Correction: The azimuthal equidistant projection is actually defined by θ = a and r = 90° - b. I got b mixed up with phi. This video was made for 3blue1brown's summer of math exposition contest. It was selected as one of the top 100, and that has really helped kickstart my channel, so a huge thank you goes out to 3blue1brown. Note: If the title changes, it's because I'm testing out different video titles to see what works the best. In this video, I explain map projections: what they are, how they work, and the math behind them. This video connects 2 of my favorite subjects: math and geography. If you want to do further research, this wikipedia page is very helpful: https://en.wikipedia.org/wiki/List_of_map_projections The images of each projection come from that page, and are made by Daniel R. Strebe. Chapters: 0:00 Intro 0:41 The Problem 3:37 Equirectangular Projection 7:15 Literal Projection 9:49 Mercator Projection 18:51 Sinusoidal Projection 21:24 Lambert Equal Area Projection 23:33 Azimuthal Equidistant Projection 24:30 Compromises 29:30 Perspectives 31:21 Conclusion