Do there exist planar curves that can be constructed from geometric figures beyond conic sections?

Question

Conic sections are algebraic curves that can be constructed from different cross-sections of a cone in $\mathbb{R}^3$. Do there exist other geometric shapes in $\mathbb{R}^3$ besides the cone whos cross-sections can yield algebraic curves analogous to conic sections? Are there higher dimensional equivalents for 3D curves or sheets in 3D that can be defined in by some geometric figure like a cone (or potential alternative geometric figures) in $\mathbb{R}^4$?

Maybe I can be a bit more concrete. Conic sections are an example of a two-variable second degree polynomial of the form $$ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. $$ I can construct a similar two-variable higher-degree polynomials as such $$ \text{Curve}_n(x,y) = A_0x^n + A_2x^{n-1}y + \ldots + A_{n-1}xy^{n-1} + A_ny^n + \text{Constant} = 0 $$ Does there exist something analogous to a "cone" for such a polynomial? What if I add another variable $z$?

Answer 1

Let us consider a torus (2nd figure) with a vertical axis of revolution with $c$ = main radius, $d$ = radius of revolving circle ($c>d$ for a non-self intersecting torus).

The sections by vertical planes at distance $f$ from the axis of revolution are called "spiric sections".

For certain combinations of $c,d,f$ (see Appendix below) we obtain Cassinian ovals (first figure). Recall: A Cassinian oval $(C_b)$ with foci $F_1, F_2$ is the set of points $M$ such that $MF_1.MF_2=b^2$.

Please note

• 1. that this locus has 2 connected components if $b<\frac{F_1F_2}{2}$,
• 2. That the lemniscate obtained in the limit case where $b=\frac{F_1F_2}{2}$ is a Booth's lemniscate (not in general a Bernoulli's lemniscate as I thought at first),
• 3. the similarity with the definition of an ellipse (with product of lengths instead of their sum).

Reference: https://www.lucamoroni.it/toric-sections/

Appendix: When vertical sections of a torus are Cassinian ovals ?

The torus equation is

$$(\sqrt{x^2 + y^2} - d)^2 + z^2 = c^2$$

Expanding the square, isolating the square root and squaring again, we can give it the fourth degree implicit polynomial equation:

$$(x^2 + y^2 + z^2-c^2)^2=4d^2(x^2+y^2)\tag{1}$$

Besides, by definition, the Cassinian oval implicit equation in a vertical plane parallel to any plane with equation $y=k$ is

$$\sqrt{(x-a)^2+z^2}\sqrt{(x+a)^2+z^2}=b^2$$

Squaring this relationship and expanding it, we get:

$$(x^2+z^2+a^2-2ax)(x^2+z^2+a^2+2ax)=b^4$$

$$(x^2+z^2+a^2)^2-4a^2x^2=b^4\tag{2}$$

It remain to express that the section of (1) by vertical plane with equation $y=f$ has equation:

$$(x^2 + f^2 + z^2-c^2)^2=4d^2(x^2+f^2)\tag{3}$$

Therefore (2) and (3) can be identified under the following conditions:

$$f^2-c^2=a^2, \ \ a=d, \ \ b^2=2df.$$

meaning that if $c,d$ are given,

$$f=\sqrt{c^2+d^2} \ \ \text{and} \ \ b=\sqrt{2df}$$

Therefore, for a given torus, there is exactly one vertical section that can be a Cassinian oval ; moreover, due to the fact that $f>c$, this section has a single component.

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Edit: Another example, this time with a cubic curve called the folium of Descartes, with equation:

$$x^3+y^3-3xy=0$$

that can be considered as the intersection of surface defined by $$z=-\operatorname{atan}(x^3+y^3-3xy)$$ with plane $z=0$ (see Fig.) (with the interest that the surface is bounded ($-\dfrac{\pi}{2}<z<\dfrac{\pi}{2}$).

We could as well have taken $z=x^3+y^3-3xy$ or $z=|x^3+y^3-3xy|$, etc.

Answer 2

A 3D generalization of the conic curves are the quadric surfaces, of general equation

$$Ax^2+By^2+Cz^2+2Dxy+2Eyz+2Fzx+2Gx+2Hy+2Iz+J=0.$$

There are nine different types, not counting the degenerate ones (I would not count the sphere as different of the ellipsoid).

They can be seen as sections of an hypercone by an hyperplane,

$$\begin{cases}Ax^2+By^2+Cz^2+Jw^2+2Dxy+2Eyz+2Fzx+2Gxw+2Hyw+2Izw=0,\\w=1.\end{cases}$$

A cross-section of a quadric by a plane is a conic. Nothing new.

The intersection of two quadrics is a 3D quartic curve, forming a hair-raisingly complex family.