In response to a question on Stack Exchange:[br] [url=http://math.stackexchange.com/questions/264446/the-fastest-way-to-obtain-orientation-from-this-ellipse-formula]http://math.stackexchange.com/questions/264446/the-fastest-way-to-obtain-orientation-from-this-ellipse-formula[/url][br]_________________[br]Given a closed loop of rigid wire in the shape of an ellipse, mark a point on the wire with a red dot.[br][br][b]Proposition:[/b] to rotate the wire about its center.

BLUE− Initial solution: Track θ in the full 360° range.[br]GREEN− Problem: the standard equation is discontinuous under rotation.[br]BLACK− Parametric solution.[br](Note: the rotation vector w has length 1; it does not scale the ellipse).[br][br][br][br]___________________________[br]Ellipse Rotation (2 of 3)[br]1. Relating the standard form and parametric equations.[url]http://www.geogebratube.org/material/show/id/45277[/url][br][b]→2. Resolving rotation in the full range, 0 ≤ θ ≤ 2π[/b][br]3. Relationship between the coefficients and rotation. Limit cases of tan(θ). Lengths and orientation of the half-axes a, b. [url]http://www.geogebratube.org/material/show/id/45924[/url]