To remind you, our main goal is to demonstrate homotopies between two objects. But before we do that, we have to actually have an STL file of the object at each instant in the homotopy!

We decided a simple example to start off with would be the homotopy between a sphere with a hole in it and a point. You can think of this homotopy as if you were peeling an orange from the top, shrinking the peel as you go along until you are left with just the bottom point of the sphere. Difficult to imagine? To visualize it, I wrote up some Mathematica code. Here’s the nice-looking version:

And here is the copy-pastable version:

sphere[\[Theta]_, \[Phi]_] := {Cos[\[Theta]] Sin[\[Phi]],
Sin[\[Theta]] Sin[\[Phi]], Cos[\[Phi]]};
Animate[ParametricPlot3D[(1 - \[Lambda]) sphere[\[Theta], \[Phi]] +
\[Lambda] {0, 0, -1}, {\[Theta], 0,
2 \[Pi]}, {\[Phi], \[Pi], (1 - \[Lambda]) 2 \[Pi]}, PlotRange -> 1,
Mesh -> True, ColorFunction -> Hue], {\[Lambda], 0, .5},
AnimationDirection -> ForwardBackward, SaveDefinitions -> True,
AnimationRunning -> True]

This results in the following nice animation:

If we replace the “{[Lambda], 0, .5}” with “{[Lambda], {.01, .05, .1, .15, .2, .25, .3, .35, .40, .45, .49}}”, then the sphere will generate at discrete times during which we can export the files as an STL and 3D print our models!

However, we ended up working with another program, Fusion 360, and using that to export some test objects. There will be more on that in the next post!

Thanks for reading,

Jonathan

### Like this:

Like Loading...

## About jonathangerhard

I am a first-year PhD student in mathematics at the University of Michigan and I got my BS in mathematics in 2017 from James Madison University. I started this blog because I love math!

Pingback: My 3D Printing Project – A Summary | Mathematical ADD

Pingback: 3D Printing Graphs, Knots, and…Books? | Mathematical ADD