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Could someone tell me where I can find the generic equation of the $\alpha$-isoptic of a conic section of this form: $$\Gamma: ax^2+2bxy+cy^2+2dx+2ey+f=0$$ I searched in many links and PDFs but I found nothing.

($\alpha$-isoptic curve is the locus of the points through which the tangents to the conic pass and which intersect forming an angle $\alpha$)

Math Attack
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    "plane spiric curves": https://mathcurve.com/courbes2d.gb/isoptic/isoptic.shtml – brainjam Jun 14 '23 at 14:57
  • @brainjam Interesting reference. When $\alpha=\pi/2$ and the curve is an ellipse, the result is known to be the "orthoptic circle" or "Monge circle" of the ellipse. See this question. – Jean Marie Jun 14 '23 at 23:18
  • @JeanMarie also "director circle". I'm guessing OP knows about the isoptic equations for axis aligned and origin centered conics $ax^2+cy^2+f=0$ but wants the ones for general conics $ax^2+2bxy+cy^2+2dx+2ey+f=0$. – brainjam Jun 15 '23 at 02:00

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I solved it by myself. Omitting the steps the formula is this:

Let be $I_1,I_2$ and $I_3$ the invariants of the conic section $$\Gamma(x,y):ax^2+2bxy+cy^2+2dx+2ey+f=0$$ If $$\mathbf{A}=\begin{pmatrix}a&b&d\\b&c&e\\d&e&f\end{pmatrix}\qquad\mathbf{B}=\begin{pmatrix}a&b\\b&c\end{pmatrix}$$ We define $$I_3=\det(\mathbf{A})\qquad I_2=\det(\mathbf{B})\qquad I_1=\text{tr}(\mathbf{B})$$ And $$(x_0,y_0)=\left(\dfrac{cd-eb}{b^2-ac}, \dfrac{ae-db}{b^2-ac}\right)$$ is the center of $\Gamma(x,y)$.

The $\alpha$-isoptic curve of $\Gamma(x,y)=ax^2+2bxy+cy^2+2dx+2ey+f$ is: $$\left[\tan(\alpha)\frac{ I_2}{2}\left((x-x_0)^2+(y-y_0)^2+\frac{I_1 I_3}{I_2^2}\right)\right]^2=-I_3\cdot(ax^2+2bxy+cy^2+2dx+2ey+f)$$

Math Attack
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  • [+1] nice use of invariants! (for other readers, see background at e.g. https://archive.org/details/analyticalconics0000spai/page/58/mode/2up) – brainjam Jun 16 '23 at 15:34