When setting up the sphere, I accidentally stumbled into and solved two problems from Dorrie's [i]100 Great Problems of Elementary Mathematics:[/i][br][math]\;\;[/math]Ellipse from conjugate radii.[br][math]\;\;[/math]Ellipse inscribed in a parallelogram.

The parametric equation gives the direction of increasing [i]t[/i] from [b]a[/b] → [b]b[/b] (Draw the segment AB, and follow it from A to B: [i]that way[/i]). [br][br]Now consider point C, with radius [b]c[/b]. Each of the possible pairs (±[b]c[/b], ±[b]d[/b]) is a pair of conjugate radii. I take the principal conjugate following the direction of increasing [i]t.[/i] This is an assumption which describes the problem I wish to solve. Direction remains a free choice, to be resolved in the context of the problem at hand. Like this:[br][br]In the Sphere problem, I adopted a right-handed, rectangular coordinate system. To begin, I manipulate the trihedron by allowing each axis to slide along one of the meridians through the coordinate axes. Naming the axes in alphabetical order, I can distinguish the correct conjugate:[br][br]e.g. on the ellipse determined by x, the vectors y and z are conjugate. Choose y at pleasure on the ellipse, and z is the direction of positive rotation. When the user manipulates the figure, I test the signs of the cross products among the axes to determine whether an axis has flipped sides front/back. z can always be uniquely assigned.