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In the real projective plane $\mathbb{R}\mathbb{P}^2$ where points are represented by homogenous coordinates $[x,y,z]$, there is a degenerate conic consisting of a real double line. This conic is of equation $x^2=0$. I am looking to determine its corresponding conic envelope.

In Jurgen Richter-Gebert's Perspectives on Projective Geometry, he says there are two such envelopes, one consiting of all lines passing through one point, and the other consisting of all lines passing through two points. Is this correct, or is it a misunderstanding?

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    $\begingroup$ What do you mean in this context by "envelope of $x^2=0$", (which is nothing else that the $y$ axis) ? Usualy this term is used as "envelope of a family of curves..." $\endgroup$ – Jean Marie Aug 21 at 12:20
  • $\begingroup$ @JeanMarie Jurgen Richter-Gebert defines it as the set of lines $\left\{ l \in \mathcal{L}_{\mathbb{R}\mathbb{P}^2} \mid l^T A^\triangle l = 0 \right\}$, where $A^\triangle$ is the transposed comatrix of $A$, and $A$ is the matrix of the original conic. $\endgroup$ – Shurik Goyal Aug 21 at 12:54
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    $\begingroup$ I understand : the conic curve is the envelope of these lines. $\endgroup$ – Jean Marie Aug 21 at 13:28
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    $\begingroup$ You will agree with me that the set of tangent lines to a line $L$ is reduced to this unique line. $\endgroup$ – Jean Marie Aug 21 at 13:30
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    $\begingroup$ I have no idea... In such degenerate cases, one can as well stick to definitions... sometimes better than intuition... $\endgroup$ – Jean Marie Aug 21 at 14:53
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I think you might have misunderstood the text here. A double line doesn’t have a unique dual: it has an infinite number of them, each consisting of the lines through each of two points on the line (which do not have to be distinct). Each of these dual conics can be obtained as the limiting case of the duals of a particular family of nondegenerate conics. For instance, given any two distinct points on the line one can obtain the corresponding two-point dual conic as the limiting case of the duals of the family of ellipses that have those points as common foci. Richter-Gebert spends much of section 9.6 on this. Indeed, it’s a major motivation for

Definition 9.5. A primal/dual pair of conics is given by a pair $(A,B)$ of real symmetric nonzero $3\times3$ matrices such that there exists a factor $\lambda\in\mathbb R$ with $AB=\lambda E$.

Note that $\lambda$ can be zero, which is in fact the case for a double line. For your canonical double line $x^2=0$, then, we have $$A=\begin{bmatrix}1&0&0\\0&0&0\\0&0&0\end{bmatrix}$$ and any nonzero matrix of the form $$B=\begin{bmatrix}0&0&0\\0&a&b\\0&b&c\end{bmatrix}$$ is dual to it. A rank-one $B$ (which occurs when $b^2=ac$) represents a double point, while a rank-two $B$ represents two distinct points. In the latter case, you can recover the two points by splitting the conic, using the line $(1,0,0)^T$ as the “intersection point” for the algorithm.

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  • $\begingroup$ Thank you. Would it be correct to say that up to projective transformation, there are essentially three distinct dual conics for a double line, and that these cases can be seen as degeneration processes from other conics? The first being from a complex ellipse giving two distinct complex points, the second from a real ellipse giving two distinct real points, and the third from a degenerate conic containing two distinct lines giving a unique point. $\endgroup$ – Shurik Goyal Aug 23 at 11:29
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    $\begingroup$ @ShurikGoyal Yes. That’s basically what the table on p.162 of my edition of R-G’s book says. Note that the two complex points nonetheless lie on a real line. $\endgroup$ – amd Aug 23 at 19:57

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