A more flexible set of axes. I will adopt this system for now.

Each axis slides along one meridian. Arrows indicate the direction of positive rotation.[br][br]I adopt the following notation:[br]Capital letters X, Y, Z denote the 3-dimensional vectors:[br]X Y Z are mutually perpendicular unit vectors forming a right-handed coordinate system.[br][br]Barred letters are projections:[br] [math] \;\;\;{\rm \bar x}, {\rm \bar y}, {\rm \bar z}[/math][br]are the projections of X Y Z on the view plane (the coordinates of [math]{\bf \bar x}[/math] are the ordinary GGB coordinates.)[br][br]I have begun with a coordinate-free representation. I submit without proof that the axes are bound by the following constraints:[br] [math] \;\;\;1.\;|{\rm \bar x}|² + |{\rm \bar y}|² + |{\rm \bar z}|² = 2 [/math][br] [math] \;\;\; 2.\; |{\rm \bar x}×{\rm \bar y}|= \sqrt{1-|{\rm \bar z}|²}[/math][br] [math] \;\;\;\;\;\; |{\rm \bar y}×{\rm \bar z}|= \sqrt{1-|{\rm \bar x}|²}[/math][br] [math] \;\;\;\;\;\; |{\rm \bar z}×{\rm \bar x}|= \sqrt{1-|{\rm \bar y}|²}[/math][br][br]The two constraints can also be stated this way: Let [math]{\rm \bar x}[/math] determine an ellipse with center O, major axis the unit perpendicular to [math]{\rm \bar x}[/math], and minor axis of [math]\sqrt{1-|{\rm \bar x}|²}[/math]. Then [math]({\rm \bar y},{\rm \bar z})[/math] are conjugate radii on the ellipse. [br]And likewise [math] ({\rm \bar z},{\rm \bar x}), ({\rm \bar x},{\rm \bar y})[/math] are conjugate on the ellipses determined by [math]{\rm \bar y},{\rm \bar z}[/math], respectively.[br][br][b]Update:[/b] So. I made that up, to respect freedom of motion. Today I find this is Gauss' Fundamental Theorem of Normal Axonometry. [br][br]This was problem #74 in Heinrich Dorrie's [i]100 Great Problems of Elementary Mathematics.[/i].[br][br]___________________[br][b]Unit Sphere[/b][br] [list][br][*]Setup: [url]http://www.geogebratube.org/material/show/id/101282[/url][br][*][b]→ Trihedron:[/b][br][*]Base Object: [url]http://www.geogebratube.org/material/show/id/105255[/url][br][*] Spherical Coordinates {link}[br][*] Meridian, (Horizon Points)[br][*] Latitude, (Horizon Points)