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If $x^2 + y^2 = 4$ represents the section of a cone by a plane horizontal to the cone, a circle, and $y^2 - x^4 = 4$ represents the section of a cone by a plane vertical to the cone, a hyperbola, what do $x^3 + y^3 = 8$ and $y^3 - x^3 = 8$ represent? sections of a 4-dimensional version of a cone by a 3-dimensional object? What about equations like $x^3 + y^2 = 8$? What does x + y = 2 represent (my first thought was the section of a triangle by a line, but that doesn't seem to make much sense)? Do these (and by extension all equations) represent a section of one object of n dimension by another object of n-1 dimension?

Hopefully you can answer all of these questions, I put the title as being Do all equations represent a section of an n dimensional object by another object of n-1 dimension? because I think it covers all of my questions in a broader way.

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    $\begingroup$ You don’t have any examples of a function representing a section of any of the objects you named. You have equations whose solution sets are those sections. It’s a subtle point, but if you use the right words it’s more likely that any answers will answer the right question. $\endgroup$
    – David K
    Apr 16, 2019 at 21:16
  • $\begingroup$ @DavidK how would you suggest I improve it? I don't see what you're saying $\endgroup$
    – jstowell
    Apr 16, 2019 at 22:03
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    $\begingroup$ On further consideration this is probably not really confusing anyone, but if you simply changed the word "function" to "equation" each time it appears in the question, it would be a more accurate description, because $x^2+y^2=4$ (for example) is an equation. By the way, all of your equations happen to be polynomial equations, so I wonder if you actually meant to ask only about polynomial equations (which is an interesting field of study) or equations of $n-1$ variables in general (for example, $\sin(x) + y^2 = 8$). $\endgroup$
    – David K
    Apr 17, 2019 at 0:05
  • $\begingroup$ @DavidK any equation, be it trigonometric, logarithmic, anything. $\endgroup$
    – jstowell
    Apr 17, 2019 at 1:07

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In order to get started, let's look at your first example. If you're working in a two-dimensional Cartesian space, the equation $x^2 + y^2 = 4$ is a circle. Of course in two-dimensional Cartesian space you cannot make a three-dimensional figure at all, since there's no third dimension to extend into.

In three-dimensional Cartesian coordinates the equation $x^2 + y^2 = 4$ describes an infinite cylinder. The $z$ coordinate can be anything (since it was not mentioned in the equation) but the equation restricts $x$ and $y$ so that the point $(x,y,z)$ is exactly $2$ units away from the $z$-axis.

In three dimensions, if you want to describe a circle, you need two equations. One equation can just select a $z$ coordinate. The simultaneous equations $x^2 + y^2 = 4$ and $z = 2$ select a circle centered on the $z$-axis in a plane parallel to the $x,y$ plane and at a distance $2$ from the $x,y$ plane. (You can also get a circle without setting $z$ to a constant; this is merely one way to get a circle.)

The circle given by $x^2 + y^2 = 4$ and $z = 2$ is a section of the cone $x^2 + y^2 - z^2 = 0,$ consistent with your question. But it's also (trivially) a section of the cylinder $x^2 + y^2 = 4,$ a section of the sphere $x^2 + y^2 + (z - 2)^2 = 4,$ a section of the sphere $x^2 + y^2 + z^2 = 8,$ and a section of the hyperboloid $x^2 + y^2 - \frac12 z^2 = 2.$

Somewhat more generally, you can take any plane shape described by two simultaneous equations, a polynomial equation in $x$ and $y$ and an equation setting $z$ to a constant, delete the "$z=\text{constant}$" equation, and you have the equation of a kind of cylinder--not generally a circular cylinder, but these other shapes are generically called cylinders as well. If the original equations included $z=0,$ you could get a more interesting shape by deleting $z=0$ and inserting some $z$ terms (any positive power of $z$ multiplied by any constant, any non-negative power of $x,$ and any non-negative power of $y$) into the polynomial equation so it is a polynomial equation of $x,$ $y,$ and $z$ that reduces to the original polynomial equation when $z=0.$ If the original equations included $z=c,$ instead of inserting $z$ terms in the equation, insert terms using positive powers of $z - c$ multiplied by any constant, any non-negative power of $x,$ and any non-negative power of $y$. Then your original plane figure will be a section of the three-dimensional figure described by your new equation.

What will the three-dimensional figure be? It could be any number of shapes. You can easily throw enough polynomial terms into the equation to produce a shape that has never been given a name. And if you already had an unnamed shape in two dimensions then it almost sure has no name in three.

The same principles apply to $n$ dimensions. Set the last dimension constant and a polynomial equation over the other $n-1$ coordinates describes a figure in $n-1$ dimensions. This could be a section of any one of an infinite (mostly unnamed) $n$-dimensional figures.

If the equations can be general equations over the coordinates then the possibilities open up even more. But that only makes it even harder to categorize the results.

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  • $\begingroup$ What specifically does x^3 + y^3 = 8 represent? $\endgroup$
    – jstowell
    Apr 17, 2019 at 1:17
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    $\begingroup$ If you translate, rotate, and scale it sufficiently (possibly different scaling in the $x$ and $y$ directions), I think you can make it match some variety of en.wikipedia.org/wiki/Conchoid_of_de_Sluze. There might be another name for it in the exact form $x^3+y^3=k,$ but I have no idea what that would be. There are some examples of cubic curves at milefoot.com/math/planecurves/cubics.htm to give you some idea of how weird they can get. $\endgroup$
    – David K
    Apr 17, 2019 at 2:10

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