# 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.

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Up to what equivalence relation? The tags are unclear. – Qiaochu Yuan May 28 '11 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. – Remke Kloosterman May 28 '11 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. – mikhail skopenkov May 28 '11 at 16:45
Equivalence relation can be stated precisely (Laguerre equivalence) but this seems to be not important because each given surface has already (probably) simplest parametrization among its equivalence class. – mikhail skopenkov May 28 '11 at 16:51