In July it was time to get back to designing and 3D printing some more math models. I had previously seen some beautiful models printed by Dave Pfaff on the Series 1 Pro, using its ability to print in a spiral. I wanted to print a large hyperboloid of one sheet using this printing technique.

So I went back to Cinema4D to design the model. I used the Function tool to make the hyperbola $$\frac{x^2}{4} -\frac{y^2}{9}=1$$, which has asymptotes $$y=\pm \frac{3x}{2}$$. I then used the Lathe tool to revolve the hyperbola around the $$y$$-axis and create a hyperboloid of one sheet. I knew, from the asymptotes, that the surface would have an angle  greater than $$45^\circ$$ angle with the $$xz$$-plane. (Recall that in Cinema 4D, the $$y$$-axis points upwards where the $$z$$-axis usually points in mathematics.) Having a steep angle with the plane is important feature when 3D printing in a spiral. Since there won’t be any supports printed, a steep angle increases the chances of a good print.

At this point I used the surface I had created to make two different kinds of models. For the first model, I simply added a top and bottom to create a closed volume.  For the second model, I thickened the surface using the Extrude tool (2mm), again creating a closed volume. The extruded surface had raised top and bottom rims, which meant the model would not sit flat. So I selected all the point around the top (respectively bottom) rim and brought them down (resp. up) an appropriate distance. Apart from this, this thickened model needed very little doing to it. I then created a third model by adding equations and removing them from the thickened surface. This ended up being quite involved as I used the Bend Tool to make sure the equations sat nicely on the surface.

In order to get a spiral print, I took the first closed model, and adjusted the print settings on the Series 1 Pro printer. The top and bottom layers of the model are not printed; instead the nozzle goes around in a spiral, printing the surface 1 layer thick. Initially, we had trouble with both the fan and print speed settings. This meant the filament was not setting at the bottom of the model, leaving a gaps and giving a ragged appearance. After we slowed the print speed down and made sure the fan turned on after 1mm (rather than after 10mm), the prints turned out beautifully. The small model is about 13cm tall, and the large one about 20cm tall. These models can be found on Thingiverse here.

The next set of models was the hyperbolic paraboloid again, but this time I wanted to surface to look more like a saddle shape. To achieve this I went into Mathematica and use ParametricPlot3D command to plot the surface $$z=x^2-y^2$$. Instead of displaying it inside a box, I had Mathematica just show the part of the surface inside the circle $$x^2+y^2=1$$. To do this, I used the RegionFunction command.

I exported the Mathematica file as a .wrl to Cinema4D. Once there, I extruded the surface (again 2mm thick), then had to spend a significant amount of time repairing the mesh. To do this, I used the Optimize function, and also went around the surface repairing the mesh by hand (removing points, lines and faces, then using the Fill Polygon Hole tool to repair the gaps). I made two copies of the model, one with equations and one without. (I also built the equations in Cinema4D and extruded them. I then removed them from the surface using the Boole tool.) I 3D printed the resulting models on the FormLabs Form 1+ printer in clear and grey resin. The surfaces printed very well, though there were a fair number of supports to remove.

These models can be found on Thingiverse here.

# Quadratic Surfaces – Hyperboloid of One Sheet

My first successful print of a quadratic surface was a hyperboloid of one sheet.

I began this project by creating a solid hyperboloid of one sheete in Mathematica. I didn’t like this object since it was a complete solid and not the surface I was trying to create.

To make the surface I originally attempted to import a Mathematica file of the surface into Cinema 4D and then give it thickness. This quickly became a nightmare to deal with. I had to be extremely careful about how many plot points I used in Mathematica because too many created too many polygons. These polygons also overlapped and so when I tried to extrude them to give the surface thickness the normals were off and it resulted in a very jagged surface. After spending hours trying to work with my Mathematica file I decided to try to create the surface from scratch in Cinema 4D.

In order to do this I used a ‘formula spline’. In Cinema 4D ‘formula splines’ are created using parametric equations. I used the equations $$x(t)=sinh(t), y(t)=cosh(t), z(t)=0$$. I then rotated this spline 360 degrees using the Lathe tool. I then optimized the polygons in order to fully connect the object where the spline’s rotation began and end. Once this was done I was able to extrude the polygons to give the surface thickness using the polygon extrude and being sure to add caps to my extrusion.

One this was done I had my object complete. The next step was to add equations. Using the instructions for how to put equations on solids I created my equations. In order to put them on the solid I used the bend tool in Cinema 4D. Using this tool was very difficult and took a lot of playing with to make it look good. The first thing I had to do was fit the bend box and rotate it in order to bend my equation correctly. When I went to bend the formula to fit my object, I found I needed to align only the first part of the equation on the left hand side with the surface (and not the center of the formula) since the bend tool bent the equation from the left and not at the center.

The first time I printed my object I realized the equations I had put on my object were far too small. I went back to my Cinema 4D file and make them bigger to get my final object! This object can be found on Thingiverse here.