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 this Jupyter Notebook.
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 triangulation of $D$, respectively $U$.
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 [1]:
import numpy as np
from scipy.spatial import Delaunay
In [2]:
import plotly.plotly as py
py.sign_in('empet','api_key')
In [3]:
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
points3D=np.vstack((x,y,z)).T
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 triangulation of the rectangle $U$.
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 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.
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:
Now we define a function that returns data for a Plotly plot of a tri-surface:
In [4]:
from plotly.graph_objs import *
In [5]:
def standard_intensity(x,y,z):
return z
In [6]:
def plotly_triangular_mesh(x,y,z, faces, intensities=standard_intensity, colorscale="Viridis",
showscale=False, reversescale=False, plot_edges=False):
#x,y,z lists or np.arrays of vertex coordinates
#faces = a numpy array of shape (n_faces, 3)
#intensities can be either a function of (x,y,z) or a list of values
vertices=np.vstack((x,y,z)).T
I,J,K=faces.T
if hasattr(intensities, '__call__'):
intensity=intensities(x,y,z)#the intensities are computed here via the set function,
#that returns the list of vertices intensities
elif isinstance(intensities, (list, np.ndarray)):
intensity=intensities#intensities are given in a list
else:
raise ValueError("intensities can be either a function or a list, np.array")
mesh=dict(type='mesh3d',
x=x,
y=y,
z=z,
colorscale=colorscale,
reversescale=reversescale,
intensity= intensity,
i=I,
j=J,
k=K,
name='',
showscale=showscale
)
if showscale is True:
mesh.update(colorbar=dict(thickness=20, ticklen=4, len=0.75))
if plot_edges is False: # the triangle sides are not plotted
return [mesh]
else:#plot edges
#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
tri_vertices= vertices[faces]
Xe=[]
Ye=[]
Ze=[]
for T in tri_vertices:
Xe+=[T[k%3][0] for k in range(4)]+[ None]
Ye+=[T[k%3][1] for k in range(4)]+[ None]
Ze+=[T[k%3][2] for k in range(4)]+[ None]
#define the lines to be plotted
lines=dict(type='scatter3d',
x=Xe,
y=Ye,
z=Ze,
mode='lines',
name='',
line=dict(color= 'rgb(50,50,50)', width=1.5)
)
return [mesh, lines]
Call this function for data associated to Moebius band:
In [7]:
pl_RdBu=[[0.0, 'rgb(103, 0, 31)'],
[0.1, 'rgb(176, 23, 42)'],
[0.2, 'rgb(214, 96, 77)'],
[0.3, 'rgb(243, 163, 128)'],
[0.4, 'rgb(253, 219, 199)'],
[0.5, 'rgb(246, 246, 246)'],
[0.6, 'rgb(209, 229, 240)'],
[0.7, 'rgb(144, 196, 221)'],
[0.8, 'rgb(67, 147, 195)'],
[0.9, 'rgb(32, 100, 170)'],
[1.0, 'rgb(5, 48, 97)']]
In [8]:
data1=plotly_triangular_mesh(x,y,z, tri.simplices, intensities=standard_intensity, colorscale=pl_RdBu,
showscale=True, plot_edges=True)
Set the layout of the plot:
In [9]:
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,
showlegend=False,
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)
In [10]:
py.iplot(fig1, filename='Mobius-band-trisurf')
Out[10]:
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]:
pl_cubehelix=[[0.0, 'rgb(0, 0, 0)'],
[0.1, 'rgb(25, 20, 47)'],
[0.2, 'rgb(21, 60, 77)'],
[0.3, 'rgb(30, 101, 66)'],
[0.4, 'rgb(83, 121, 46)'],
[0.5, 'rgb(161, 121, 74)'],
[0.6, 'rgb(207, 126, 146)'],
[0.7, 'rgb(207, 157, 218)'],
[0.8, 'rgb(193, 202, 243)'],
[0.9, 'rgb(210, 238, 238)'],
[1.0, 'rgb(255, 255, 255)']]
In [13]:
data2=plotly_triangular_mesh(x,y,z, tri.simplices, intensities=standard_intensity, colorscale=pl_cubehelix,
showscale=True, reversescale=False, plot_edges=False)
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)
)
)))
In [14]:
py.iplot(fig2, filename='cubehexn')
Out[14]:
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.
In [15]:
from plyfile import PlyData, PlyElement
Define a function that extract from plydata the vertices and the faces of a triangulated 3D object:
In [16]:
def extract_data(plydata):
vertices=list(plydata['vertex'])
vertices=np.asarray(map(list, vertices))
nr_faces=plydata.elements[1].count
faces=np.array([plydata['face'][k][0] for k in range(nr_faces)])
return vertices, faces
We choose a ply file from a list provided here.
In [17]:
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)
In [18]:
vertices, faces=extract_data(plydata)
x, y, z=vertices.T
Get data for a Plotly plot of the graphical object read from the ply file:
In [19]:
data3=plotly_triangular_mesh(x,y,z, faces, intensities=standard_intensity, colorscale=pl_RdBu,
showscale=True, reversescale=False, plot_edges=True)
Update the layout for this new plot:
In [20]:
title="Trisurf from a PLY file<br>"+\
"Data Source:<a href='http://people.sc.fsu.edu/~jburkardt/data/ply/airplane.ply'> [1]</a>"
In [21]:
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)
)
)
))
In [23]:
py.iplot(fig3, filename='Chopper-Ply-lines')
Out[23]:
An isosurface, F(x,y,z) = c, is discretized by a triangular mesh, extracted by the Marching cubes algorithm from a volume given as a (M, N, P) array of doubles.
The scikit image function, measure.marching_cubes_lewiner(F, c)
returns the vertices and simplices of the triangulated surface.
In [24]:
from skimage import measure
X,Y,Z = np.mgrid[-2:2:40j, -2:2:40j, -2:2:40j]
F = X**4 + Y**4 + Z**4 - (X**2+Y**2+Z**2)**2 + 3*(X**2+Y**2+Z**2) - 3
vertices, simplices = measure.marching_cubes_lewiner(F, 0,
spacing=(X[1,0, 0]-X[0,0,0], Y[0,1, 0]-Y[0,0,0], Z[0,0, 1]-Z[0,0,0]))[:2]
x,y,z = zip(*vertices)
In [25]:
pl_amp=[[0.0, 'rgb(241, 236, 236)'],
[0.1, 'rgb(229, 207, 200)'],
[0.2, 'rgb(219, 177, 163)'],
[0.3, 'rgb(211, 148, 127)'],
[0.4, 'rgb(201, 119, 91)'],
[0.5, 'rgb(191, 88, 58)'],
[0.6, 'rgb(179, 55, 38)'],
[0.7, 'rgb(157, 24, 38)'],
[0.8, 'rgb(126, 13, 41)'],
[0.9, 'rgb(92, 14, 32)'],
[1.0, 'rgb(60, 9, 17)']]
In [27]:
data4=plotly_triangular_mesh(x,y,z,simplices, intensities=standard_intensity, colorscale=pl_amp,
showscale=True, reversescale=True, plot_edges=False)
fig4 = Figure(data=data4, layout=layout)
fig4['layout'].update(dict(title='Isosurface',
width=600,
height=600,
scene=dict(camera=dict(eye=dict(x=1,
y=1,
z=1)
),
aspectratio=dict(x=1, y=1, z=1)
)))
py.iplot(fig4,filename='Isosurface')
Out[27]:
In [28]:
from IPython.core.display import HTML
def css_styling():
styles = open("./custom.css", "r").read()
return HTML(styles)
css_styling()
Out[28]: