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Surface Triangulation in Python

How to make Tri-Surf plots in Python with Plotly.

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Defining and plotting triangulated surfaces

with Plotly Mesh3d

A triangulation of a compact surface is a finite collection of triangles that cover the surface in such a way that every point on the surface is in a triangle, and the intersection of any two triangles is either void, a common edge or a common vertex. A triangulated surface is called tri-surface.

The triangulation of a surface defined as the graph of a continuous function, $z=f(x,y), (x,y)\in D\subset\mathbb{R}^2$ or in a parametric form: $$x=x(u,v), y=y(u,v), z=z(u,v), (u,v)\in U\subset\mathbb{R}^2,$$ is the image through $f$,respectively through the parameterization, of the Delaunay triangulation or an user defined triangulation of the planar domain $D$, respectively $U$.

The Delaunay triangulation of a planar region is defined and illustrated in a Python Plotly tutorial posted here.

If the planar region $D$ ($U$) is rectangular, then one defines a meshgrid on it, and the points of the grid are the input points for the scipy.spatial.Delaunay function that defines the planar triangulation of $D$, respectively $U$.

Triangulation of the Moebius band

The Moebius band is parameterized by:

$$\begin{align*} x(u,v)&=(1+0.5 v\cos(u/2))\cos(u)\\ y(u,v)&=(1+0.5 v\cos(u/2))\sin(u)\quad\quad u\in[0,2\pi],\: v\in[-1,1]\\ z(u,v)&=0.5 v\sin(u/2) \end{align*} $$

Define a meshgrid on the rectangle $U=[0,2\pi]\times[-1,1]$:

In [2]:
import plotly.plotly as py
from plotly.graph_objs import *

import numpy as np
import matplotlib.cm as cm
from scipy.spatial import Delaunay

u=np.linspace(0,2*np.pi, 24)
v=np.linspace(-1,1, 8)
u,v=np.meshgrid(u,v)
u=u.flatten()
v=v.flatten()

#evaluate the parameterization at the flattened u and v
tp=1+0.5*v*np.cos(u/2.)
x=tp*np.cos(u)
y=tp*np.sin(u)
z=0.5*v*np.sin(u/2.)

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

tri.simplices is a np.array of integers, of shape (ntri,3), where ntri is the number of triangles generated by scipy.spatial.Delaunay. Each row in this array contains three indices, i, j, k, such that points2D[i,:], points2D[j,:], points2D[k,:] are vertices of a triangle in the Delaunay triangularization of the rectangle $U$.

In [6]:
print tri.simplices.shape, '\n', tri.simplices[0]
(322, 3) 
[73 96 72]

The images of the points2D through the surface parameterization are 3D points. The same simplices define the triangles on the surface.

Setting a combination of keys in Mesh3d leads to generating and plotting of a tri-surface, in the same way as plot_trisurf in matplotlib or trisurf in Matlab does.

We note that Mesh3d with different combination of keys can generate alpha-shapes.

In order to plot a tri-surface, we choose a colormap, and associate to each triangle on the surface, the color in colormap, corresponding to the normalized mean value of z-coordinates of the triangle vertices.

Define a function that maps a mean z-value to a matplotlib color, converted to a Plotly color:

In [7]:
def map_z2color(zval, colormap, vmin, vmax):
    #map the normalized value zval to a corresponding color in the colormap
    
    if vmin>vmax:
        raise ValueError('incorrect relation between vmin and vmax')
    t=(zval-vmin)/float((vmax-vmin))#normalize val
    R, G, B, alpha=colormap(t)
    return 'rgb('+'{:d}'.format(int(R*255+0.5))+','+'{:d}'.format(int(G*255+0.5))+\
           ','+'{:d}'.format(int(B*255+0.5))+')'   

To plot the triangles on a surface, we set in Plotly Mesh3d the lists of x, y, respectively z- coordinates of the vertices, and the lists of indices, i, j, k, for x, y, z coordinates of all vertices:

In [8]:
def tri_indices(simplices):
    #simplices is a numpy array defining the simplices of the triangularization
    #returns the lists of indices i, j, k
    
    return ([triplet[c] for triplet in simplices] for c in range(3))

def plotly_trisurf(x, y, z, simplices, colormap=cm.RdBu, plot_edges=None):
    #x, y, z are lists of coordinates of the triangle vertices 
    #simplices are the simplices that define the triangularization;
    #simplices  is a numpy array of shape (no_triangles, 3)
    #insert here the  type check for input data
    
    points3D=np.vstack((x,y,z)).T
    tri_vertices=map(lambda index: points3D[index], simplices)# vertices of the surface triangles     
    zmean=[np.mean(tri[:,2]) for tri in tri_vertices ]# mean values of z-coordinates of 
                                                      #triangle vertices
    min_zmean=np.min(zmean)
    max_zmean=np.max(zmean)  
    facecolor=[map_z2color(zz,  colormap, min_zmean, max_zmean) for zz in zmean] 
    I,J,K=tri_indices(simplices)
    
    triangles=Mesh3d(x=x,
                     y=y,
                     z=z,
                     facecolor=facecolor, 
                     i=I,
                     j=J,
                     k=K,
                     name=''
                    )
    
    if plot_edges is None:# the triangle sides are not plotted 
        return Data([triangles])
    else:
        #define the lists Xe, Ye, Ze, of x, y, resp z coordinates of edge end points for each triangle
        #None separates data corresponding to two consecutive triangles
        lists_coord=[[[T[k%3][c] for k in range(4)]+[ None]   for T in tri_vertices]  for c in range(3)]
        Xe, Ye, Ze=[reduce(lambda x,y: x+y, lists_coord[k]) for k in range(3)]
        
        #define the lines to be plotted
        lines=Scatter3d(x=Xe,
                        y=Ye,
                        z=Ze,
                        mode='lines',
                        line=Line(color= 'rgb(50,50,50)', width=1.5)
               )
        return Data([triangles, lines])

Call this function for data associated to Moebius band:

In [9]:
data1=plotly_trisurf(x,y,z, tri.simplices, colormap=cm.RdBu, plot_edges=True)

Set the layout of the plot:

In [10]:
axis = dict(
showbackground=True, 
backgroundcolor="rgb(230, 230,230)",
gridcolor="rgb(255, 255, 255)",      
zerolinecolor="rgb(255, 255, 255)",  
    )

layout = Layout(
         title='Moebius band triangulation',
         width=800,
         height=800,
         scene=Scene(  
         xaxis=XAxis(axis),
         yaxis=YAxis(axis), 
         zaxis=ZAxis(axis), 
        aspectratio=dict(
            x=1,
            y=1,
            z=0.5
        ),
        )
        )

fig1 = Figure(data=data1, layout=layout)

py.iplot(fig1, filename='Moebius-band-trisurf')
Out[10]:

Triangularization of the surface $z=\sin(-xy)$, defined over a disk

We consider polar coordinates on the disk, $D(0, 1)$, centered at origin and of radius 1, and define a meshgrid on the set of points $(r, \theta)$, with $r\in[0,1]$ and $\theta\in[0,2\pi]$:

In [11]:
n=12 # number of radii
h=1.0/(n-1)
r = np.linspace(h, 1.0, n)
theta= np.linspace(0, 2*np.pi, 36)

r,theta=np.meshgrid(r,theta)
r=r.flatten()
theta=theta.flatten()

#Convert polar coordinates to cartesian coordinates (x,y)
x=r*np.cos(theta)
y=r*np.sin(theta)
x=np.append(x, 0)#  a trick to include the center of the disk in the set of points. It was avoided
                 # initially when we defined r=np.linspace(h, 1.0, n)
y=np.append(y,0)
z = np.sin(-x*y) 

points2D=np.vstack([x,y]).T
tri=Delaunay(points2D)

Plot the surface with a modified layout:

In [12]:
data2=plotly_trisurf(x,y,z, tri.simplices, colormap=cm.cubehelix, plot_edges=None)
fig2 = Figure(data=data2, layout=layout)
fig2['layout'].update(dict(title='Triangulated surface',
                          scene=dict(camera=dict(eye=dict(x=1.75, 
                                                          y=-0.7, 
                                                          z= 0.75)
                                                )
                                    )))

py.iplot(fig2, filename='trisurf-cubehx')
Out[12]:

This example is also given as a demo for matplotlib plot_trisurf.

Plotting tri-surfaces from data stored in ply-files

A PLY (Polygon File Format or Stanford Triangle Format) format is a format for storing graphical objects that are represented by a triangulation of an object, resulted usually from scanning that object. A Ply file contains the coordinates of vertices, the codes for faces (triangles) and other elements, as well as the color for faces or the normal direction to faces.

In the following we show how we can read a ply file via the Python package, plyfile. This package can be installed with pip.

We choose a ply file from a list provided here.

In [13]:
!pip install plyfile
from plyfile import PlyData, PlyElement

import urllib2
req = urllib2.Request('http://people.sc.fsu.edu/~jburkardt/data/ply/chopper.ply') 
opener = urllib2.build_opener()
f = opener.open(req)
plydata = PlyData.read(f)
Requirement already satisfied (use --upgrade to upgrade): plyfile in /Library/Frameworks/Python.framework/Versions/2.7/lib/python2.7/site-packages
Requirement already satisfied (use --upgrade to upgrade): numpy>=1.8 in /Library/Frameworks/Python.framework/Versions/2.7/lib/python2.7/site-packages (from plyfile)
You are using pip version 8.0.3, however version 8.1.2 is available.
You should consider upgrading via the 'pip install --upgrade pip' command.

Read the file header:

In [14]:
for element in plydata.elements:
    print element
    
nr_points=plydata.elements[0].count
nr_faces=plydata.elements[1].count
element vertex 1066
property float x
property float y
property float z
element face 2094
property list uchar int vertex_indices

Read the vertex coordinates:

In [15]:
points=np.array([plydata['vertex'][k] for k in range(nr_points)])
points[0]

x,y,z=zip(*points)

faces=[plydata['face'][k][0] for k in range(nr_faces)]
faces[0]
Out[15]:
array([0, 1, 2], dtype=int32)

Now we can get data for a Plotly plot of the graphical object read from the ply file:

In [17]:
data3=plotly_trisurf(x,y,z, faces, colormap=cm.RdBu, plot_edges=None)

title="Trisurf from a PLY file<br>"+\
                "Data Source:<a href='http://people.sc.fsu.edu/~jburkardt/data/ply/airplane.ply'> [1]</a>"

noaxis=dict(showbackground=False,
            showline=False,  
            zeroline=False,
            showgrid=False,
            showticklabels=False,
            title='' 
          )

fig3 = Figure(data=data3, layout=layout)
fig3['layout'].update(dict(title=title,
                           width=1000,
                           height=1000,
                           scene=dict(xaxis=noaxis,
                                      yaxis=noaxis, 
                                      zaxis=noaxis, 
                                      aspectratio=dict(x=1, y=1, z=0.4),
                                      camera=dict(eye=dict(x=1.25, y=1.25, z= 1.25)     
                                     )
                           )
                     ))
                      
py.iplot(fig3, filename='Chopper-Ply-cls')
Out[17]:

This a version of the same object plotted along with triangle edges:

In [18]:
from IPython.display import HTML
HTML('<iframe src=https://plot.ly/~empet/13734/trisurf-from-a-ply-file-data-source-1/ \
     width=800 height=800></iframe>')
Out[18]:

Reference

See https://plot.ly/python/reference/ for more information and chart attribute options!

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