In response to a question on Stack Exchange: [url]http://math.stackexchange.com/questions/264446/the-fastest-way-to-obtain-orientation-from-this-ellipse-formula[/url] _________________ Given a closed loop of rigid wire in the shape of an ellipse, mark a point on the wire with a red dot. [b]Proposition:[/b] to rotate the wire about its center.
BLUE− Initial solution: Track θ in the full 360° range. GREEN− Problem: the standard equation is discontinuous under rotation. BLACK− Parametric solution. (Note: the rotation vector w has length 1; it does not scale the ellipse). ___________________________ Ellipse Rotation (2 of 3) 1. Relating the standard form and parametric equations.[url]http://www.geogebratube.org/material/show/id/45277[/url] [b]→2. Resolving rotation in the full range, 0 ≤ θ ≤ 2π[/b] 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]