Ellipse: Oriented Rotation 1
- Author:
- Ryan Hirst
Proposition:
Given an ellipse rotated by ∡θ, determine the relationships among the values a, b, θ, and the equations
i) Ax² −2Bxy + Cy² = F
ii) x²/a² + y²/b² =1
iii) parametric equation r(t) = a u + b v
Notes:
*Parameter F is not free. It is determined by A, B, C:
F = AC - B²
I used a special case of the standard form to isolate the relationship among axes, angle and coefficients.
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.
* -π/2 < arctan() < π/2. But θ may go outside this range.
To resolve the full range 0≤ θ ≤ 2π using the standard form, I need additional information. Onward--
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Ellipse Rotation, (1 of 3):
→1: Converting between standard form and parametric equations.
2. Resolve θ, 0 ≤ θ≤ 2π using atan(); continuity troubles: http://www.geogebratube.org/material/show/id/45026
3. Determine the half-axis lengths and orientation, disambiguate tan(θ) for limit cases: http://www.geogebratube.org/material/show/id/45924