Skip to main content

Enneper’s Minimal Surface

What is minimal surface

Minimal surfaces are important in physics, chemistry, and architecture. Minimal surfaces can be regarded as surfaces of minimal surface area for given boundary conditions. For a given closed curve \(C\) in \(R^3\), the problem is to how find the surface \(S\) of least area spanning. In mathematics, minimal surfaces are defined as surfaces with zero mean curvature. In the particular case when \(C\) lies in a two-dimensional plane, the minimal surface is simply the planar region bounded by \(C\). However, the general problem of minimal surface is difficult to solve. The easiest way to physically study minimal surfaces is to look at soap films. In the late nineteenth century, the Belgian physicist Joseph Plateau conducted a number of soap film experiments. He showed that a minimal surface can be obtained in the form of a soap film stretched on a wire framework. The mathematical boundary value problem for minimal surfaces is therefore also called the Plateau problem. A minimal surface parametrized as \(S=(u,v,h(u,v))\) therefore satisfies Lagrange's equation: \[(1+h^2_v)h_{uu}-2h_uh_vh_{uv}+(1+h^2_u)h_{vv}=0\] It is not easy to solve the nonlinear, second-order partial differential equation with specified boundary conditions (determined by the given closed curve \(C\)). The problem of minimal surface is considered part of the calculus of variations. With the aid of a computer, numerical algorithms can be used to find approximate solutions. Enneper’s minimal surface can be parametrized as following: \[\left\{\begin{split} \begin{aligned} x(u, v) &= u( 1 - \frac{1}{3} u^2 + v^2 )\\ y(u, v) &= v( 1 - \frac{1}{3} v ^2 +u^2 )\\ z(u, v) &= u^2 – v^2\\ \end{aligned} \end{split}\right.\] where u and v are coordinates on a circular domain of radius \(R\), i.e. \(u= r cos\theta, v= r sin\theta, 0 \le \theta \le 2\pi, 0\le r \le R\). For \(R \lt 1\) the surface is stable and has a global area minimizer.

An example of Enneper’s minimal surface

In our example, let us take \(R=0.8\), and \(u=0.8*cos\theta\), \(v=0.8*sin\theta\), where \(0\le \theta \le 2\pi\). The boundary of the Enneper’s minimal surface is defined by the functions: \[\left\{\begin{split} \begin{aligned} x(\theta) &= 0.8*cos\theta( 1 - \frac{1}{3}*0.64*(cos\theta)^2 +0.64*(sin\theta)^2 ) \\ y(\theta) &= 0.8*sin\theta( 1 - \frac{1}{3} 0.64*(sin\theta)^2 +0.64*(cos\theta)^2) \\ z(\theta) &= 0.64*cos2\theta \\ \end{aligned} \end{split}\right.\] The shape of the frame we chose is shown in the following picture.
The bubble on our frame is an example of Enneper’s minimal surface, which shape is shown in the following picture.

Why to use this example

This example of the minimal surface was chosen for the following reasons:
  1. The mathematical form of Enneper’s minimal surface is concise and easy to understand. The Enneper’s minimal surface is special because it is a complete minimal surface in \(R^3\) for which the integral of the Gaussian curvature over the whole surface is \(-4\pi\).
  2. Minimal surfaces have wide applications, especially in architecture. Among the surfaces having the same boundary minimal surface is the surface of the least area. Its weight is therefore less and the amount of material is reduced to a minimum. Minimal surface is used for light roof constructions, form-finding models for tents, nets and air halls. In the following picture, the form of the bamboo shell structure takes on the mathematical Enneper’s minimal surface.
    http://www.iaacblog.com/programs/animated-systems-_camboo-pavilion/
In the above example, we investigate Enneper’s surface. This helps us to better understand the concept of minimal surfaces. The 3D print of boundary curve is shown in the following picture:

Comments

Post a Comment

Popular posts from this blog

Knot 9-31

Knot 9-31 A knot is mathematics is defined as a closed, non-self-intersecting curve that is placed in three dimensions and cannot be the "unknot". The main difference between a knot in the real world and a known in mathematics is that a knot in mathematics does not contain any extra strands. The example following will help visualize this. Today, I have specifically chosen knot 9-31 from the Knot Atlas. This knot is very unique and contains some very interesting properties that we are going to look into. The Crossing Number For my knot today, I chose knot 9-31 from the Knot Atlas. This knot contains 9 crossings! The Unknotting Number The unknotting number is exactly what is sounds like. This is the minimum number of times the knot must be passed through itself to untie it. Luckily, the Knot Atlas is super useful and provides the unknotting number for us, but I still...
Post a Comment
Read more

Do Over: Integration for Over Regions in the Plane

Introduction Earlier in the semester, we visited the topic of integration for over region regions in the plane. This was our first experience with taking integration in the third dimension. To do this, we had to use double integrals to calculate the volume of a region between two surfaces. This topic seems relatively straightforward and anyone who has taken a higher calculus class knows of double integrals. Today, we are going to revisit this topic for a couple of reasons. Why Double Integrals Again? As mentioned before, I am revisiting this topic for a couple of reasons. One of the main reasons is due to how the 3D print turned out. Due to the function, I chose, the final rectangular prism was stand alone. When printing, this resulted in a sloppy prism that lacked structural integrity. The image below is a reference to my original design that includes the prism that had the issue. The main cause of ...
Post a Comment
Read more

Middle of Mass Measured from Moment

The center of mass of an object is the point that lies in the average position of mass in that object. For a normal polygon, that point is in the middle of the shape. However, if the shape is irregular, has concave angles, or has holes, the center of mass can be far from the middle of the object. Two things come into factor when determining the center of mass of an object. Weight and distance. Obviously, if one side of an object has more mass than the other, the center of mass will be on the heavier side. But, distance plays a role, too. If two sides of an object have the same mass, but one is farther from the middle, the center of mass will be on the side that has farther mass. Multiplying both of these factors together gives the moment of an object. Dividing the moment by the overall mass gives the location of the center of mass.   The reason I chose my shape is to mimic how those toy balancing birds work. The toy is a model bird with its wings outstretched. The ...
Post a Comment
Read more