We define two ellipses by the polynomial equations:
Ax2 + By2 + Cx + Dy + E = 0 |
(1) |
Fx2 + Gy2 + Hx + Iy + J = 0 |
(2) |
From eq. (1) we get the value of y: |
![Value of y](/TrigoCalc/Circles2/Ellipse2/Eq/eq1.png) |
Substitute the value of y into equation (2) yields the equation:
After multiplying all terms by 4B2 and open paranthesis we get the equation
In order to simplepy the equation we will define the following constants:
δ = |
2BDI − 2D2G + 4BEG − 4B2J |
φ = |
4BCG − 4B2H |
θ = |
4ABG − 4B2F |
μ = |
2BI − 2DG |
And we get the equation: |
![Equation with x](/TrigoCalc/Circles2/Ellipse2/Eq/eq1c.png) |
Taking the squares of bothe sides we get:
D2μ2 − 4ABμ2x2 − 4BCμ2x − 4BEμ2 = x4θ2 + x2φ2 + δ2 + 2θφx3 + 2θδx2 + 2φδx
After arranging the equation by powers we get a forthe power of x which can be solved mathematically:
x4θ2 + x32θφ + x2(φ2 + 2θδ + 4ABμ2) + x(4BCμ2 + 2φδ) + (δ2 − D2μ2 + 4BEμ2) = 0
|