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A year ago, I asked about properly triangulating a periodic shape on a plane: the annulus (Getting a proper Delaunay triangulation of an annulus (using python)).

I now want to expand this to triangulating a cylinder (or in general, any periodic surface). I try a straightforward extension of the 2D code:

from scipy.spatial import Delaunay
NZ = 14
NTheta = 14

R = 1 #radius of cylinder 
L = 3 #length of cylinder 

#define base rectangle (u,v)
u=np.linspace(0, 2*np.pi, NTheta) #periodic direction
v=np.linspace(0, L, NZ)
# u=u[:-1] #leave out one point
u,v=np.meshgrid(u,v)
u=u.flatten()
v=v.flatten()

#evaluate the parameterization at the flattened u and v
x=R*np.cos(u)
y=R*np.sin(u)
z=v

#define 2D points, as input data for the Delaunay triangulation of U
points2D=np.vstack([u,v]).T
tri = Delaunay(points2D, incremental=True)#triangulate the rectangle U
triSimplices = tri.simplices

xyz0 = np.vstack([x,y,z]).T

I create a cylinder via parameterisation, and obtain the triangulation via scipy.spatial.Delaunay() of the base domain -- the rectangle. Obviously, this triangulation does not know about the periodicity. I can see this clearly by moving the last row, and plotting: enter image description here

To correct this, I try a straightforward extension of the 2D solution-- I add an extra point in 3D, re-triangulate and remove the unwanted simplices. 

Tri1 = Delaunay(points2D) #triangulate the rectangle U
Tri2 = Delaunay(xyz0)

## we add a central (0,0,L/2) point to xy0 to fill it up with triangles
last_pt = xyz0.shape[0]
xy1 = np.vstack((xyz0,(0,0,L/2)))  # add ctr point
Tri3 = Delaunay(xyz1)
print(Tri3.points.shape, Tri3.simplices.shape)
print(Tri1.points.shape, Tri1.simplices.shape)
print(Tri2.points.shape, Tri2.simplices.shape)

## remove the simplices that contain the central point
mask = ~(Tri3.simplices==last_pt).any(axis=1)
triSimplices = Tri3.simplices[mask,:]

However, the extension of the 2D code to 3D seems to have a big problem -- triangulations in 3D give tetrahedra, not triangles! Moreover, it seems to be more sensitive to the choice of the central point. In short, I am stuck.

So, what is the proper way of triangulating such a periodic surface?

ap21
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  • What is your expected output an open or closed cylinder? Are you aware of the [`matplotlib.tri.Triangulation`](https://matplotlib.org/api/tri_api.html) method that implements Delaunay triangulation for matplotlib ? [Here](https://stackoverflow.com/a/53931367/3067485) an example of usage. – jlandercy Feb 18 '19 at 15:41
  • @jlandercy I am looking for a closed cylinder. Yes, I am aware of `matplotlib`'s method, but I can't see how that will do anything different unless I specifically ask it to do so. – ap21 Feb 18 '19 at 16:06
  • What is wrong with the initial triangulation? – lrineau Feb 19 '19 at 08:11
  • @lrineau It doesn't reflect the fact that the surface is periodic. The triangulation is a discrete representation of a continuous surface, so it should reflect any periodicities as well. How do I make it do so? – ap21 Feb 19 '19 at 19:27

1 Answers1

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The following short script creates the vertices and also uses scipy to create the triangles. I then reassign the last column

# Create a 2d map of the vertices
num_tri_high = 10
num_tri_in_circum = 9
xx = np.linspace(0, 10, num_tri_in_circum + 1)
yy = np.linspace(0, 10, num_tri_high)
xxx,yyy = np.meshgrid(xx,yy)

# Triangulate vertices
receiver_tri = Delaunay(np.transpose(np.vstack((xxx.flatten(), yyy.flatten()))))
triangles = receiver_tri.simplices

# Set last column equal to 1st (which effectively closes the gap)
tot = (num_tri_in_circum + 1) * num_tri_high
for i in range(num_tri_high+1):
    triangles[triangles == (num_tri_in_circum+1) * i + num_tri_in_circum] = (num_tri_in_circum+1) * i

# Wrap into a cylinder
radius = 2
xp = radius*np.sin(xxx.flatten()/np.max(xxx.flatten())*np.pi*2)
yp = radius*np.cos(xxx.flatten()/np.max(xxx.flatten())*np.pi*2)
zp = yyy.flatten()
tow_scene = mlab.triangular_mesh(xp.flatten(),yp.flatten(),zp.flatten(), triangles, representation='wireframe', color=(0,0,0), opacity=1)
mlab.show()

Open cylinder

Willem
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