Re: Re: Modeling the Orbitsphere

Luther, I tend to agree with you. I dont see why the summation should
necessarily extend to root 2 pi when it would make sense to rotate thru pi
to get to the other side - so to speak.

But then look at it this way:

In normal cartesian coordinates, if a point a is moved up the y axis by some
distance r, the distance traveled is then r. If another point b is moved
along the axis x by r, then the distance travelled is also r. But the
distance between a and b if both are moved simultaneously along their
respective axes is obviously going to to be sqrt(r^2+r^2) = sqrt(2)*r. In
other words, the length of the line ab along the plane of ab in units of r
is going to be sqrt(2) times r. Please forgive the apparently childlike
explanation, but this is partially me explaining to myself too.

Now, similarly with angles. If a plane (orbitsphere or whatever) is rotated
by angle a in the plane of axis y, while at the same time being rotated by
angle a in the plane of axis x, then the total angle of rotation to get back
to the starting point (albeit with the z axis now flipped 180 degrees) is
also going to be Sqrt(2) times the angle a, but, and this to me is the key,
ALONG THE PATH OF OF SOME SOME ARBITRARY POINT DESCRIBED - in this case on a
great circle, in agular increments as measured at the origin. In other
words, each time you move thru delta a in one direction and at the same time
delta a at 90 degrees to that direction, the effective angle moved is then
sqrt(2) times delta a, and thus the total angle moved must then be to a
maximum of sqrt(2)*pi. But this is in terms of the combined angle as moved
thru both axes.

Now this is where I start to get really confused. Because the incremetal
step chosen for n in going from n =1 to some maximum, then maximum would