Let
Equation of Asymptotic lines
LetIn asymptotic line, the directions
or
Hence, differential equation of asymptotic line is
This completes the proof.
Prove that normal curvature in a direction perpendicular to an asymptotic line is twice the mean curvature.
SolutionLet
Let,
or
Adding (i) and (ii), we get
or
or
This completes the solution.
Show that asymptotic lines of general surface of revolution is
SolutionLet
Also, fundamental coefficients of the surface are
Now, the differential equation of asymptotic lines is
or
or
This completes the proof.
Necessary and sufficient condition for on be asymptotic line is
ProofLet
Now, necessary and sufficient condition for the curve to be asymptotic is
This completes the proof.
Condition for Asymptotic Lines to be Orthogonal
LetAlso, equation of double family of curves on the surface is
Comparing (i) with (ii), we get
Now, condition for the asymptotic line to be orthogonal is
or
This completes the proof.
Show that asymptotic directions are orthogonal if and only if the surface is minimal.
SolutionLet
or
This shows that, asymptotic directions are orthogonal.
The necessary and sufficient condition for parametric curves to be asymptotic lines is
ProofLet
Also, equation of parametric curves on the surface is
Now, necessary and sufficient condition for parametric curves to be asymptotic lines is, (i) and (ii) must be identical
This completes the proof.
Show that parametric curves of right helicoids are asymptotic lines.
SolutionThe position of arbitrary point on the right helicoids is
Then, the fundamental coefficients of the surface is
In this case, we have
Hence, on right helicoids, the parametric curves are asymptotic lines.
Show that osculating plane on asymptotic line is tangent plane to the surface.
ProofLet
Then
equation of osculating plane on asymptotic line is
Also, equation of tangent plane to the surface is
Here
Differentiating of (i) w. r. to. s, we get
or
In asymptotic line
Thus, we have
or
or
From (iii) and (iv), we can write
Thus, (i) and (ii), are same.
This completes the proof.
Show that all straight lines on a surface are asymptotic lines.
SolutionLet
Then, C is trraight line if and only if
Also
Differentiating of (ii) w. r. to. s, we get
or
In straight line
Thus, we have
or
or
or
or
This is equation of asymptotic line.
Thus, all straight lines on a surface are asymptotic lines.
Curvature and Torsions of Asymptotic Lines
LetThen expression of curvature for asymptotic line C is
Operating dot product on both side by
or
or
or
For asymptotic line, we have
Next, the torsion of asymptotic line C is
Operating dot product on both side by
or
or
or
For asymptotic line, we have
Thus, torsion of asymptotic line is,
Hence, curvature and torsion of asymptotic lines are
Theorems of Beltrami and Ennper
Torsion of asymptotic line isProof
Let
Now, torsion of C is
or
or
or
or
or
Without loss of generality, we take asymptotic line along parametric curve, then
Thus
or
or
Next, torsion of asymptotic line is
- Case 1: For asymptotic line along
curve, we have
Thus
Here
or
or
Thus
(B)
- Case 2: For asymptotic line along
curve, we have
Thus
Here
or
or
Thus
(C)
Using (A), we get,
This completes the proof.
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