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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 $z(x,y)=\pm 2\sqrt{2}\sqrt{x+\sqrt{x^2+y^2}}$.

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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: $\mathbf{r}(u,v)=\frac{1}{1+u^2+v^2}\left( u^4-3 u^2v^2-3u^2, 4 u^3 v, 4 u^3 \right)$

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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: $\mathbf{r}(u,v)=\frac{1}{1+u^2+v^2}\left( 2 uv(u^2-v^2-1), u^2 (3v^2-u^2-1), 4 u^2 v \right)$

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

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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
up vote 2 down vote accepted

If Surface I yet has no name, I would christen it Winged Victory. :-)
           Surface 1

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

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thanks. i doubt these are official names:-) – mikhail skopenkov Jun 2 '11 at 17:58

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