Stereographic Projection - Jazmin Ortiz

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Stereographic ProjectionJazmin OrtizOctober 18, 2023George Mason University Math 401: Mathematics Through 3D PrintingStereographic Projection is essentially a map projection that we obtain by projecting the points on a sphere (P) to the points in a plane (P’). It is used to represent three-dimensional objects or spaces in two dimensions. It is a perspective projection, which in this case, we see through a specific point on the sphere, the north pole, onto the projection plane. It is considered a smooth, bijective function, but it is not isometric, which means that it does not preserve distance, nor is it equiareal, which means that it does not preserve the area. Stereographic projection is mainly seen in fields like mathematics, crystallography, cartography, geology, photography, and astronomy. It was originally known as planisphere projection, which was studied by Hipparchus and Ptolemy. During the eras of Hipparchus and Ptolemy, stereographic projection was most importantly used in the representation of celestial charts.In order to create an image, in Mathematica, on the plane to then project this image stereographically onto the sphere, I first established my shape function to be shape[t]:={Cos[5t],Sin[5t]}+{0.4Cos[t],0.4Sin[t]}. I also established the stereo projection function that we will use in order to plot it onto the sphere, which was stereo[{x,y_}]:={((2x))⁄((1+x^2+y^2)),((2y))⁄((1+x^2+y^2)),((-1+x^2+y^2))⁄((1+x^2+y^2))}. I set the scale factor to 20, the thickness to 2, and the radius to 1. Then, I created a Parametric Plot of my shape function, from 0 to 2 Pi, and in the plot style tube: ParametricPlot[{shape[t]},{t,0,2Pi},PlotRange->{{-1.5,1.5},{-1.5,1.5}},PlotStyle->{Tube[0.1]},AspectRatio->1,Axes->False]Then, I scaled my shape function using the set scale from before and the stereo functions, which I then used in order to create my Parametric 3D Plot from -10Pi to 5Pi:shape3D[t__ ]≔scalestereo[starFunction[t,5,1,0.4]]curve=ParametricPlot3D[shape3D[t],{t,-10Pi,5Pi},PlotRange->All,PlotPoints->100,PlotStyle->Tube[radius]]Finally, I exported my final curve that was on the plane and then was projected onto a sphere and created the needed stl file in order to start printing. I printed on a Tina2 Basic 3D Printer and the print time was around 3 hours with normal supports. Overall, I faced a few challenges when I started creating my image that I wanted to project. I wanted to make fancy shapes, like hearts all around, but could not figure it out, so I tried to stick with simple curves. I wish I could have made more curves in order to cover more of the plane, but I ran out of time. I also had to reprint because my first print was too small and I broke it when I tried removing the supports.Citations:https://mathworld.wolfram.com/StereographicProjection.htmlhttps://en.wikipedia.org/wiki/Stereographic_projection