h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Very well done....but applies to plane trigonometry. The Universe has more general relations. Look at a triangle on the surface of the Earth. View a triangle formed by a section of the equator and two lines from the North pole- it has 2 right angles and a third included angle. Other geometries (torus etc.) are more complex yet. This was great but should say Plane.

By definition, a triangle is a polygon with three points. Whether expressed in two dimensions or three, it will always lie on a flat plane. The triangle you refer to is not a polygon. It is the intersection of a spherical shell and three planes.

Cheers,
Ricky P.

You're both right. The trig here is in Euclidean geometry. The trig the other guy was referring to is in spherical (Riemannian geometry). They're both valid geometries, and trig in each is valid too. Perhaps someone might do an entry in spherical trig: there's lots of info to do it...maybe I might do it, lol

well, we know how to get from cartesian coords to spherical coords, if that helps

assuming Y is vertical:
theta = arctangent(X / Z)
phi = arctangent(Y / (the square root of (X^2 + Z^2)))
rho = the square root of (X^2 + Y^2 + Z^2)

and spherical to cartesian is:
X = rho * cosine(phi) * cosine(theta)
Y = rho * sine(phi)
Z = rho * cosine(phi) * sine(theta)

i don't know why i told you guys this, i just thought you might want to know
>8^B