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A general conic has the form $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$. I understand that there are certain properties of this equation that make it special and allow us to classify the different types of curves it can represent (ellipses, parabolas, and hyperbolas, distinguished by their eccentricity), all of which can be derived by slicing cones. But why does it stop there? Are there not interesting properties of an equation like

$$ Ax^3 + Bx^2y^2 + Cy^3 + Dx^2y + Exy^2 + Fx^2 + Gxy + Hy^2 + Ix + Jy + K = 0 $$

Or is there not a generalized equation for these curves such as $$ C = \sum_{i = 1}^n(A_ix^i + B_ix^{i-1}y^{i-1} + C_iy^i + D_ix^{i-1}y^{i-2} + E_ix^{i-2}y^{i-1}) $$

Basically my question is what is the name for curves of this general form? Are they important, do they have interesting properties, have they been studied thoroughly? I've never seen equations like these discussed before anywhere, so I was wondering if they're considered mathematically "important" like general conics are.

user3002473
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  • Be reassured, the cubics (and also quartics and higher degree) have been well studied. http://en.wikipedia.org/wiki/Cubic_plane_curve But these topics are too much advanced and complicated for normal curricula. –  Aug 13 '14 at 12:24

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Yes they are important, and actually there is a lot of theory about them.

They are indeed called Algebraic curves, because they are described by one polynomial equation (the "algebra" part) in two variables (and so they are curves in the plane).

Their theory was largely developed through the centuries, since they are object you can actually draw, and there are many and many textbooks about them.

You can start by looking at the corresponding Wikipedia page about Algebraic curves and at the related links at the end of it.

(Un)fortunately for you the theory about those general curves is much more hard and complicate than the one for conics.

dadexix86
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  • The Wikipedia page certainly doesn't look like something I could jump right into like conics. Could you suggest prior things I should learn about before trying to tackle this subject? – user3002473 Aug 13 '14 at 12:28
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    Easier than that article? At the moment nothing comes to my mind (probably it's because I had an entire course in my Bachelor about Algebraic curves and that article is just really basic stuff).

    I suggest that we wait for someone with a bit more knowledge than me to drop here a title of a textbook about them :)

     – dadexix86 Aug 13 '14 at 12:34
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The accepted answer here suggests that the jump in generality, abstraction, and difficulty in the study of algebraic curves may dissuade a beginner.

A gentler introduction can be found in older texts such as Salmon, A Treatise on the Higher Plane Curves: Intended as a Sequel to A Treatise on Conic Sections, 1879. The wordy title indicates that it is intended as a sequel to Salmon’s Conic Sections (also on archive.org), and thus speaks to the OP's situation. Although this is not the book for students wishing to learn modern methods it is a good place to get some intuition, motivation, and examples.

brainjam
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