Conic Rotation 3: half-axes
- Ryan Hirst
Rotating Ellipses and Hyperbolas: θ, a and b. Construction of the ratio tan(θ), from an equation in the form Ax² + Bxy + Cy² =±1.
When the conic flips between ellipse and hyperbola, the orientation of axes (major, minor) flips 90°.
The major axis is unbounded under manipulation of A,B,C. The minor axis is not; it transitions smoothly between ellipse and hyperbola. Adopting it as a reference point would also avoid the axis flip.
The values A=C, B=0, B²-4AC = 0 need limit definitions. For tan(θ), they are nicely determined geometrically: consider always segment f. I found it easier to tackle a, b with Algebra. At least one of the three ratios in the "Limits" section (expand both "+"s) is always defined. Click one to see the undefined case.
Conic Rotation, 3 of ?
1. Ellipse, 1: Conversion between the Standard Form and parametric equations: http://www.geogebratube.org/material/show/id/45277
2. Ellipse, 2: Resolving discontinuity under rotation. http://www.geogebratube.org/material/show/id/45026
*3. Determine the half-axis lengths and orientation, disambiguate limit cases