The next quadratic surfaces I printed were an elliptic paraboloid and a regular paraboloid.

For the elliptic paraboloid I imported the surface from Mathematica.

I then optimized the polygons, extruded them by 0.20 cm to give the surface thickness. After that I used the boole tool to make the edge flat and added an equation through the surface.

I created the regular paraboloid from scratch in Cinema 4D using the same process as the cone.  I used the formula spline $$x(t)=t, y(t)=t^2, z(t)=0$$ and then used the lathe tool with 60 rotation segments to rotate it 360 degrees. I optimized the polygons and extruded them to give the surface thickness. I also made sure to “boole” the edge to make it flat and added an equation.

I printed both paraboloids on the same build bed with the MakerBot 2X printer. They can be found on Thingiverse here and here.

The next quadratic surface I made was a cone. This is a particularly challenging object since the cone’s two halves meet at a single point in the center. In order to deal with this printing challenge, I originally attempted to create the object and add a certain amount of thickness to the middle so it would print correctly.

The first cone I made using a method similar to my other quadratic surfaces by using a spline with the formula $$x(t)=t, y(t)=1.5*t, z(t)=0$$ (it was important to use the * otherwise Cinema 4D did not multiply) from $$t=-2$$ to $$t=2$$ and rotating it with the lathe tool 360 degrees with 72 subdivisions. Then I optimized the object by 0.02cm. With the first cone I made I reversed the normals so they were facing outwards from the cone surface. When I extruded the surface by 0.25cm this automatically gave center of the object more thickness. Unfortunately I realized that while it gave the center thickness, it also offset the lines of the cone so they didn’t match up, which is not what we wanted. In order to fix this problem I did the same thing but did not reverse the normals (so they were facing the inside) and extruded the surface to give the cone thickness.

With this second cone I removed the center using the Boole Tool, and a cylinder of radius$$=0.25$$cm and height$$=2\cdot 0.25\cdot 1.5=0.75$$cm. We did this in order to preserve the lines of the cone and give it the support the object needed. After removing this cylinder from the center I then added one of the same dimensions in its place. We knew this print had a high potential to fail but decided to print it as a test to see what we might need to change in our design. We left it overnight and it was a huge mess the next morning. Clearly this design didn’t work and we needed to rethink it.

I decided that creating both sides of the cone in one object was just not going to work. Instead I decided to create both halves separately and then connect them using a cylinder made from a sheet of clear overhead plastic (an idea that came from Henry Segerman’s Calculus Surfaces). In order to do this I used the same formula spline except this time from only $$t=-2$$ to $$t=0$$ and copied and rotated it to create the second half. On one half I put the equation for the cone $$\frac{z^2}{4}=x^2+y^2$$ and immediately ran into trouble. I had used too many subdivisions (72) and the object was not accepting the Boole with the equation. After creating many different cones with different subdivisions I found that 60 worked. Once this problem was solved I added the equations to one of the halves of my cone and printed it. When I added the equation I put it all the way through the surface and not just imprinted on it since the object had very little thickness to it.

The print was successful but showed a few flaws in my design. One was the equations and how they were a little too big and how the fraction parts of it needed to be downsized to match the other parts of the equation. This was easy to fix in Adobe Illustrator by changing the font size in the fractions from 36 to 24. I also made sure the numbers in my equation were not italicized and just the variables were. The other design flaw I found was that the bottom of the cone was not completely flat and was angled from when I extruded the surface to give it thickness. To fix this I used a cylinder and “booled” the bottom of both cones to make them flat. I then reprinted my object and had great results. These objects can be found on Thingiverse here.

My next successful quadratic surface was an ellipsoid. This surface I simply imported from Mathematica and then added equations to it, using the same process as described in my post on the hyperboloid of one sheet.

The first ellipsoid I made in was $$\frac{x^2}{16}+\frac{y^2}{25}+\frac{z^2}{4}=1$$. When I put my .STL file into the MakerBot Desktop program I noticed that the program created supported that went up to the equation because it was on a curved surface and cut into the object. I decided to print the object with these supports and the equation on the side of ellipse and no raft.

The first print I did of the ellipsoid I cancelled the print early on so that I could inspect the sides. The surface looked a bit melty, where the filament had shrunk. We decided this was fine and to try to print it again. The second time I printed this object at about half way through the print it fell over and we found it covered in a stringy mess of filament.

Taking this failure and the melty-ness of the ellipsoid, we decided to create a new ellipsoid that was a little rounder. I followed the same process as for the first ellipsoid using the equation $$\frac{x^2}{4}+\frac{y^2}{6}+\frac{z^2}{3}=1$$

This time when I printed it I decided to put the equations on the top of the ellipse and use a raft. It printed perfectly.

After my success with my second ellipsoid I decided to try to print my first ellipsoid again and this time with a raft. The object never fell over and printed perfectly. These ellipsoids can be found on Thingiverse here and 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.

# Tetrahedra

I made two tetrahedra, both of which demonstrate the strategy behind setting up triple integrals. One tetrahedron came from our calculus textbook, another came from Professor Keller. The former is defined by the equations $$x + 2y + z – 2, x = 2y, z = 0$$, and $$x = 0$$ (Example 5 of 12.5 in Stewart’s Essential Calculus), and the latter by $$y=-6, z=0, z=x+4$$, and $$2x+y+z=4$$. When I first tried to print the Tetrahedron from the textbook, the equation on the bottom face did not appear. Later, Dave Pfaff told us that it was because the object was inside out in Cinema 4D. I fixed the issue by reversing the normals on the object.

I then tried to print the object in addition to a set of round coordinate axes, but one of the axes was either knocked out of position by the left extruder (the unused one) or it curled up because it cooled (or maybe both). Later, I tried to print the tetrahedron with Prof Keller’s tetrahedron, but they curled up at the ends because the cooling of the plastic is exacerbated when the length increases. Next, I used a raft to print the two shapes, but the equations looked awful. So then I used some ABS juice on the half of the build-plate that contained the sharp vertex of Professor Keller’s tetrahedron and the print came out well, except for a messy-looking number “4” and some stringy filament on one of the faces (see blue shape).

Finally, I printed Professor Denne’s tetrahedron along with another set of coordinate axes (see black objects). Since there was some juice left over on the build-plate, the only end that curled up was at the origin of the coordinate axes, and the rest of the objects looked great.  These tetrahedra can be found on Thingiverse here and here.

# Coordinate Axes

I made a set of Coordinate Axes to go along with my two wedge solids from Multivariable Calculus.  The axes will also work well for my Double Riemann Approximations, my tetrahedra (coming soon), and other objects that I will print in the future.  The first issue with the axes was figuring out how to connect each part  – I tried using six rectangular prisms and Magic-Merge to join them together, but that did not work.  Later, with the help of Dave Pfaff, I extruded the coordinate axes out of a cube at the origin.  I then was able to imprint the labels for each of the axes much more easily. The axes can be found on Thingiverse here.

# Approximating a Volume by Rectangular Prisms

Next, I modeled the volume under the curve $$z = 16 – x^2 – 2y^2$$, using a double Riemann sum approximation. Here, in the $$xy$$-plane, each approximation contains $$m$$ squares along the $$x$$-axis and $$n$$ squares along the $$y$$-axisIn this case, I used $$m = n = 4$$ and $$m = n = 8$$.  In order to print the solids, I modeled them on Cinema 4D.  They were extremely thin (the height was much greater than the width and length) (see Cinema 4D pictures), so I divided the height of each approximation by 4.  I then printed them simultaneously, which took 8 hours and 45 minutes.

The models turned out well, except in the $$m = n = 8$$ case the printer did not print one of the rectangular prisms because I accidentally placed two rectangular prisms in the same place.   I then printed a c
orrected version as well as a model of the original solid (height divided by 4 once again).  On the original solid, I imprinted the equation onto the bottom, which does not look great because it should be imprinted deeper into the solid.  We considered using Magic Merge because the first print of the $$m = n = 8$$ approximation printed some vertical lines, but when we fixed the model in Cinema 4D, the MakerBot printed smooth faces on each side.

The three objects can be found on Thingiverse  here, here, and here.

# Three Intersecting Cylinders

My latest project was to create three intersecting cylinders and the area of their intersection. I created these objects similarly to the two intersecting cylinders and the Steinmetz solid. For the volume common to all three cylinders I used a Mathematica code and imported the object into Cinema 4D.

For the three cylinders I created them from scratch in Cinema 4D using the ‘Tube’ object. Creating the three cylinders was very straightforward. After my fail from the two cylinder object I also wanted to create the three intersecting cylinders cut in half, so the inside was visible. This proved to be a little more complicated. I used a ‘Cube’ and ‘Boole’ with each tube in order to get half cylinders. Then I was able to merge them together using a very similar process to putting equations on solids in Cinema 4D.

I have yet to print the half cylinder object, but I did print the inside intersection. The first time I printed it I ran into a problem that also occurred with the Steinmetz object. Both solids seemed to have ‘melty’ sides. The angle of the object was such that no supports were needed and yet gravity seemed to affect the plastic and cause some issues. We think that the plastic filament cooled and shrunk during printing causing the deformations.

In order to fix this I decided to reprint the inside intersection with a different orientation. this required more supports but we had much less melting on the sides. I plan to try reprinting the Steinmetz object with another orientation to see if I can eliminate the deformed sides.

The original orientation is on the left and the new orientation is on the right. The objects can be found on Thingiverse here and here.

# Solid of Revolution – Comparing Methods #2

To get my solid, I rotated the area between the curves $$y=x$$ and $$y=x^2$$ around the line $$y=1.25$$.  I then approximated it by cylindrical shells.  The width of each of my shells was 1/16 and the height was $$x^2-x$$, where the $$x$$-value was chosen from the right endpoint of each interval.

One of the issues with the print comes from the fact aht  the top shell does not connect smoothly to the shell below it.  Initially I thought that this resulted from an error in the printer.  When I looked at the Cinema 4D file, however, I discovered that the top shell does not make contact with the shell below it because its height is less than 1/16 (the bowl shape is very flat there).  Additionally, the shell second from the top does not make contact with the shell below it, but the height difference is not noticable.  Another issue with the print was that the leftover supports look bad.  The inside of the object was filled with them, and even after clearing out most of the supports, it still looks terrible.  We are thinking about printing the object on a different printer, such as the Form Labs printer (which uses stereolithography) or the ProJet 260 (which uses powder), in order to get the inside part of the object to look clean.