Robert
January 10, 2020, 8:43am
21
How does spherical geometry differ from what you are looking for?
Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry that is not Euclidean. Two practical applications of the principles of spherical geometry are navigation and astronomy.
In plane (Euclidean) geometry, the basic concepts are points and (straight) lines. On a sphere, points are defined in the usual sense. The equivalents of lines are not defined in the usual sense of "straight line" in Euclidean geometry, but in the sense of "the shortest...
See also
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a m
The...
r_speir
January 10, 2020, 2:26pm
22
PdotdQ:
You can easily write it down just by making the usual coordinate transformations. Also, don’t mistake the radial coordinate , with 1-form basis drdr and the curvature parameter rr , which is just a parameter that determines the curvature of a circle. The second is just a parameter and not a “radial coordinate”.
I can do that. However, literature would be nice to refer to in a write-up.
Are you referring to this “r” in the parameterization equations for a half circle?
Also I must keep in mind that I need to imbed this sphere into 4D. It does grow.
PdotdQ
January 10, 2020, 5:56pm
23
I am talking about something like this:
The line element for a one-dimensional circle that is not embedded into anything is:
ds^2 = r^2 \textrm{d}\theta^2 \; ,
The r here is just a parameter that has no meaning of “radius”. There is no radius, and there could not be any radius, as we are in one-dimension. In addition, there is no “radial coordinate”. The only reason people write that parameter down with the suggestive letter r is because when we embed the one-dimensional circle into the very special 2-dimensional space R^2, then that r becomes the radius that we know from elementary school geometry.
