# Tutte's Barycenter Method applied to Isotopies

Éric Colin de Verdière,
Michel Pocchiola, and
Gert Vegter.
**Tutte's Barycenter Method applied to
Isotopies**.
*Computational Geometry: Theory and Applications*, vol. 26, no 1,
pp. 81-97, 2003.
This page aims at:

giving the data of two examples for which the method described in
Section 2 fails;
giving 3-dimensional coordinates for Starbird's example, as
described in Section 3.
## Two examples for which the method of Section 3 fails

We define below two embeddings of triangulated planar graphs. For each
of these embeddings, we compute a (non-necessarily positive)
equilibrium stress using the lift z=x^2+y^2. We then try to get an
isotopy using linear interpolation between this stress and the stress
where all weights corresponding to an edge equal 1. The following
examples show that this method does not always yield an isotopy.
These examples have been found by generating randomly triangulated
embeddings and detecting planarity failure (program in C++ using the
LEDA library and Numerical Recipes), then checking the result by exact
calculations using Mathematica.

Here is the format of the data: each line corresponds to a vertex of
the embedding and contains successively the number of the vertex, its
coordinates x and y, and the list of its neighbors.
#### Our minimal example

In this counter-example, the situation is very close to a
degeneracy. One can check by calculation the planarity of the initial
embedding and the non-planarity at some stages of the process. There
are 4 exterior vertices and 2 interior vertices.
`
`1 -500 900 2 5 6 4

2 -850 900 1 3 5

3 -950 -900 4 6 5 2

4 0 -400 1 6 3

5 -900 -699 6 1 2 3

`6 -800 -300 1 5 3 4
`

#### The example given in the manuscript

5 exterior vertices, 3 interior vertices.
`
`1 -681.67 314.31 5 2 8 6

2 -938.19 -391.67 7 8 1 3

3 419.75 -833.89 4 8 7 2

4 841.39 52.42 5 6 8 3

5 712.91 332.73 1 6 4

6 733.43 99.34 5 1 8 4

7 128.62 38.94 8 2 3

`8 277.47 156.82 1 2 7 3 4 6
`

#### An example which gives an isotopy, as in the most common case

## Three-dimensional coordinates for Starbird's example

The initial and final
embeddings of the graph C are available in OOGL (`vect`)
format, to be viewed for example using Geomview (these two files are commented
and thus also human-readable).

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Last modified: Thu Feb 27 17:36:23 MET 2003
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