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

The following would be my version of a solution. It is similar to @Paul's solution in that it takes the triangles, groups them by face they belong to and joins them to a single face.

The difference would mainly be that this solution does not use nx or simpy. Many of the necessary operations are performed by reindexing, extensive use of unique and some linear algebra.
The order of the vertices of the final faces is determined by ConvexHull. I think this should not be a limitation, as (I think that) any half space intersection should result in convex shapes only. However, I also added another method which can be used if the shapes are not convex (based on the idea from this question).

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

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)
simplices = hull.simplices

org_triangles = [verts[s] for s in simplices]

class Faces():
    def __init__(self,tri, sig_dig=12, method="convexhull"):
        self.method=method
        self.tri = np.around(np.array(tri), sig_dig)
        self.grpinx = list(range(len(tri)))
        norms = np.around([self.norm(s) for s in self.tri], sig_dig)
        _, self.inv = np.unique(norms,return_inverse=True, axis=0)

    def norm(self,sq):
        cr = np.cross(sq[2]-sq[0],sq[1]-sq[0])
        return np.abs(cr/np.linalg.norm(cr))

    def isneighbor(self, tr1,tr2):
        a = np.concatenate((tr1,tr2), axis=0)
        return len(a) == len(np.unique(a, axis=0))+2

    def order(self, v):
        if len(v) <= 3:
            return v
        v = np.unique(v, axis=0)
        n = self.norm(v[:3])
        y = np.cross(n,v[1]-v[0])
        y = y/np.linalg.norm(y)
        c = np.dot(v, np.c_[v[1]-v[0],y])
        if self.method == "convexhull":
            h = ConvexHull(c)
            return v[h.vertices]
        else:
            mean = np.mean(c,axis=0)
            d = c-mean
            s = np.arctan2(d[:,0], d[:,1])
            return v[np.argsort(s)]

    def simplify(self):
        for i, tri1 in enumerate(self.tri):
            for j,tri2 in enumerate(self.tri):
                if j > i: 
                    if self.isneighbor(tri1,tri2) and \
                       self.inv[i]==self.inv[j]:
                        self.grpinx[j] = self.grpinx[i]
        groups = []
        for i in np.unique(self.grpinx):
            u = self.tri[self.grpinx == i]
            u = np.concatenate([d for d in u])
            u = self.order(u)
            groups.append(u)
        return groups


f = Faces(org_triangles)
g = f.simplify()

ax = a3.Axes3D(plt.figure())

colors = list(map("C{}".format, range(len(g))))

pc = a3.art3d.Poly3DCollection(g,  facecolor=colors, 
                                   edgecolor="k", alpha=0.9)
ax.add_collection3d(pc)

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