec{r}=\vec{r}( u,v )\) be a surface and \( \vec{r}=\vec{r}( s )\) be asymptotic line on it.

Now, torsion of C is

\(\tau =[ \vec{N},\vec{N}',\vec{t}]\)

or \(\tau =[ \vec{N},d\vec{N},d\vec{r}] \frac{1}{ds^2}\)

or \( \tau =\{ \vec{N}.(\vec{N}_1 du+\vec{N}_2 dv) \times (\vec{r}_1 du+\vec{r}_2 dv) \} \frac{1}{ds^2} \)

or \( \tau =\{[\vec{N},\vec{N}_1, \vec{r}_1] du^2 + [\vec{N},\vec{N}_1, \vec{r}_2]dudv+[\vec{N},\vec{N}_2, \vec{r}_1] dudv +[\vec{N},\vec{N}_2, \vec{r}_2] dv^2 \} \frac{1}{ds^2} \)

or \( \tau =[\vec{N},\vec{N}_1, \vec{r}_1] (\frac{du}{ds})^2 + [\vec{N},\vec{N}_1, \vec{r}_2]\frac{du}{ds}\frac{dv}{ds}+[\vec{N},\vec{N}_2, \vec{r}_1] dudv+[\vec{N},\vec{N}_2, \vec{r}_2] (\frac{dv}{ds})^2 \)

or \( \tau =\frac{EM-FL}{H} (\frac{du}{ds})^2 +( \frac{FM-GL}{H}+\frac{EN-FM}{H} )\frac{du}{ds}\frac{dv}{ds}+\frac{FN-GM}{H}(\frac{dv}{ds})^2\)

Without loss of generality, we take asymptotic line along parametric curve, then

\( L=0,N=0,M \ne 0\)

Thus

\( K=\frac{LN-M^2}{H^2}=-\frac{M^2}{H^2}\)

or \(\frac{{{M}^2}}{{{H}^2}}=-K \)

or \(\frac{M}{H}=\sqrt{-K}\) (A)

Next, torsion of asymptotic line is

\( \tau =\frac{EM}{H}(\frac{du}{ds})^2 +\frac{-GM}{H}(\frac{dv}{ds})^2=\frac{M}{H}( E (\frac{du}{ds})^2- G(\frac{dv}{ds})^2 )\)

- Case 1: For asymptotic line along \( v\) curve, we have \(du=0\)

Thus

\( \tau =-\frac{M}{H} G(\frac{dv}{ds})^2\)

Here

\( Edu^2+2Fdudv+Gdv^2=ds^2\)

or \( Gdv^2=ds^2\)

or \( G(\frac{dv}{ds})^2=1\)

Thus

\( \tau =-\frac{M}{H}\) (B)

- Case 2: For asymptotic line along \( u \) curve, we have \(dv=0\)

Thus

\( \tau =\frac{M}{H} E (\frac{du}{ds})^2 \)

Here

\( Edu^2+2Fdudv+Gdv^2=ds^2\)

or \( Edu^2=ds^2\)

or \( E(\frac{du}{ds})^2=1\)

Thus

\( \tau =\frac{M}{H}\) (C)

\( \tau= \pm\frac{M}{H} \) (D)

Using (A), we get,

\( \tau= \pm \sqrt{-K} \)

This completes the proof.

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