Finding the Coefficients for a Conic in a Rotated System
Key Questions
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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:
#A x^2 + B x y +C y^2 + Dx + Ey + F = 0#
(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
#theta# , where#u = x cos theta + y sin theta#
#v = -x sin theta + y cos theta# or to substitute in the general conic equation,
#x = u cos theta - v sin theta#
#y = u sin theta + v cos theta# If you use the specific angle
#theta# where#cot (2 theta) = (A-C)/B# 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
#x^2 + y^2 - 25# is the same circle as#3x^2 + 3y^2 - 75 = 0# .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:
http://www.stewartcalculus.com/data/ESSENTIAL%20CALCULUS/upfiles/topics/ess_at_13_ra_stu.pdf
dansmath
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Mar 4 2015
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The general equation of a conic section is in the form
#Ax^2 + Bx + Cy^2 + Dy + E = 0# When
#A# or#C# is 0, the equation is that of a parabola.When
#A# and#C# have the same value (including signs), the equation is that of a circle.When
#A# and#C# have the same sign, the equation is that of an ellipseWhen
#A# and#C# have different signs, the equation is that on a hyperbola
seph
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Oct 19 2014