The author reflects on how projective space and homogeneous coordinates, once abstract and unmotivated during his student years, became natural tools that eliminate exceptional cases and hand-wavy references to objects "at infinity." He advocates for a bottom-up approach in mathematics education, where abstractions are motivated by concrete problems. The projective plane, he explains, can be understood as a sphere with antipodal points identified or as a square with opposite edges glued in an impossible way. However, its true value lies in making geometry more elegant by removing special cases—like vertical lines in standard geometry—through the use of homogeneous coordinates.
Programmers, especially in graphics and cryptography, often rediscover these concepts independently when handling edge cases like the "point at infinity" in elliptic curve algorithms. The post ends with a humorous nod to Dr. Strangelove, playing on the phrase "How I Learned to Stop Worrying and Love...".
Source: Hacker News · Summarized by HeadlinesBriefing