What is a minimal surface?
A minimal surface is simply a surface with zero mean curvature. Mean curvature H of a surface S is an extrinsic measure of curvature that locally describes the curvature of an embedded surface in some ambient space. Zero mean curvature means on a minimal surface the curvature along the principal curvature planes are equal and opposite at every point.
More definitions and ways of looking at minimal surfaces from class:
From class (11-3-21) a surface is minimal if and only if every point of a surface "M" has a neighborhood bounded by a simple closed curve, which has the least area among all surfaces having the same boundary. From class (11-1-21) a surface is a minimal surface if and only if the mean curvature at every point is zero. Another way of looking at it from class (11-1-21) is that the mean curvature of a surface \(M\) at \(p\) is \(\frac{1}{2\pi}\int_{0}^{2\pi}K(\theta)\,dx\) where \(\theta\) is an angle for a standard plane that contains the normal vector, and \(K\) is the absolute value of the derivative of the unit tangent to the parameterization of a curve, over the absolute value of the paramaterization of a curve \[K=\frac{|T'(t)|}{|r'(t)|}=curvature.\]
For this project we will be choosing a frame representing the mean curvature (of all the normal curves) of a minimal surface and dipping it in bubble solution. The bubble solution will attach to the frame to form a bubble that represents the minimal surface over the frame selected.
What frame will I be using and what will the bubble make on the frame when it is dipped in bubble solution?
The shape of the frame I chose is a Schwarz P-surface. I created everything 100% by myself in ONSHAPE and OPENSCAD using pictures online as reference to show the frame that will be dipped in bubble surface,
and the actual surface as well as what the bubble should create when it attaches to the frame:
I predict the bubble solution will attach to the frame to construct a shape very similar to the pictures of the Schwarz P-Surface in ONSHAPE I created.
Why I chose the design I did:
It was the only minimal surface I could find that I could attempt to make myself without feeling like I did the minimal effort but also without having to watch and read more than four hours of OPENSCAD curriculum (only 3 and a half this time). I also think the frame and full surface will be good decoration for my desk.To actually dip it in solution and maintain the shapes integrity I will add a stick to the wire frame or drop it down using a string.
A minimal surface is simply a surface with zero mean curvature. Mean curvature H of a surface S is an extrinsic measure of curvature that locally describes the curvature of an embedded surface in some ambient space. Zero mean curvature means on a minimal surface the curvature along the principal curvature planes are equal and opposite at every point.
More definitions and ways of looking at minimal surfaces from class:
From class (11-3-21) a surface is minimal if and only if every point of a surface "M" has a neighborhood bounded by a simple closed curve, which has the least area among all surfaces having the same boundary. From class (11-1-21) a surface is a minimal surface if and only if the mean curvature at every point is zero. Another way of looking at it from class (11-1-21) is that the mean curvature of a surface \(M\) at \(p\) is \(\frac{1}{2\pi}\int_{0}^{2\pi}K(\theta)\,dx\) where \(\theta\) is an angle for a standard plane that contains the normal vector, and \(K\) is the absolute value of the derivative of the unit tangent to the parameterization of a curve, over the absolute value of the paramaterization of a curve \[K=\frac{|T'(t)|}{|r'(t)|}=curvature.\]
For this project we will be choosing a frame representing the mean curvature (of all the normal curves) of a minimal surface and dipping it in bubble solution. The bubble solution will attach to the frame to form a bubble that represents the minimal surface over the frame selected.
What frame will I be using and what will the bubble make on the frame when it is dipped in bubble solution?
The shape of the frame I chose is a Schwarz P-surface. I created everything 100% by myself in ONSHAPE and OPENSCAD using pictures online as reference to show the frame that will be dipped in bubble surface,
I predict the bubble solution will attach to the frame to construct a shape very similar to the pictures of the Schwarz P-Surface in ONSHAPE I created.
Why I chose the design I did:
It was the only minimal surface I could find that I could attempt to make myself without feeling like I did the minimal effort but also without having to watch and read more than four hours of OPENSCAD curriculum (only 3 and a half this time). I also think the frame and full surface will be good decoration for my desk.To actually dip it in solution and maintain the shapes integrity I will add a stick to the wire frame or drop it down using a string.
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