What is the closest mathematics is to a general theory of curves? I ask this because today Wildberger (I like infinity, but I watch his videos because his format is appealing to me) made a video in which he talked about quadratics, cubics and other curves in two real variables (Stuff like parabolas, hyperbolas and ellipses) and he mentioned how even though we knew all about quadratics long ago, it took until Newton for mathematicians to seriously study and classify cubic curves (third degree curves in two real variables). And he mentioned that Newton found 72 different types of cubic curve (as opposed to the 3 types of quadratics) and then he mentioned some other algebraic curves like the quartic lemniscate of Bernoulli.
But all of that happened at the beginnings of Calculus. What about today? Do we have a general theory of curves? Suppose I gave you a 50th degree curve in two variables, is there some obscure paper I could find that would classify that curve for me and tell me interesting properties?
I don't think that's really it (but if I'm wrong please post examples). No description of differential geometry even mentions algebraic curves specifically, though I see how the techniques of differential geometry can be applied to the study of algebraic curves.
What I am asking about is just a paper or perhaps a sub-field of a sub-field of algebra that has a general method of classification of algebraic curves in two real variables. Like Newton did for the case n=3 and previously Descartes did for n=2.
That's not really it. Let me give you an easy example:
In typical analytic geometry, they teach you about quadratics (also known as conics) and you learn stuff like being able to recognize when you have a circle, ellipse, parabola or hyperbola and from this you get a general idea of how to graph the curve and stuff like that.
Now, I want something like that but in the general case. I know this is probably non-trivial, but given that nowadays algebraists study much more ambitious subjects one would think that either this is a resolved problem or something similar to polynomial congruences happened. As in, we don't have general theorems, but we have general techniques.
To go back to my 50th degree algebraic curve, suppose I wanted to find a classification of 50th degree curves. Would I need to start from scratch or are there already some kind of papers on the general study of algebraic curves that I could use as a guide?
Luke Reyes
sorry man can't help you >pic is my face when I can't help you
Parker Young
It's fine. Do you study math? It's weird, right? Back in the day, this was the core subject of study of the greatest mathematicians. Analytic geometry revolutionized math, placing algebra in equal footing to geometry. Juggernauts like Newton were obsessed with these classifications theorems (and I must mention, after Newton, another guy did the same work and instead recognized over 200 types of cubics) but nowadays no one speaks of this. At some point in math history this type of math died. Wildberger did mention in his video that after people started asking questions about the arc-length of these curves, people started studying "Elliptic curves" instead. So was that it? Did Elliptic curves replace the other curves in popularity? I don't get it.
In high school, you only learn about conics, and even as a college student I have only studied very specific cubics. I feel like there is a hole in math history that people don't even teach about.
Brody Martin
>Wildberger did mention in his video that after people started asking questions about the arc-length of these curves, people started studying "Elliptic curves" instead. So was that it? Did Elliptic curves replace the other curves in popularity?
Computing arc length leads to elliptic Integrals and trying to evaluate these elliptic integrals leads to elliptic curves.
The main reason Elliptic Curves are so popular is because these have a group structure.
Christian Martin
You're asking about plane algebraic geometry.
Bentley Stewart
Yeah, that seems to be it. Is there a general theory of classification?
Eli Phillips
Not a general one, but theres tons of work in that direction. It won't be real curves in R^2 either, it will be complex curves in CP^2 which are much more comfortable and expressive
Samuel Foster
I watched a handful of infinity this weekend. It was awful. I wouldn't trust the content further than it mentions actual problems/fields.
Evan Davis
I see. I guess that makes sense. It would be fun if there was an underlying pattern across all algebraic curves but the fact that just for n=3 there are like 200 different types of cubic curves hints at how completely random this stuff us.
Daniel Robinson
Potentially dumb answer: algebraic geometry?
Jackson Allen
Yes. Classical algebraic geometry in particular is what OP wants. Try Fulton's Algebraic Curves.
Cameron Williams
Differential equations are sufficient for describing most curves, and calculus offers some closed form smooth solutions. Algebraic geometry, which was mentioned before, offers a more precise approach, but is more difficult to analyse. There are also some topological proofs for curves, but I think they also assume smoothness.
Joshua Turner
>And he mentioned that Newton found 72 different types of cubic curve (as opposed to the 3 types of quadratics)
That wasn't newton but some sand nigger in the middle east.
Mason Baker
There's a number of ways to go about this, mainly algebraic topology, differential geometry and algebraic geometry, depending on what you mean by classify. If you mean curves defined by polynomial equations, you can look at classical algebraic geometry (Dolgachev's "Classical Algebraic Geometry: A Modern View" is a future classic, you can look there but be warned, it's tough). It has been a staple of mathematics for centuries and is, in some sense, still very active (see the minimal model program for example).
Andrew Bailey
he means algebraic (polynomial) curves. the differentiable viewpoint is very easy: every curve is uniquely determined by its curvature and torsion (up to a euclidean motion)
Oliver Moore
I see, thanks for the recommendations. What I'd really like to know if a structural reason is known for why there are only 3 conics and yet over 200 cubics (and who knows how many quartics). It can't be just random. It is obviously because as we incease n, the amount of terms that count as nth degree increase so we have more freedom in defining curves, but how much does this freedom affect the number of types of curves? And other stuff like lower and upper bounds for the number of types of curves of nth degree would be very nice.
It is also very nice that the second degree algebraic curves defined by Descartes correspond precisely to the conics defined and studied by the greeks. Is there a structural reason for this? Is there an analog for cubics, quartics, etc? So many interesting questions that I can't find answers to on the internet.
Parker Hall
I was going to recommended this exact same book.
William Rivera
There's only one conic though
Aaron Brooks
Hyperbolas, parabolas, ellipses.
Liam Thompson
But they all look like ellipses to me!
Alexander Foster
Why, instead of assuming everyone who says something you dont understand is wrong, dont you take a minute to google?
