A minimal surface is a type of surface in differential geometry that has minimal area for a given boundary. Mathematically, minimal surfaces are characterized by having zero mean curvature everywhere.
The principles of minimal surfaces are applied in architecture and structural design, such as in the design of lightweight, efficient structures. Understanding minimal surfaces helps in studying the behavior of materials and membranes, especially in the context of minimal energy configurations.
Let \(S:\vec{r}=\vec{r}(u,v)\) be a surface. Now S is called minimal surface if mean curvature of S is zero. In such minimal surface,
\(\kappa _a+\kappa _b=0\) at all points.
Some example of minimal surfaces are: plane surface, helicoids and Enneper surface.
Theorem
Necessary and sufficient condition for a surface to be minimal is \(EN-2FM+GL=0\)Proof
Let \(S:\vec{r}=\vec{r}(u,v)\) be a surface.
Now, necessary and sufficient condition for a surface to be minimal is
\(\mu=0\)
or \(\frac{\kappa _a+\kappa _b}{2}=0\)
or\( \kappa _a+\kappa _b=0 \)
or \(\frac{EN-2FM+GL}{EG-F^2 }=0 \)
or \(EN-2FM+GL=0 \)
This completes the proof.
Example
Show that \(e^z \cos x=\cos y\) is minimal surface.Solution
The parametric equation of the surface \(e^z \cos x=\cos y\) is
\(\vec{r}=(x,y,\log \cos y-\log \cos x) \) (i)
By computing the fundamental coefficients, we get
\( E=\sec^2 x, F=-\tan x \tan y, G=\sec^2 y\)
\(L=\frac{\sec^2 x}{H}, M=0, N=-\frac{\sec^2 y}{H} \)
Now
\(EN-2FM+GL\)
=\( (\sec^2 x)(-\frac{\sec^2 y}{H})-2(-\tan x \tan y)(0)+(\sec^2 y)(\frac{\sec^2 x}{H})\)
=\(0 \)
This completes the solution.
Exercise
- Write two example of minimal surface
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