Printing a Gorgeous Goblet with the Help of Maple
Part 1: Generating the 3D Model in Maple
Along with using Tinkercad to create stunning egg lamps, I used Maple, a powerful math software, to design a unique 3D goblet. Unlike Tinkercad, Maple does not have built in shapes that one can modify to obtain their desired object. Rather, Maple allows me to graph functions so that I can see the 2D version of the goblet before I revolve the functions about the z-axis in order to generate the 3D model. Thus, using Maple was more challenging than building objects in Tinkercad, but Maple has an abundance of commands that enables one to generate a 3D model of a complex object. The goal of this goblet project was to not only successfully design and 3D print the goblet, but to also learn how to use the various Maple commands and to calculate the volume of the goblet using calculus methods.Before opening Maple, I had to ponder about the shape of the goblet as I wanted the stem to consist of several different functions, ellipses, or circles so that I can challenge myself. After looking at some goblets for inspiration, I finally had a design in mind and I drew it out. The bowl of my goblet has a curved lip and comprises of a total of four curves; meanwhile, the stem is made up of four curves and a line, and the base is made up of a single curve and a line. My sketch of the goblet is displayed below.
The next step in my goblet project was to determine the functions and the equations of the ellipse and circles that give me the outline of my goblet. To begin this process, I shifted a parabola upward to start constructing the bowl. Then, I shifted an upside down parabola up and to the right to create the curved lip. In Maple, I named both of these functions and used the solve command to determine where the functions meet. I adjusted the interval of the two functions to connect them, which gave me the inner curve of the bowl. To get the outer curve, I started with the same two parabolas, but I shifted both of them down and to the right as shown below. Once again, I used the solve command to determine where they meet and adjusted their intervals accordingly. I also used the plot and display commands often to graph the functions in order to ensure that the curves stay true to my goblet drawing.
I continued to use the same approach to construct the stem of the goblet. I named the functions, used the solve command, and utilized the plot, implicit plot, and display commands constantly to keep track of my progress. To begin the stem, I used a square root function to connect the bowl and the circle, which actually appears to be a semicircle after altering the interval. Next, I simply linked the semicircle and the ellipse with a vertical line. The base of the goblet, which is a semicircle, is connected to the ellipse by another square root function and a horizontal line is used to complete the bottom of the base. My goblet was not done yet. The last thing I did was finish the bowl by joining the inner and the outer curve with a vertical line.
Once I finished the 2D version of the goblet, I had to generate the 3D model in Maple, which was difficult at first. I had a 2D graph and somehow I needed to revolve the curves that shaped the goblet around the z-axis. My first attempt at this was a failure because it revolved the curves around the x-axis and it did not result in the shape of a goblet. For my second attempt, I used the surface of revolution command, which seemed to work as it gave me the structure of the goblet, but in the end, I chose to use the plot3d command. Because I used the plot3d command, I had to parametrize the curves as well as represent them in three components. For every curve,the first component was x times cos(t) with t being the parameter. The second component was x times sin(t) and the last component was the curve itself. To make it easier to revolve the ellipse and the circles, I solved the equations for y, resulting in square root functions that allowed me to use the plot3d command. For vertical lines, it was a little different. Instead of multiplying the first two components by x, I multiplied them by the vertical line itself and the last component was z, where I determined the interval for z.
Finally, after completing all the math, it was time to 3D print my goblet. I was able to export the STL file from Maple; however, the file did not open in Cura. Thus, I had to use MeshLab, a software for editing 3D triangular meshes, to repair the file. According to MeshLab’s website, the geometric elements of a 3D object can be considered “wrong” by other software, which prevents the file from opening. In my case, when I loaded the goblet into MeshLab, I received the warning, “mesh contains 13680 vertices with NAN coords and 0 degenerated faces.” MeshLab not only corrects that issue, but it also unifies duplicated vertices. After correcting the problems, I exported the file from MeshLab and I was able to open it in both Cura and Flashprint. I then printed prototypes of the goblet, made slight adjustments, and printed the final version.
To accomplish the goal of this goblet project, my last task was to calculate the volume of the bowl. After two years of computing the volume of a curve revolved an axis, I felt confident in my calculus skills to compute it first by hand. I recognized that the disk method can be used to compute the volume, but I had to use two integrals because the bowl consists of two curves. I began by solving the two equations of the inner bowl for x in terms of y. Next, I found the limits of integration for both curves. I then set up the two integrals and solved them, getting a final answer of 7Pi/3. Maple verified my result.
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| By using the Volume of Revolution command, I was able to compute the volume of my goblet in Maple. |
Part 2: Printing a Goblet with a Unique Characteristic
The Lycurgus cup is no ordinary cup. It is a glass cup from the 4th century that is currently displayed in the British Museum and has the special effect of changing color from red to green depending on the illumination angle. According to this journal article, it is possible to achieve a similar color changing effect with 3D printing plastic. Throughout the paper, they explain the chemistry behind the color changing effect and describe the process of making the dichroic solution that will make the 3D printing plastic reflect a brown color and transmit a purple color. It is a fascinating effect; therefore, I was interested in making my goblet have the same effect.
The article states that to begin the synthesis of the solution, 0.5 mL of a 34 mM citrate solution in distilled water is added to a 100 mL boiling solution of 0.25 mM HAuCl4 in distilled water. The solution was being stirred vigorously and kept boiling for five minutes, which is when the solution should appear to have a brown reflection. They then left the solution to cool to 50 °C at which they started the AuNP–PVA fabrication process by adding small pieces of PVA 3D printable filament to the dichroic solution to obtain a final concentration of 0.1% (w/w) of AuHCl4 in PVA. All of the PVA filament was dissolved before they transferred the solution to plastic petri dishes and left it in a ventilated oven until all the water had evaporated, resulting in the hard AuNP–PVA plastic. In order to prepare the plastic for printing, they shredded it and extruded it. The final result is stunning.
Perhaps in the near future I will be able to achieve this with my goblet.
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