Conic Sections- Full Cone

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The 4 inch model is meant to display the conic sections in mathematics on a 3D scale. The conic sections come from 2 cones, stacked on top of each other (tip to tip), being cut by planes in a 3D space, and the resulting cuts create 2D graphs. With the help of this model, the origin of these conic shapes can be seen more clearly and manipulated by mathematicians and students. Those who use the model can see the different types of shapes that are created as well as how they relate to each other in the cones they were created from. The use of conic sections in the classroom is one of the Maine mathematics standards as dictated by the Common Core Mathematics Standards. Standards CCSS.MATH.CONTENT.HSG.GPE.A.1, 2, and 3; all discuss the idea of conic sections graphically as well as through equations. Yet the conceptual ideas of the conic sections and where they come from is not as easy to understand. Even if students are given a set of equations and how to work with them, it doesn’t help the students see how it relates to cones. By having this model in the classroom, students can see where the planes cut the cone to make the different 2D shapes from a 3D model. (A. Reynolds - MAT 363, December 2014) Print Settings Printer Brand: SeeMeCNC Printer: Rostock MAX Rafts: No Supports: No Notes: The model has a tendency to pop off the bed when printing. So use the hairspray or tape and keep the bed hot! Taping it down after the base circle prints would be smart. How I Designed This The design process of this model began by using Rhinoceros, or Rhino. Two cones were constructed so that the tops, or the tipped points, were intersected at the point of the top. Then, planes cut through the model at certain angles to create the 2D graphical representations. The parabola was cut so that it hit only 1 cone from a slanted vertical angle. The hyperbola was cut so that the plane hit both cones at a slanted vertical angle. The ellipse was made by cutting one cone at a slanted horizontal angle. The circle and the point were created by the construction of the two cones together, where you can slice one cone at a straight, horizontal line across to get the circle and the point is the intersection of the two cones. The last conic section, the line, was the first problem in the design process. The lines were made by creating a skeleton of the 2 cones and removing the Rhino designed cones. Once the ellipse, the hyperbola, and the parabola were created, the intersection of the plane and the cones were made in Rhino. The planes were then removed, leaving only the curves created by the slicing planes. Then, the cones themselves were deleted from the model, leaving only “floating” curves that were created. We then constructed 4 lines from the two circles left from the cones, creating a skeleton of a cone. The model then had all of the components required of the conic sections and was theoretically printable. The model was then piped to create thickness in the model so that it could be 3D printed. (AR, Dec. 2014) Custom Section Mathematical Questions/Challenges When the model was failing to print, we realized that the printer was attempting to print on a point of singularity. The middle point, technically, is a single point on the entire plane. Even though the model added width with the piping, there still was such a small surface area that the printer was struggling to build up from it. The problem that was discovered was that in order for the model to be printable, there were two issues to consider. The first issue was how big to get the intersection so that it was strong enough to support the top half of the cones. When we increased the pipe width, the surface area increased, which should solve the problem of the thinness in the model. Yet when we increased the piping in order to increase the surface area, the weight from the top half of the model also increased. This created the problem that the weight was still too large, even for the increased surface area. As we attempted to increase the surface area, the weight increased. The pressure from the top half of the cone was too great for the smaller surface area to support. We then considered how to increase the surface area on the intersection between the two cones without increasing the weight as well. The angle between the two cones also affected the surface area as well. As the angle increased, the surface area on the point also increased because the cross sections of the intersection became wider. This should have, in theory, not increased the weight too much in order to balance out the weight and pressure problem. Yet when it was attempting to print, the model was still considered to be too thin in multiple areas. We realized that Rhino will pipe items by doing a multiple radius pipe. This method tapers the pipe so that it is smaller on one end. The idea to do a multiple radius piping helped also to decrease the weight on the top of the model. Hopefully, all of these new variables will help make the next print a little bit larger. The size of the last print is a good individual size, but if we wanted a larger model to print with there will have to be more mathematics done. (AR, Dec. 2014)