Plot 3D convex closed regions in matplot lib

Question

I am trying to plot in 3D a polytope defined by a set of inequalities. Essentially, I try to reproduce the functionality of this matlab plotregion library in matplotlib.

Answer 1

Pretty sure there is nothing native in matplotlib. Finding the faces that belong together is not particularly hard, though. The basic idea implemented below is that you create a graph, where each node is a triangle. You then connect triangles that are co-planar and adjacent. Finally, you find the connected components of the graph to determine which triangles form a face.

import numpy as np
from sympy import Plane, Point3D
import networkx as nx


def simplify(triangles):
    """
    Simplify an iterable of triangles such that adjacent and coplanar triangles form a single face.
    Each triangle is a set of 3 points in 3D space.
    """

    # create a graph in which nodes represent triangles;
    # nodes are connected if the corresponding triangles are adjacent and coplanar
    G = nx.Graph()
    G.add_nodes_from(range(len(triangles)))
    for ii, a in enumerate(triangles):
        for jj, b in enumerate(triangles):
            if (ii < jj): # test relationships only in one way as adjacency and co-planarity are bijective
                if is_adjacent(a, b):
                    if is_coplanar(a, b, np.pi / 180.):
                        G.add_edge(ii,jj)

    # triangles that belong to a connected component can be combined
    components = list(nx.connected_components(G))
    simplified = [set(flatten(triangles[index] for index in component)) for component in components]

    # need to reorder nodes so that patches are plotted correctly
    reordered = [reorder(face) for face in simplified]

    return reordered


def is_adjacent(a, b):
    return len(set(a) & set(b)) == 2 # i.e. triangles share 2 points and hence a side


def is_coplanar(a, b, tolerance_in_radians=0):
    a1, a2, a3 = a
    b1, b2, b3 = b
    plane_a = Plane(Point3D(a1), Point3D(a2), Point3D(a3))
    plane_b = Plane(Point3D(b1), Point3D(b2), Point3D(b3))
    if not tolerance_in_radians: # only accept exact results
        return plane_a.is_coplanar(plane_b)
    else:
        angle = plane_a.angle_between(plane_b).evalf()
        angle %= np.pi # make sure that angle is between 0 and np.pi
        return (angle - tolerance_in_radians <= 0.) or \
            ((np.pi - angle) - tolerance_in_radians <= 0.)


flatten = lambda l: [item for sublist in l for item in sublist]


def reorder(vertices):
    """
    Reorder nodes such that the resulting path corresponds to the "hull" of the set of points.

    Note:
    -----
    Not tested on edge cases, and likely to break.
    Probably only works for convex shapes.

    """
    if len(vertices) <= 3: # just a triangle
        return vertices
    else:
        # take random vertex (here simply the first)
        reordered = [vertices.pop()]
        # get next closest vertex that is not yet reordered
        # repeat until only one vertex remains in original list
        vertices = list(vertices)
        while len(vertices) > 1:
            idx = np.argmin(get_distance(reordered[-1], vertices))
            v = vertices.pop(idx)
            reordered.append(v)
        # add remaining vertex to output
        reordered += vertices
        return reordered


def get_distance(v1, v2):
    v2 = np.array(list(v2))
    difference = v2 - v1
    ssd = np.sum(difference**2, axis=1)
    return np.sqrt(ssd)

Applied to your example:

from scipy.spatial import HalfspaceIntersection
from scipy.spatial import ConvexHull
import scipy as sp
import numpy as np
import matplotlib.pyplot as plt
import mpl_toolkits.mplot3d as a3
import matplotlib.colors as colors


w = np.array([1., 1., 1.])


# ∑ᵢ hᵢ wᵢ qᵢ - ∑ᵢ gᵢ wᵢ <= 0
#  qᵢ - ubᵢ <= 0
# -qᵢ + lbᵢ <= 0
halfspaces = np.array([
                    [1.*w[0], 1.*w[1], 1.*w[2], -10 ],
                    [ 1.,  0.,  0., -4],
                    [ 0.,  1.,  0., -4],
                    [ 0.,  0.,  1., -4],
                    [-1.,  0.,  0.,  0],
                    [ 0., -1.,  0.,  0],
                    [ 0.,  0., -1.,  0]
                    ])
feasible_point = np.array([0.1, 0.1, 0.1])
hs = HalfspaceIntersection(halfspaces, feasible_point)
verts = hs.intersections
hull = ConvexHull(verts)
faces = hull.simplices

ax = a3.Axes3D(plt.figure())
ax.dist=10
ax.azim=30
ax.elev=10
ax.set_xlim([0,5])
ax.set_ylim([0,5])
ax.set_zlim([0,5])

triangles = []
for s in faces:
    sq = [
        (verts[s[0], 0], verts[s[0], 1], verts[s[0], 2]),
        (verts[s[1], 0], verts[s[1], 1], verts[s[1], 2]),
        (verts[s[2], 0], verts[s[2], 1], verts[s[2], 2])
    ]
    triangles.append(sq)

new_faces = simplify(triangles)
for sq in new_faces:
    f = a3.art3d.Poly3DCollection([sq])
    f.set_color(colors.rgb2hex(sp.rand(3)))
    f.set_edgecolor('k')
    f.set_alpha(0.1)
    ax.add_collection3d(f)

# # rotate the axes and update
# for angle in range(0, 360):
#     ax.view_init(30, angle)
#     plt.draw()
#     plt.pause(.001)

Note

Upon reflection, the function reordered probably needs some more work. Pretty sure this will break for weird / non-convex shapes, and I am not even 100% sure that it will always work for convex shapes. Rest should be fine though.