# Names of certain surfaces

Are there any generally used names for the following algebraic and nonalgebraic surfaces? Any references to literature where the surfaces are studied are also appreciated.

Surface I. Implicit equation: $z^2(z^2-16x)=64y^2$. Parametric form: $\mathbf{r}(u,v)=\left( u^2-v^2, 2 u v , 4 u \right)$ It is a graph of a harmonic function $z(x,y)=\pm 2\sqrt{2}\sqrt{x+\sqrt{x^2+y^2}}$.

Surface II. Implicit equation: $(x^2 + y^2 + z^2) z^4 - 2 (8 x^2 + 9 y^2 + 9 z^2) x z^2 - 27 (y^2 + z^2)^2=0$ Parametric form: $\mathbf{r}(u,v)=\frac{1}{1+u^2+v^2}\left( u^4-3 u^2v^2-3u^2, 4 u^3 v, 4 u^3 \right)$

Surface III. Implicit equation: $z^2(y^2+z^2)\left(z^2-4y-4\right)^2 + x^2 \left(64 y^3 - 24 (y-3) y z^2 - 6 (y+6) z^4 + z^6\right)-27 x^4 z^2 =0$ Parametric form: $\mathbf{r}(u,v)=\frac{1}{1+u^2+v^2}\left( 2 uv(u^2-v^2-1), u^2 (3v^2-u^2-1), 4 u^2 v \right)$

Surface IV. Parametric form: $\mathbf{r}(u,v)=\left((u^2 - v^2)\left(1 - \frac{1}{u^2 + v^2}\right) - \ln(u^2 + v^2), 2 u v \left(1 - \frac{1}{u^2 + v^2}\right), 4 u\right)$

An example of a kind of an answer I would be happy to get: the surface $\left(y^2+z^2\right)(1-z)=x^2z$ is a parabolic horn cyclide; see mathworld article.

• Up to what equivalence relation? The tags are unclear. Commented May 28, 2011 at 14:47
• From an algebraic geometry point of view: the surfaces I-III are rational surfaces. I.e., they contain an open that is isomorphic with an open in $\mathbb{P}^2$. Several rational surfaces come with a name, but most of them are smooth projective (del Pezzo surfaces, quadric surfaces), whereas your surfaces are singular. Examples of rational surfaces "come with a name" are cones over rational curves, surfaces swept out by a morphism between two curves. You can check whether your surfaces are of one of these types. Commented May 28, 2011 at 16:05
• This question is not quite mathematical: it is asked whether specialists (in algebraic or differential geometry, or variational calculus) could recognize some famous surfaces among given ones. Commented May 28, 2011 at 16:45
• All the image links in this post are broken, but my attempts to fix them were unsuccessful. Commented May 28, 2017 at 12:39
• The links seem to be broken... Commented May 28, 2017 at 21:58

If Surface I yet has no name, I would christen it Winged Victory. :-)

I would call Surface I the "Whitney Umbrella"

You may wish to check whether your surfaces are equivalent to any from this list.