Parabola from Vertex and Two Points on the Arc
- Author:
- Ryan Hirst
Notes:
- Finding the roots of a cubic equation: {link}
- Orientation: Ialways let open upward (in the direction of ). From a given value of tan θ , I take 0 < θ < π . The angles are signed. I have used complex rotation, not because I like being abstract, but because (for example) complex multiplication by corresponds always to "rotation by angle α". More on Signed Angles:http://www.geogebratube.org/material/show/id/99837 Complex Rotation: http://www.geogebratube.org/material/show/id/115348
- Equation (3) usually has ONE real solution, and the parabola is uniquely determined by the constraints. When there are instead THREE real solutions, I choose the value of θ which is closest to the previous value. This can always be done so that the axes rotate smoothly (no leaps). As a result, if A and C begin on opposite arms of the parabola, they will stay on opposite arms. Put another way, v will always point inside the triangle ΔVAC).