Tangent Quadrics in Real 3-Space¶
In this article, we derive the polynomial conditions for quadrics in 3-space to contain a collection of points and to be tangent to a collection of lines and planes. When the number of such figures equals nine, there exist a finite number of quadrics which satisfy those conditions. The number of such quadrics is given by the corresponding entry in Schubert’s triangle.
Our article serves as a first step towards showing that they can all be real. The following Jupyter notebook computes the number 104 (above in blue). The code is presented as a Jupyter notebook in julia and the main computations rely on the package HomotopyContinuation.jl.
The functions which construct the polynomial systems, verify the nondegeneracy of solutions, and reproduce the certification computations may
be downloaded here
TangentQuadricsCode.zip. Simply unzip the folder and open TangentQuadrics.ipynb in a Jupyter notebook.
You may also run this file online yourself by clicking the link below.
The certification files may be downloaded here
Project page created: 21/10/2020
Code contributors: Taylor Brysiewicz, Claudia Fevola, and Bernd Sturmfels
Jupyter Notebook written by: Taylor Brysiewicz, 21/10/2020
Corresponding author of this page: Taylor Brysiewicz, Taylor.Brysiewicz@mis.mpg.de