Skip to main content

Minimal Surfaces

Minimum surfaces can be described in many equivalent ways. Today, we are going to focus on minimum surfaces by defining it using curvature. A surface is a minimum surface if and only if the mean curvature at every point is zero. This means that every point on the surface is a saddle point with equal and opposite curvature allowing the smallest surface area possible to form.

Curvature helps define a minimal surface by looking at the normal vector. For a surface in R3, there is a tangent plane at each point. At each point in the surface, there is a normal vector perpendicular to the tangent plane. Then, we can intersect any plane that contains the normal vector with the surface to get a curve. Therefore, the mean curvature of a surface is defined by the following equation.

Where theta is an angle from a starting plane that contains the normal vector.

For this week’s project, we will be demonstrating minimum surfaces with a frame and soap bubbles!

How It Works

Minimum surfaces are just the minimal surface for any given boundary condition. Similarly, bubbles form the smallest surface area possible which contains a certain volume. Therefore, we can create a frame which will represent the boundary, then place the frame in bubble solution causing the bubble to form the smallest area possible between the boundaries.  

The Frame

For my example, I chose to make the shape of the frame two linked circles. The first circle lays horizontally and the second circle, which is 3/4th the size of the first circle, lays vertically. However, unlike linked circles we have seen in the past, these circles intersect each other on a point along the edge of the frame rather than floating together.


(Note: The L shape part on the left is not part of the object, it acts as a handle to dip the frame into the bubble solution)

Once placed into the bubble solution, the solution should create the minimum surface of the frame. My guess for the shape of the bubble is that the solution will form with the inner vertical surface and connect with the parts of the outer circle, specifically the parts parallel where the inner circle stops in the middle. I have not had the opportunity to test the frame yet, but on Friday we will be able to see what the minimal surface is!

 Why I Chose This Shape

When first brainstorming what shape to use, I was thinking of ways to manipulate the original example of two same sized circles. Instead, I wanted to use three circles spaced evenly in a row and have the middle circle have a larger radius. However, I realized this would probably give us a very similar result to the original demonstration. So then I started over using the two-circle example, I wanted to know what surface would form if I changed the angle of one of the circles. This led to my frame of a horizontal circle intersected with a vertical surface on a point along the edges. This is also known as link circles, which we have looked at in past examples. Linked circles were a shape I was curious about in the past, so this was a perfect fit for learning more about them.

Comments

Post a Comment

Popular posts from this blog

Do Over: Integration Over a Region in a Plane

Throughout the semester we have covered a variety of topics and how their mathematical orientation applies to real world scenarios. One topic we discussed, and I would like to revisit, is integration over a region in a plane which involves calculating a double integral. Integrating functions of two variables allows us to calculate the volume under the function in a 3D space. You can see a more in depth description and my previous example in my blog post, https://ukyma391.blogspot.com/2021/09/integration-for-over-regions-in-plane_27.html . I want to revisit this topic because in my previous attempt my volume calculations were incorrect, and my print lacked structural stability. I believed this print and calculation was the topic I could most improve on and wanted to give it another chance. What needed Improvement? The function used previously was f(x) = cos(xy) bounded on [-3,3] x [-1,3]. After solving for the estimated and actual volume, it was difficult to represent in a print...
Post a Comment
Read more

Do Over: Ruled Surfaces

Why to choose this project to repeat For the do over project, I would like to choose the ruled surfaces. I don't think my last project was creative, and the 3D printed effect was not very satisfactory. In the previous attempts, all the lines are connected between a straight line and a circle. This connection structure is relatively uncomplicated. The printed model has too many lines, resulting in too dense line arrangement. The gaps between lines are too small, and the final effect is that all the lines are connected into a curved surface, which is far from the effect I expected. What to be improved In this do over project, I would like to improve in two aspects. Firstly, a different ruled surface is chosen. In the previous model, one curve is a unit circle on the \(x-y\) plane, and the ruled surface is a right circular conoid. In this do over project, it is replaced by two border lines. Each borderline is in the shape of an isosceles right triangl...
Post a Comment
Read more