# Geometry package

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The geometry package is multipackage providing functions to manipulate geometrical entities in 2D and 3D.

## Tutorials

### Meshing Octave

Under construction

This tutorial shows the workflow to generate a triangular mesh inside an arbitrary region. This tutorial requires that you install the package fpl and msh (which requires Gmsh installed in your system). Alternatively, the core function `delaunay` could be used (the tutorial explains how) but the result aren't so pretty.

The first part of the tutorial requires an interesting shape. If you have Inkscape you can use the previous tutorial to load it into octave. Here I will be using this SVG.

 Code: Loading the file as polygon compatible with geometry package ```octavesvg = svg ("octave.svg").normalize(); ids = octavesvg.pathid(); P = octavesvg.path2polygon (ids{1}, 12)(1:end-1,:); P = bsxfun (@minus, P, centroid (P)); ```

Now we have our SVG as a polygon compatible with the Geometry package format. You can plot the polygon using the function `drawPolygon`.

 Code: Plotting a polygon compatible with geometry package ```drawPolygon (P,'-o'); ```

As you can see the polygon has lots of points. We need to simplify the polygon in order to obtain a mesh of reasonable size. Otherwise gmsh will have problems meshing and the result could be huge (or a segmentation fault :( ). THe package geometry comes with a simplification function but as of version 1.4.0, this function is very naive and wont fix this problem. If you know how to simplify polygons you can contribute! We are going to reduce the amount of points of the polygon in a drastic and destructive way (i.e. the shape may be considerably damaged). The codes to do so follows, in the future a better procedure will be published here.

 Code: Symplification of a polygon compatible with geometry package ```Ps = P; n = 1e6; thrsd = 0.38; while n>size (Ps,1) n = size (Ps,1); ds = sqrt (sumsq (Ps(1:end,:) - Ps([2:end 1],:),2)); Ps = Ps(ds>mean (ds)*(1-thrsd),:); end whos P Ps ```

The next step is to mesh the interior of the polygon. To do this we could just call `delaunay` on the polygon and be done with it, but in general such mesh wont be so nice (you will need to add interior points). A very effective way of generating a good mesh is to use the package msh, which requires Gmsh installed in your system. The function `data2geo` in the Geometry package makes our work very easy:

 Code: Generating mesh for plot with msh package ```pkg load msh filename = tmpnam (); meshsize = sqrt (mean (sumsq (diff (P, 1, 1), 2)))/2; data2geo (P, meshsize, "output", [filename ".geo"]); T = msh2m_gmsh (filename); ```

To plot the generated mesh we use the function `pdemesh` from the fpl package.

 Code: Plotting mesh with fpl package ```pkg load fpl pdemesh (T.p, T.e, T.t) ```