I'm still working on re-deriving Takeno's results for the line element given in his paper. I'm slogging through the work having to remember where to divide by factors of the radius squared to get back to differential angle elements and where to multiply by factors of the speed of light squared to transform from time elements to distance elements. The following is a rather abstruse set of instructions on how I'm doing all of this, left mostly for me should I ever forget, or need to do it again. I'm not sure if it's even all correct yet, so a warning... if you're not into spending your Sunday morning reading distracted notes on differential calculus stop now :) Despite all the differential calculus shown in the board work in the picture, the line element basically describes how the Pythagorean theorem for finding the length of a line works in a given coordinate system. The familiar example from middle/high school is that that length squared equals the...