I can take an algebraic curve, and I can draw its first polar. By this, I mean that I can take an arbitrary point not on the algebraic curve, and then I can identify all the points on that algebraic curve whose tangents go through my arbitrary point, and then I can generate a new algebraic curve that touches all those points. This is what I understand the first polar curve of a given algebraic curve to be (and of course it depends on my chosen arbitrary point).
Like much of my recent education, this has been presented and I can do it, and now I have two curves, one of them being the polar curve of the other, and I have no idea why I've done this. The best I have so far is from Wiki, which tells me "It is used to investigate the relationship between the curve and its dual, for example in the derivation of the Plücker formulas." So it seems I can use this polar line to learn things about the original algebraic curve. But that's as far as I can get; I don't know WHAT things I can learn about the original algebraic curve. WHAT can I learn now about the original algebraic curve?
I know that the Plücker formulae give me absolute limits on what algebraic curves can exist, by relating algebraic curves and their duals; but what's the link to the first polar curve of a given algebraic curve? Or should I be asking about the link between the first polar curve and the Plücker formulae?
I'm definitely spotting a trend in what I'm learning recently. Lots of (apparently arbitrary) definitions and ways to generate lines, very little by way of what they mean and what the consequences are. Sad times for me.