Göttingen Collection of Mathematical Models and Instruments

# Cubic surface with a biplanr double point

### Model 145

 Category: A IV 12-26

### Description

Cubic surface with a biplanar double point B3 whose tangent planes are conjugate imaginary. Gypsum.

The eqation of this algebraic surface of order 3 was created by a fit of a 3D scan of the model, it is \begin{align*} f(x,y,z)= & 3,3 \cdot 10^{-6}\, x^{3} + 6,6 \cdot 10^{-5}\, x^{2} y - 5,4 \cdot 10^{-6}\, x y^{2} \\ - & 2,2 \cdot 10^{-5} \, y^{3} - 1,9 \cdot 10^{-7} \, x^{2} z - 5,8 \cdot 10^{-6} \, x y z \\ + & 2,9 \cdot 10^{-7}, y^{2} z - 1,1 \cdot 10^{-6} , x z^{2} - 1,4 \cdot 10^{-6} \, y z^{2} \\ + & 4,7 \cdot 10^{-6} \, z^{3} + 5,1 \cdot 10^{-4} \, x^{2} + 4,5 \cdot 10^{-4} \, x y \\ + & 6,1 \cdot 10^{-4} \, y^{2} + 4.8 \cdot 10^{-5} \, x z + 1,3 \cdot 10^{-4} \, y z - 8,3 \cdot 10^{-4} \, z^{2} \\+ & 1,1 \cdot 10^{-3} \, x - 3,1 \cdot 10^{-3} \, y + 5\cdot 10^{-2} \, z - 1. \end{align*}

The surface has a singularity, rather a biplanar double point $B_3$, whose tangent planes at that point are imaginary conjugated.

It contains 3 real straight lines with real asymptotic points. The model was created by Carl Rodenberg in 1881 as part of his dissertation. Later, he produced and sold models for higher education in mathematics.

Text written by: Sönke Pförtner

Showcase of this model is Case number 4

### References

Schilling, Martin(Hrg.): Catalog mathematischer Modelle, Leipzig(Verlag von Martin Schilling) 1911, 7.Auflage, No.54. p. 117.

Fischer, Gerd(Hrg.): Fotoband: Mathematische Modelle / Mathematical Models, mit 132 Fotografien, Braunschweig/Wiesbaden (Vieweg) 1986, picture(s) 21. .

Fischer, Gerd(Hrg.): Mathematical Models, Commentary, Braunschweig/Wiesbaden(Vieweg) 1986. .

Rodenberg. Modelle von Flächen 3. Ordnung . Online version