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I have a very limited exposure to projective geometry, but I'm having fun exploring the concept of duality. In particular I'd like to know if this naive method of finding a "dual" curve to a given curve $\Gamma$ (in the real projective plane) could be correct. Here's the idea. Let $P(t)$ be a point of $\Gamma$ with (homogeneous) coordinates $(a(t),b(t),1)$, and let $P_t^*$ be the line dual to $P$, i.e. \begin{equation} P_t^*:\qquad a(t)x+b(t)y+1=0 \end{equation} Now, and this is where problems could arise, I'd like to assume that the set \begin{equation} P^*=\{ a(t)x+b(t)y+1=0\quad|\quad t\in\mathbb{R}\} \end{equation} is the set of tangents to the dual curve $\Gamma^*$. In this case, I could find the equation for $\Gamma^*$ by considering the following limit: \begin{equation} \lim_{h\rightarrow 0}P_t^*\cap P_{t+h}^* \end{equation} My question is: is this method plausible or is it complete nonsense? I have checked it only for the case of a parabola $\Gamma$ given by $P(t)=(t,at^2,1)$, which yields a "dual" parabola $\Gamma^*$ given by $P(t)=(t,\frac{t^2}{4a},1)$.

I am afraid that the assumption of $P^*$ being a set of tangents is way too strong and basically unmotivated. If it's the case, is there an elementary method of obtaining a dual curve from a given one?

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  • $\begingroup$ * sits on a rock and waits * $\endgroup$ – marco trevi Apr 15 '16 at 13:39
  • $\begingroup$ * bounties to the sharks * $\endgroup$ – marco trevi Apr 18 '16 at 16:54

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