Crest of the Askr Kingdom

thingiverse

This object was the final project for George Mason University’s Math 401: Mathematics Through 3D Printing. The subject for this project was projecting from two dimensions to three dimensions, but the only assistance given to us was a set of 3 equations to convert from Cartesian to Spherical points on a unit sphere. The result is this Crest of the Kingdom of Askr from the mobile game Fire Emblem: Heroes. By shining a light through this object, a shadow of the original image can be seen without distortion. This object was printed on an Ultimaker S5 using generic PLA. Default print settings were used, and the print took approximately 11 hours. It took three attempts to print successfully. The initial print had the tubes too thin, making the object too fragile. Fraying also occurred, and a large portion of my print was never made. The second attempt failed due to a printer part breaking. I’m unsure what caused these issues. To make this object, I used Mathematica and created two functions. One was to convert two two-dimensional points into a simple linear function of t. The other was to take t values and a corresponding t function and generate a single three-dimensional point on the unit sphere. The inputs of these functions would come from a matrix of points from the crest's corners. To cut on development time, I used For Loops to take these points and input it into the function to create functions of t, then took the t function as an input to the 2D to 3D conversion function. There are some technical details that need to be discussed. If the image you wish to project to the sphere is in several pieces, you should change the thickness of the tubes if the pieces are close enough so that they intersect. Otherwise, you may need connections. The points used may require scaling. I had values between +/- 220. If I did not scale these points down, all the tubes would exist in an extremely tight area around the north pole of the sphere. I found the best scaling multiplier to be 1/300, but results may vary. Try to keep maximum value between 0.7 and 1 after scaling. Finally, be wary of vertical lines generated using endpoints of the same x value. You would obtain a divide by zero error using my code.