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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