How do you find the coefficients for a conic in a rotated system?
This is a good question, but a little bit vague. I'll try to guess:
When you say, "a conic," do you (1) have an equation and you want to rotate the axes to get a simpler equation? Or do you (2) have a graph or some points that the curve goes through?
The general conic equation is a relation between x and y:
(1) • If you want to rotate this curve to "get rid of" the xy-term, you can substitute coordinates u and v, rotated by angle
or to substitute in the general conic equation,
If you use the specific angle
then you will get an equation in u and v that has no xy-term.
(2) • If you just have some points that the conic goes through, you need five points to determine a unique conic. You then have 5 equations in the 6 variables A, B, C, D, E, F.
They are scalable so this determines a unique conic. For example
Note: If you have six or more points, they may not all lie on the same conic.
// dansmath strikes again! //
p.s. Here's the story, and an example, from Stewart's Calculus: