stylistic:tone, stylistic:style_transfer, linguistic:translation
Conversion to canonical form

In the case of an ellipse or hyperbola, the equation

can be converted to canonical form in transformed variables x ~, y ~ {\displaystyle {\tilde {x}},{\tilde {y}}} as

or equivalently

where λ 1 {\displaystyle \lambda _{1}} and λ 2 {\displaystyle \lambda _{2}} are the eigenvalues of the matrix (A B / 2 B / 2 C) {\displaystyle \left({\begin{matrix}A&B/2\\B/2&C\end{matrix}}\right)} — that is, the solutions of the equation

— and S {\displaystyle S} is the determinant of the 3 × 3 matrix above, and Δ = λ 1 λ 2 {\displaystyle \Delta =\lambda _{1}\lambda _{2}} is again the determinant of the 2 × 2 matrix. In the case of an ellipse the squares of the two semi-axes are given by the denominators in the canonical form.