a.k.a., Space 11: How to do Interstellar Navigation
Various antecedents you may want to have peered at first:
Today's post is about Hyperbolic Geometry, wherein you learn what those "Warning, Evil, Don't Look" columns are about.
It's now safe to look; well okay, no it isn't, but too late! AHAHAHAHAHAHAHA.
Hyperbolic geometry is basically Geometry On Drugs and we know that's never going lead anywhere good.
To be fair, Spherical Geometry is arguably also on drugs, but at least it's easier to explain in that, having had lots of experience with basketballs and whatnot, you already know what a sphere is. Having a concrete place for the "points" to live, I can then tell you
- what "lines" are (great circles, or planes slicing the sphere through the origin / center of the sphere),
- how to measure "distance" along a "line" segment (measure angle between endpoints from the center of the sphere),
- how to measure "angles" between "lines" (the planes will intersect; there's an angle there; done), and
- what "circles" are (they're um, circles, … or, if you like, planes that don't necessarily go through the origin, or cones coming out of the origin; whatever works for you),
and then you're basically good to go, ready to do all of the geometry/trigonometry you could ever want, once you've heeded my warnings that Certain Things Will Be Different (no such thing as "parallel", triangles add up to 180 plus area instead of just 180, do not feed them after midnight, etc…).
Unfortunately, the place where we're Doing Geometry today is this inside-out Hyperboloid Sheet Thing with a fucked up metric, … and if you've actually seen one of those in real life, I will be very surprised, especially since it's not something that can exist in ordinary 3D space. Oddly enough, it will end up relating to something you do have day-to-day experience with, namely (cue reverb and James Earl Jones voice)… Your Future,… but I'm not sure how much help that's going be in visualizing it.
( bring on the drugs...Collapse )
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