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We all know what a conic section is - a circle, a hyperbola, an ellipse, or a parabola.

But what about the cubic curve? Does it not slice through some other 3D shape? If so, what is that called? What other curves can we find in it?

If there IS no answer, or mathematicians of this day and age haven't found it yet, please let me know in these comments, as I'm still very very curious. I'd like to know if it doesn't exist!

soupynoodles
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  • Even if this involves 4 dimensions I'd still be very curious – soupynoodles Aug 22 '15 at 18:11
  • The title could be improved ---- "conic" specifically means a curve that we get by slicing a cone. You could ask instead (for example) "What surface can we slice to obtain a cubic curve?" – Schemer Aug 22 '15 at 18:42
  • Done :) Thanks for the suggestion! – soupynoodles Aug 22 '15 at 19:25
  • @soupynoodles You mean like central cubics A x^3 + B y^3 + C x^2 y + D x y^2 = 0 etc..? Some algebraic curves? Wild guessing, I suspect surfaces of revolution of positive and negative Gauss curvature may give third degree sections at some places. We know toric sections. Even we dont know about slanted sections of a catenoid, hyperboloid. Some closed loop ovals of higher degree like Cartesian ovals..can they belong to the needed category? – Narasimham Aug 22 '15 at 19:28
  • @Narasimham Well i never really heard of those curves, but Relapsarian just gave an answer to my basic question. Thank you for enlightening me about all these curves, I look forward to learning about them on Google! – soupynoodles Aug 22 '15 at 19:31
  • It is possible to find space curve of section numerically if inclination of cutting plane to axis of symmetry and the meridian are given. – Narasimham Aug 22 '15 at 20:58
  • @Narasimham thanks! does this apply to trigonometric as well as algebraic curves? – soupynoodles Aug 22 '15 at 21:03
  • @Narasimham also, what can we do if the curve has no symmetry at all? – soupynoodles Aug 22 '15 at 21:04

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Any (algebraic) curve can be thought of as a slice of a 3D shape. Suppose your curve is defined by an equation $f(x,y)=0$: then define a polynomial $F(x,y,z)$ by some formula

$$F(x,y,z) = f(x,y) + z \cdot ( \text{ anything you like} ). $$

Then the equation $F(x,y,z)=0$ defines a 3D shape whose slice $\{z=0\}$ is exactly the curve you started with.

In the particular case you ask about, the nicest answer is to view cubic curves as slices of cubic surfaces. If you search for that term, you'll see that cubic surfaces are much-loved by algebraic geometers for their beautiful, regular geometric properties. (Warning: to get the nice answer, one needs to think of surfaces in projective space.)

For the last part of the question, there are lots of other kinds of curves inside cubic surfaces. For example, probably the most famous property of these surfaces is that they always (again, in the projective world) contain exactly 27 lines. If we take a slice by a plane containing a line, we get the line together with a conic. Any slice by a plane that doesn't have one of the 27 lines in it will give a plane cubic curve, but if you look at the picture in the affine plane over the real numbers (as in the original context of your question) there is a multitude of types of these. (Newton classified them and found something like 72 distinct types, but if memory serves he missed a few cases.) One could think of this big list as the analogue of the four types of conic sections you mention.

Finally let me mention that both in the conic and cubic case, you can get many, many other curves in your surfaces by taking slices by things other than planes --- that is, intersections with surfaces of higher degree. But that is probably taking us too far from the original question.

Schemer
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  • Hi @Relapsarian, thanks for your response! You mentioned "algebraic curves" - does this mean that trigonometric curves (like sine, cosine, tan, sec, cot) are not included? I tried looking up 'trigonometric slices' but I really couldn't find anything. Once again, thanks for a very elaborate and detailed response! :) – soupynoodles Aug 22 '15 at 19:34
  • @soupynoodles: actually $f$ could be anything in my discussion, including something involving trigonometric functions. (I just said "algebraic" because that's the natural context for the things you were asking about --- conics and cubics.) If you want to pass to projective geometry, though, you really do need to stick to polynomials, that is, algebraic curves. – Schemer Aug 23 '15 at 09:34
  • Alright, thanks @Relapsarain. You have given me a lot to Google and research about :) If there's anything related to this that I should know, please do feel free to tell me! Thanks again – soupynoodles Aug 23 '15 at 09:42