Asymptotic Lines and related Theorems

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Let $S:\overrightarrow{r}=\overrightarrow{r}(u,v)$ 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.

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Equation of Asymptotic lines

Let $S:\overrightarrow{r}=\overrightarrow{r}(u,v)$ 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
$Ld{u}^2+2Mdudv+Nd{v}^2=0$
This completes the proof.

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

Solution
Let $S:\overrightarrow{r}=\overrightarrow{r}(u,v)$ 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.

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Show that asymptotic lines of general surface of revolution is $f_{11}d{u}^2+u{f}_{1}d{v}^2=0$

Solution
Let $\overrightarrow{r}=\overrightarrow{r}(u,v)$ be general surface of revolution, then position of arbitrary point on the surface is
$\overrightarrow{r}=(u\cos v,u\sin v,f(u))$
Also, fundamental coefficients of the surface are
$L=\frac{u{f}_{11}}{H},M=0,N=\frac{u^2{f}_1}{H},H^2=u^2(1+f_1^2)$
Now, the differential equation of asymptotic lines is
$Ld{u}^2+2Mdudv+Nd{v}^2=0$
or$\frac{u{f}_{11}}{H}d{u}^2+\frac{u^2{f}_1}{H}d{v}^2=0$
or$f_{11}d{u}^2+u{f}_{1}d{v}^2=0$
This completes the proof.

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

Proof
Let $\overrightarrow{r}=\overrightarrow{r}(u,v)$ be a surface then
$d\overrightarrow{N}={\overrightarrow{N}}_{1}du+{\overrightarrow{N}}_{2}dv$
$d\overrightarrow{r}={\overrightarrow{r}}_{1}du+{\overrightarrow{r}}_{2}dv$
Now, necessary and sufficient condition for the curve to be asymptotic is
$Ld{u}^2+2Mdudv+Nd{v}^2=0$
$({\overrightarrow{N}}_{1}du+{\overrightarrow{N}}_{2}dv)\cdot ({\overrightarrow{r}}_{1}du+{\overrightarrow{r}}_{2}dv)=0$
$d\overrightarrow{N}\cdot d\overrightarrow{r}=0$
This completes the proof.

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Condition for Asymptotic Lines to be Orthogonal

Let $\overrightarrow{r}=\overrightarrow{r}(u,v)$ be a surface and C be an asymptotic line on it. Then equation of asymptotic line is
$Ld{u}^2+2Mdudv+Nd{v}^2=0$ (i)
Also, equation of double family of curves on the surface is
$Pd{u}^2+2Qdudv+Rd{v}^2=0$ (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.

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

Solution
Let $\overrightarrow{r}=\overrightarrow{r}(u,v)$ be a surface. Assume that the surface is minimal. Then,
$\mu =0$
or$EN-2FM+GL=0$
This shows that, asymptotic directions are orthogonal.

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

Proof
Let $\overrightarrow{r}=\overrightarrow{r}(u,v)$ be a surface and C be an asymptotic line on it. Then differential equation of asymptotic line is
$Ld{u}^2+2Mdudv+Nd{v}^2=0$ (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.

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Show that parametric curves of right helicoids are asymptotic lines.

Solution
The position of arbitrary point on the right helicoids is
$\overrightarrow{r}=(ucosv,usinv,cv)$ (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.

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

Proof
Let $\overrightarrow{r}=\overrightarrow{r}(u,v)$be a surface and C be an asymptotic line on it.
Then
equation of osculating plane on asymptotic line is
$(R-\overrightarrow{r})\overrightarrow{b}=0$ (i)
Also, equation of tangent plane to the surface is
$(R-\overrightarrow{r})\overrightarrow{N}=0$ (ii)
Here
$\overrightarrow{N}.\overrightarrow{t}=0$ (iii)
Differentiating of (i) w. r. to. s, we get
$\frac{d\overrightarrow{N}}{ds}.\overrightarrow{t}+\overrightarrow{N}.\kappa \overrightarrow{N}=0$
or $\frac{d\overrightarrow{N}}{ds}.\frac{d\overrightarrow{r}}{ds}+\overrightarrow{N}.\kappa \overrightarrow{N}=0$
In asymptotic line
$\frac{d\overrightarrow{N}}{ds}.\frac{d\overrightarrow{r}}{ds}=0$
Thus, we have
$\frac{d\overrightarrow{N}}{ds}.\frac{d\overrightarrow{r}}{ds}+\overrightarrow{N}.\kappa \overrightarrow{N}=0$
or $\overrightarrow{N}.\kappa \overrightarrow{N}=0$
or $\overrightarrow{N}.\overrightarrow{N}=0$ (iv)
From (iii) and (iv), we can write
$\overrightarrow{N}=\overrightarrow{b}$
Thus, (i) and (ii), are same.
This completes the proof.

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

Solution
Let $\overrightarrow{r}=\overrightarrow{r}(u,v)$ be a surface and C be a curve on it.
Then, C is trraight line if and only if
$\kappa =0$ (i)
Also
$\overrightarrow{N}\overrightarrow{t}=0$ (ii)
Differentiating of (ii) w. r. to. s, we get
$\frac{d\overrightarrow{N}}{ds}.\overrightarrow{t}+\overrightarrow{N}.\kappa \overrightarrow{N}=0$
or $\frac{d\overrightarrow{N}}{ds}.\frac{d\overrightarrow{r}}{ds}+\overrightarrow{N}.\kappa \overrightarrow{N}=0$
In straight line
$\kappa =0$
Thus, we have
$\frac{d\overrightarrow{N}}{ds}.\frac{d\overrightarrow{r}}{ds}+\overrightarrow{N}.\kappa \overrightarrow{N}=0$
or $\frac{d\overrightarrow{N}}{ds}.\frac{d\overrightarrow{r}}{ds}=0$
or $d\overrightarrow{N}.d\overrightarrow{r}=0$
or $({\overrightarrow{N}}_{1}du+{\overrightarrow{N}}_{2}dv).({\overrightarrow{r}}_{1}du+{\overrightarrow{r}}_{2}dv)=0$
or $Ld{u}^2+2Mdudv+Nd{v}^2=0$
This is equation of asymptotic line.
Thus, all straight lines on a surface are asymptotic lines.

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Curvature and Torsions of Asymptotic Lines

Let $\overrightarrow{r}=\overrightarrow{r}(u,v)$ be a surface and $\overrightarrow{r}=\overrightarrow{r}(s)$ be asymptotic line on it.
Then expression of curvature for asymptotic line C is
${\overrightarrow{t}}^{\prime }=\kappa \overrightarrow{N}$
Operating dot product on both side by $\overrightarrow{N}$, we get
$\kappa ={\overrightarrow{t}}^{\prime }.\overrightarrow{N}$
or $\kappa ={\overrightarrow{t}}^{\prime }.(\overrightarrow{b}\times \overrightarrow{t})$
or $\kappa =[{\overrightarrow{t}}^{\prime },\overrightarrow{b},\overrightarrow{t}]$
or $\kappa =[\overrightarrow{t},{\overrightarrow{t}}^{\prime },\overrightarrow{b}]$
For asymptotic line, we have
$\overrightarrow{N}=\overrightarrow{b}$.v Thus, the curvature of asymptotic line is,
$\kappa =[\overrightarrow{t},{\overrightarrow{t}}^{\prime },\overrightarrow{N}]$
Next, the torsion of asymptotic line C is
${\overrightarrow{b}}^{\prime }=-\tau \overrightarrow{N}$
Operating dot product on both side by $\overrightarrow{N}$, we get
$\tau =-{\overrightarrow{b}}^{\prime }.\overrightarrow{N}$
or $\tau =-{\overrightarrow{b}}^{\prime }.(\overrightarrow{b}\times \overrightarrow{t})$
or $\tau =-[{\overrightarrow{b}}^{\prime },\overrightarrow{b},\overrightarrow{t}]$
or $\tau =[\overrightarrow{b},{\overrightarrow{b}}^{\prime },\overrightarrow{t}]$
For asymptotic line, we have
$\overrightarrow{N}=\overrightarrow{b}$.
Thus, torsion of asymptotic line is,
$\tau =[\overrightarrow{N},{\overrightarrow{N}}^{\prime },\overrightarrow{t}]$
Hence, curvature and torsion of asymptotic lines are
$\kappa =[\overrightarrow{t},{\overrightarrow{t}}^{\prime },\overrightarrow{N}]$ and $\tau =[\overrightarrow{N},{\overrightarrow{N}}^{\prime },\overrightarrow{t}]$

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Theorems of Beltrami and Ennper

Torsion of asymptotic line is $\tau =\pm \sqrt{-K}$, where K is the Gaussian Curvature.
Proof
Let $\overrightarrow{r}=\overrightarrow{r}(u,v)$ be a surface and $\overrightarrow{r}=\overrightarrow{r}(s)$ be asymptotic line on it.
Now, torsion of C is
$\tau =[\overrightarrow{N},{\overrightarrow{N}}^{\prime },\overrightarrow{t}]$
or $\tau =[\overrightarrow{N},d\overrightarrow{N},d\overrightarrow{r}]\frac{1}{d{s}^2}$
or $\tau =\{\overrightarrow{N}.({\overrightarrow{N}}_{1}du+{\overrightarrow{N}}_{2}dv)\times ({\overrightarrow{r}}_{1}du+{\overrightarrow{r}}_{2}dv)\}\frac{1}{d{s}^2}$
or $\tau =\{[\overrightarrow{N},{\overrightarrow{N}}_1,{\overrightarrow{r}}_1]d{u}^2+[\overrightarrow{N},{\overrightarrow{N}}_1,{\overrightarrow{r}}_2]dudv+[\overrightarrow{N},{\overrightarrow{N}}_2,{\overrightarrow{r}}_1]dudv+[\overrightarrow{N},{\overrightarrow{N}}_2,{\overrightarrow{r}}_2]d{v}^2\}\frac{1}{d{s}^2}$
or $\tau =[\overrightarrow{N},{\overrightarrow{N}}_1,{\overrightarrow{r}}_1](\frac{du}{ds}{)}^2+[\overrightarrow{N},{\overrightarrow{N}}_1,{\overrightarrow{r}}_2]\frac{du}{ds}\frac{dv}{ds}+[\overrightarrow{N},{\overrightarrow{N}}_2,{\overrightarrow{r}}_1]dudv+[\overrightarrow{N},{\overrightarrow{N}}_2,{\overrightarrow{r}}_2](\frac{dv}{ds}{)}^2$
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
$\tau =\frac{EM}{H}(\frac{du}{ds}{)}^2+\frac{-GM}{H}(\frac{dv}{ds}{)}^2=\frac{M}{H}(E(\frac{du}{ds}{)}^2-G(\frac{dv}{ds}{)}^2)$

1. Case 1: For asymptotic line along $v$ curve, we have $du=0$
   Thus
   $\tau =-\frac{M}{H}G(\frac{dv}{ds}{)}^2$
   Here
   $Ed{u}^2+2Fdudv+Gd{v}^2=d{s}^2$
   or $Gd{v}^2=d{s}^2$
   or $G(\frac{dv}{ds}{)}^2=1$
   Thus
   $\tau =-\frac{M}{H}$ (B)
   
2. Case 2: For asymptotic line along $u$ curve, we have $dv=0$
   Thus
   $\tau =\frac{M}{H}E(\frac{du}{ds}{)}^2$
   Here
   $Ed{u}^2+2Fdudv+Gd{v}^2=d{s}^2$
   or $Ed{u}^2=d{s}^2$
   or $E(\frac{du}{ds}{)}^2=1$
   Thus
   $\tau =\frac{M}{H}$ (C)
   

Combining both, we get
$\tau =\pm \frac{M}{H}$ (D)
Using (A), we get,
$\tau =\pm \sqrt{-K}$
This completes the proof.