Let me begin here, and incrementally improve the representation. I don't want 3D coordinates to "happen" to me. I may pick them out of a list, but I think there is a better way. My coordinates will describe an intrinsic property of physical objects (orientation), and should be consistent from object to object. So, I have introduced coordinates in this order 1. orient an object in a natural way 2. describe that orientation with math. I call these [i]object[/i] or [i]local[/i] coordinates. Like this:

[list] [*]Declare the FRONT of the object. →The principal (facing) vector is X. [*]If the object moves forward (in X), the SIDE is the direction perpendicular to X, and also [i]lying in the plane of motion[/i]. →The second vector in the not-up plane is Y. [*]→UP is Z. [/list] By convention, we choose X Y Z by positive rotation. [b]Example 1:[/b] I stand upright and face forward. The direction I face is X. My LEFT ARM is Y, and Z is up. Now I walk around. My plane of motion on the surface of the earth is the XY plane, and Z is up. [b]Examples 2-n:[/b] Car. Airplane. Boat. Flying guitar. bicycle. sled. squirrel. ambulating cabinet. pram. poodle. moon. This is consistent with classical differential geometry, which is also to say all of classical physics (mechanics). ([i]Why?[/i]) I can always conform another representation to this system with a (series of elementary) matrix multiplication(s). So that is what I will do. ________ [b]Unit Sphere[/b] [list] [*][b] →Setup[/b] [*] Trihedron: [url]http://www.geogebratube.org/material/show/id/107064[/url] [*] Base Object:[url]http://www.geogebratube.org/material/show/id/105255[/url] [*] Spherical Coordinates {link} [*] Meridian, (Horizon Points) [*] Latitude, (Horizon Points) [/list]

The axes are continuous under transformation: Drag points A, B. The vector u1 will not flip from side to side, but changes smoothly. When is this condition violated? The axes moves briskly when the figure approaches a circle. And in fact this transformation is undefined for |a| = |b|. For example, try typing SetValue[A, O + prp (B-O)]. (I use a matrix [math]\;\;\; {\small {\rm prp} =\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}}\;[/math] to rotate vectors 90° counterclockwise. ) Slumberland will address this shortly. To preserve continuity, there is a correct answer: preserve the last good directions of u1, u2, whenever the transformation is undefined. If continuity does not matter, the choice may be arbitrary. Is that all? [i]No.[/i] I say, we can still bring about an instantaneous rotation of the axes by 90°. How? In context, this may or may not be descriptive of the problem. Time to define the limit cases.