[b]Proposition:[/b][br]Given an ellipse rotated by ∡θ, determine the relationships among the values a, b, θ, and the equations[br]i) Ax² −2Bxy + Cy² = F[br]ii) x²/a² + y²/b² =1[br]iii) parametric equation r(t) = a u + b v

Notes:[br]*Parameter F is not free. It is determined by A, B, C:[br] F = AC - B²[br]I used a special case of the standard form to isolate the relationship among axes, angle and coefficients.[br]TO DO: connect it with natural solution of the differential equation dy/dx = (a1x−a2y) /(a2x + b2y), where F is free (the constant of integration) but the coefficients are determined by the angle. [br][br]* -π/2 < arctan() < π/2. But θ may go outside this range.[br]To resolve the full range 0≤ θ ≤ 2π using the standard form, I need additional information. Onward--[br][br]________________[br]Ellipse Rotation, (1 of 3):[br][b]→1: Converting between standard form and parametric equations.[/b][br]2. Resolve θ, 0 ≤ θ≤ 2π using atan(); continuity troubles: [url]http://www.geogebratube.org/material/show/id/45026[/url][br]3. Determine the half-axis lengths and orientation, disambiguate tan(θ) for limit cases: [url]http://www.geogebratube.org/material/show/id/45924[/url]