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I'm defining a quadric surface as "a 3D surface which is a graph of an algebraic equation of degree 2, i.e. a quadratic equation". Since more than one variable can be squared, should I change the last part to "i.e. an equation with at least one quadratic term", "a quadratic equation with 2 or more variables", or something else? I'd appreciate suggestions to further improve it in other ways. Basically I'm looking for a definition that is simple enough for a high school student to understand but covers all cases/examples. (ChatGPT didn't give me a satisfactory answer.)

Edit: Are cylindrical surfaces a subset of quadric surfaces or are they considered to be their own category of surfaces?

Nate
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    Why don't you use the definition given in Quadric? – Gonçalo Jul 06 '25 at 22:49
  • @Gonçalo Other sources list additional examples like the double-napped cone and cylinders. – Nate Jul 07 '25 at 07:59
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    In complex projective 3-space it’s simple, the quadrics are projectively equivalent to one of $x^2+y^2+z^2+w^2=0, x^2+y^2+z^2=0, x^2+y^2=0, x^2=0$ the latter two are reducible. The former two are the full rank, smooth quadric, and the quadric with a singular point. Getting all the real affine versions amount to changing sign in front of the squares and moving the plane at infinity. The cylinder is a cone with the singularity at infinity, so your question reduces to whether to allow the singular irreducible quadrics in the definition, which you do, and more, considering all degree two equations – Jan-Magnus Økland Jul 08 '25 at 11:00
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    I am surprised to see how reasonable questions get downvoted while often there is a rush to answer questions that look like homework assignments and have no obvious interest. – Alexey Jul 08 '25 at 21:09
  • write $P=[x,y,z,1]$ and then for any real symmetric $4\times 4$ matrix $M$ the quadratic form $P M P^T = 0$ is a quadratic surface. – user619894 Jul 09 '25 at 08:38
  • $5x^2 + 5y^2 + 8z^2 − 2xy + 8yz + 8zx + 12x − 12y + 6 = 0$ is a cylinder as we see from putting it in the form $5(x+z)^2-2(x+z)(y+z)+5(y+z)^2+12(x+z)-12(y+z)+6=0$ – Jan-Magnus Økland Jul 09 '25 at 09:51

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Since there is no universally accepted definition of a quadric surface I suggest going for essentially the wikipedia definition http://en.m.wikipedia.org/wiki/Quadric:

A quadric surface is an implicit surface defined by a general second-degree equation in $x$, $y$, and $z$: $$Ax^2 + By^2 + Cz^2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0$$ where $A, B, \ldots, J $ are real numbers, and at least one of $ A, B, C $ is nonzero.

Note that this excludes certain degenerate cases such as the union of the planes $x=0$ and $y=0$, which is the equation $xy=0$. It includes union of planes such as $(x-y)(x+y)=x^2-y^2$. However, to get a definition of a quadric surface that precisely includes or exludes degenerate cases, you would either need to list a lot of of cases with various conditions on the coefficients or alternatively involve a matrix definition (as in the comments), which I assume is no good for high schoolers and not worth it.

Jap88
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  • This definition allows for cones/cylinders. And other plane pairs like $(x+z)(y+z)=0$ are allowed. Even the double planes like $x^2=0$ are allowed. More naturally, one of $A,..,F$ nonzero and conditions coming from the determinants of the matrix formulation to stratify the types? – Jan-Magnus Økland Jul 10 '25 at 10:10
  • Yes, but I think that is fine in this context. I assume that no matrix algebra can be involved and that a certain amount of simplification in the definition can be allowed for highschoolers. – Jap88 Jul 10 '25 at 10:13
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Restricting to surfaces in ${\Bbb R}^3,$ the wikipedia entry, in saying there are nine quadric surfaces, implicitly goes for the a surface (dimension two), maybe singular, but not reducible or non-reduced definition.

The problem is we have a lot of work to go from the equations of degree 2 to get at these nine. $x^2+y^2+z^2+1=0$ has no real points and as such is no real surface.

The precise conditions on the coefficients are cumbersome, so actually listing the nine (plus the reducible and non-reduced cases to make eleven) cases might be preferable. But knowing that it comes from a simpler complex projective world is also valuable.

  • The smooth ones

(the non-example above) From $x^2+y^2+z^2+w^2=0$ the no real points one.

  1. From $x^2+y^2+z^2-w^2=0$ in the ${\Bbb R}^3$ from $w=1$ we get by scaling the variables the ellipsoids.

  2. From $x^2+y^2-z^2+w^2=0$ we get the hyperboloid of two sheets

  3. From $x^2-y^2+z^2-w^2=0$ which is also $(x+y)(x-y)+(z+w)(z-w)=0$ we get the hyperboloid of one sheet.

4&5. From moving these so that one focus is at infinity we get the elliptic and hyperbolic paraboloids.

  • The singular ones:
  1. The cone over the conic sections with singular point in the finite

7-9. The cone over the conic sections with singular point at infinity.

  • The reducible ones (the plane pairs)

$x^2-y^2=(x+y)(x-y)=0$

  • the non-reduced ones (the double planes)

$x^2=0,$ remembering that $(x+y+z+1)^2=x^2+y^2+z^2+2xy+2xz+2yz+2x+2y+2z+1=0$ is such a case.

Jan-Magnus Økland
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  • Isn't a hyperboloid of one sheet just $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}$ = 1? – Nate Jul 16 '25 at 04:23
  • @Nate yes, however the suggestive way to show that it’s also ${\Bbb P}^1\times {\Bbb P}^1$ is to write your equation as scaled to $x^2+y^2-z^2-1=0$ and use $(x+z)(x-z)-(w-y)(w+y), w=1$ or $x’z’-y’w’=0.$ btw, a hyperboloid of one sheet exists on every set of three distinct skew lines, where these three become part of one ruling. Also the tangent plane at any point picks out one line of each ruling – Jan-Magnus Økland Jul 16 '25 at 07:44
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A quadratic surface is a three-dimensional surface defined by a second-degree polynomial equation in three variables (typically x, y, and z).

Think of it as the 3D extension of the conic sections you might know from 2D geometry - just as circles, ellipses, parabolas, and hyperbolas are defined by quadratic equations in two variables, quadratic surfaces are their three-dimensional counterparts.

The general mathematical form of a quadratic surface is:

            Ax² + By² + Cz² + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0

where A, B, C, D, E, F, G, H, I, and J are constants, and at least one of A through F is non-zero.

Examples in math include an ellipsoid

                 (x-h)²/a² + (y-k)²/b² + (z-l)²/c² = 1.

If a = b = c, you get a perfect sphere. A paraboloid can open upward like a bowl (elliptic paraboloid) or have a saddle shape (hyperbolic paraboloid).

Avaadakedavra
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