Trying to plot an ellipse or general conic section.
From this GaTech open textbook on linear algebra, I can find the equation of the rotated and off-origin centered ellipse (last example in the link) in the form: $$ Ax^2 + By^2 + Cxy + Dx + Ey + F = 0 $$ However, trying to plot the resulting equation vs the dataset of points it was generated from to compare whether they match becomes the challenging part.
From this site on ellipses, and this question it appears the plotable, parametric results I am looking for have the form:
$$
x = x_c+(a \cosθ \cosϕ−b \sinθ \sinϕ) \\
$$
$$
y = y_c+(a \cosθ \sinϕ+b \sinθ \cosϕ)
$$
where:
$x_c = $ ellipse center in x
$y_c = $ ellipse center in y
$a = $ ellipse major axis length
$b = $ ellipse minor axis length
$θ = $ ellipse rotation angle
$ϕ = $ parameter, which ranges from 0 to 2π radians
From the Ben's answer in this question, it appears there is a well known transformation from the parametric components into the general form (unfortunately I want the opposite), using: $$ A = \frac{\cos^2(\theta)}{a^2} + \frac{\sin^2(\theta)}{b^2} $$ $$ B = \frac{\sin^2(\theta)}{a^2} + \frac{\cos^2(\theta)}{b^2} $$ $$ C = \frac{\sin(2\theta)}{a^2} - \frac{\sin(2\theta)}{b^2} $$ $$ D = -\frac{2 x_c \cos^2(\theta)}{a^2} - \frac{y_c\sin(2\theta)}{a^2} - \frac{2 x_c \sin^2(\theta)}{b^2} + \frac{y_c \sin(2\theta)}{b^2} $$ $$ E = -\frac{x_c \sin(2\theta)}{a^2} - \frac{2y_c\sin^2(\theta)}{a^2} + \frac{x_c \sin(2\theta)}{b^2} - \frac{2y_c \cos^2(\theta)}{b^2} $$ $$ F = \frac{x_c^2 \cos^2(\theta)}{a^2} + \frac{x_c y_c \sin(2\theta)}{a^2} + \frac{y_c^2 \sin^2(\theta)}{a^2} + \frac{x_c^2 \sin^2(\theta)}{b^2} - \frac{ x_c y_c \sin(2\theta)}{b^2} + \frac{y_c^2 \cos^2(\theta)}{b^2} - 1 $$ From bubba's answer in the same question and this answer it appears the way to get to some form of parameterized equation involves using a u,v substitution like: $$x = u+v$$ $$y = u-v$$ $$(u+v)(u-v)=u^2-v^2$$ where u,v apparently are of the form: $$u = (\cos\alpha, \sin\alpha)$$ $$v = (-\sin\alpha, \cos\alpha)$$ Somewhere in here I'm getting lost though, as I think what I really want are: $$x_c = f(A, B, C, D, E, F)$$ $$y_c = f(A, B, C, D, E, F)$$ $$a = f(A, B, C, D, E, F)$$ $$b = f(A, B, C, D, E, F)$$ $$θ = f(A, B, C, D, E, F)$$ $$ϕ = f(A, B, C, D, E, F) ?$$ Last one might not be related to A, B, C, D, E, F
Does anyone know what the parameter conversion of the above is, or perhaps just explain the part I'm missing of how $u$, $v$ subbing turns into $x_c$, $y_c$, $a$, $b$, $θ$, $ϕ$ ?
Edit 12/18/2022 After posting, found Osmund Francis answer in this question that appears to answer the final goal, in not the process. This question and answer that got linked appears to show evaluation of an example using eigenvalues and eigenvectors, rather than general form, yet that may work. Still parsing.
