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In this article I present a present a reimplementation in pure Python of Joseph O'Rourke's incremental 3D convex hull algorithm from his book Computational Geometry in C.

A convex hull in pure Python

This is the second, rather off topic, article on computational geometry in this blog. The previous article presented an implementation of the marching tetrahedrons algorithm.
The goal of this article is to provide object oriented, pythonic code to compute the convex hull of a collection of 3D points. The code is contained in a single Python module that may be downloaded from GitHub.
A sample of how to use this module is shown below, where we create a a roughly spherical cloud of points, calculate its convex hull and print this hull in STL format to stdout. The resulting object is shown in the image (as seen in Blender).

from random import random
from chull import Vector,Hull

sphere=[]
for i in range(2000):
 x,y,z = 2*random()-1,2*random()-1,2*random()-1
 if x*x+y*y+z*z < 1.0:
  sphere.append(Vector(x,y,z))
h=Hull(sphere)
h.Print()

Difference from the original implementation

The original code restricted the coordinates of points to integers, here there is no such restriction. This might result in errors if the coordinates are large.

In this atricle we show a simple implementation of the Marching Tetrahedrons algorithm in Python.

Marching Tetrahedrons

The listing povided below is a straightforward reimplementation in Python of the ideas and code presented by Paul Bourke. The image

shows the result of sampling a simple lobed function (rendered in Blender). I might implement the code directly in Blender but for now we export to an STL file that can be read by almost any 3D package. Note that the exported triangles do not have normals that point in a uniform direction, in fact we do no export any normals at all. You have to recalculate the normals in your 3D package before you can render the resulte with smooth shading.

class Vector: # struct XYZ
 def __init__(self,x,y,z):
  self.x=x
  self.y=y
  self.z=z
 
 def __str__(self):
  return str(self.x)+" "+str(self.y)+" "+str(self.z)
  
class Gridcell: # struct GRIDCELL
 def __init__(self,p,n,val):
  self.p   = p   # p=[8]
  self.n   = n   # n=[8]
  self.val = val # val=[8]

class Triangle: # struct TRIANGLE
 def __init__(self,p1,p2,p3):
  self.p = [p1, p2, p3] # vertices

# return triangle as an ascii STL facet  
 def __str__(self):
  return """facet normal 0 0 0
outer loop
vertex %s
vertex %s
vertex %s
endloop
endfacet"""%(self.p[0],self.p[1],self.p[2])

# return a 3d list of values
def readdata(f=lambda x,y,z:x*x+y*y+z*z,size=5.0,steps=11):
 m=int(steps/2)
 ki = []
 for i in range(steps):
  kj = []
  for j in range(steps):
   kd=[]
   for k in range(steps):
    kd.append(f(size*(i-m)/m,size*(j-m)/m,size*(k-m)/m))
   kj.append(kd)
  ki.append(kj)
 return ki

from math import cos,exp,atan2

def lobes(x,y,z):
 try:
  theta = atan2(x,y)         # sin t = o 
 except:
  theta = 0
 try:
  phi = atan2(z,y)
 except:
  phi = 0
 r = x*x+y*y+z*z
 ct=cos(theta)
 cp=cos(phi)
 return ct*ct*cp*cp*exp(-r/10)
 
def main():

 data = readdata(lobes,5,41)
 isolevel = 0.1

 #print(data)
 
 triangles=[]
 for i in range(len(data)-1):
  for j in range(len(data[i])-1):
   for k in range(len(data[i][j])-1):
    p=[None]*8
    val=[None]*8
    #print(i,j,k)
    p[0]=Vector(i,j,k)
    val[0] = data[i][j][k]
    p[1]=Vector(i+1,j,k)
    val[1] = data[i+1][j][k]
    p[2]=Vector(i+1,j+1,k)
    val[2] = data[i+1][j+1][k]
    p[3]=Vector(i,j+1,k)
    val[3] = data[i][j+1][k]
    p[4]=Vector(i,j,k+1)
    val[4] = data[i][j][k+1]
    p[5]=Vector(i+1,j,k+1)
    val[5] = data[i+1][j][k+1]
    p[6]=Vector(i+1,j+1,k+1)
    val[6] = data[i+1][j+1][k+1]
    p[7]=Vector(i,j+1,k+1)
    val[7] = data[i][j+1][k+1]
   
    grid=Gridcell(p,[],val)
    triangles.extend(PolygoniseTri(grid,isolevel,0,2,3,7))
    triangles.extend(PolygoniseTri(grid,isolevel,0,2,6,7))
    triangles.extend(PolygoniseTri(grid,isolevel,0,4,6,7))
    triangles.extend(PolygoniseTri(grid,isolevel,0,6,1,2))
    triangles.extend(PolygoniseTri(grid,isolevel,0,6,1,4))
    triangles.extend(PolygoniseTri(grid,isolevel,5,6,1,4))
    
 export_triangles(triangles)

def export_triangles(triangles): # stl format
 print("solid points")
 for tri in triangles:
  print(tri)
 print("endsolid points")
 
def t000F(g, iso, v0, v1, v2, v3):
 return []

def t0E01(g, iso, v0, v1, v2, v3):
 return [Triangle(
 VertexInterp(iso,g.p[v0],g.p[v1],g.val[v0],g.val[v1]),
 VertexInterp(iso,g.p[v0],g.p[v2],g.val[v0],g.val[v2]),
 VertexInterp(iso,g.p[v0],g.p[v3],g.val[v0],g.val[v3]))
 ]

def t0D02(g, iso, v0, v1, v2, v3):
 return [Triangle(
 VertexInterp(iso,g.p[v1],g.p[v0],g.val[v1],g.val[v0]),
 VertexInterp(iso,g.p[v1],g.p[v3],g.val[v1],g.val[v3]),
 VertexInterp(iso,g.p[v1],g.p[v2],g.val[v1],g.val[v2]))
 ]

def t0C03(g, iso, v0, v1, v2, v3):
 tri=Triangle(
 VertexInterp(iso,g.p[v0],g.p[v3],g.val[v0],g.val[v3]),
 VertexInterp(iso,g.p[v0],g.p[v2],g.val[v0],g.val[v2]),
 VertexInterp(iso,g.p[v1],g.p[v3],g.val[v1],g.val[v3]))
 return [tri,Triangle(
 tri.p[2],
 VertexInterp(iso,g.p[v1],g.p[v2],g.val[v1],g.val[v2]),
 tri.p[1])
 ]

def t0B04(g, iso, v0, v1, v2, v3):
 return [Triangle(
 VertexInterp(iso,g.p[v2],g.p[v0],g.val[v2],g.val[v0]),
 VertexInterp(iso,g.p[v2],g.p[v1],g.val[v2],g.val[v1]),
 VertexInterp(iso,g.p[v2],g.p[v3],g.val[v2],g.val[v3]))
 ]

def t0A05(g, iso, v0, v1, v2, v3):
 tri = Triangle(
 VertexInterp(iso,g.p[v0],g.p[v1],g.val[v0],g.val[v1]),
 VertexInterp(iso,g.p[v2],g.p[v3],g.val[v2],g.val[v3]),
 VertexInterp(iso,g.p[v0],g.p[v3],g.val[v0],g.val[v3]))
 return [tri,Triangle(
 tri.p[0],
 VertexInterp(iso,g.p[v1],g.p[v2],g.val[v1],g.val[v2]),
 tri.p[1])
 ]

def t0906(g, iso, v0, v1, v2, v3):
 tri=Triangle(
 VertexInterp(iso,g.p[v0],g.p[v1],g.val[v0],g.val[v1]),
 VertexInterp(iso,g.p[v1],g.p[v3],g.val[v1],g.val[v3]),
 VertexInterp(iso,g.p[v2],g.p[v3],g.val[v2],g.val[v3]))
 return [tri,
 Triangle(
 tri.p[0],
 VertexInterp(iso,g.p[v0],g.p[v2],g.val[v0],g.val[v2]),
 tri.p[2])
 ]

def t0708(g, iso, v0, v1, v2, v3):
 return [Triangle(
 VertexInterp(iso,g.p[v3],g.p[v0],g.val[v3],g.val[v0]),
 VertexInterp(iso,g.p[v3],g.p[v2],g.val[v3],g.val[v2]),
 VertexInterp(iso,g.p[v3],g.p[v1],g.val[v3],g.val[v1]))
 ]

trianglefs = {7:t0708,8:t0708,9:t0906,6:t0906,10:t0A05,5:t0A05,11:t0B04,4:t0B04,12:t0C03,3:t0C03,13:t0D02,2:t0D02,14:t0E01,1:t0E01,0:t000F,15:t000F}

def PolygoniseTri(g, iso, v0, v1, v2, v3):
 triangles = []

 #   Determine which of the 16 cases we have given which vertices
 #   are above or below the isosurface

 triindex = 0;
 if g.val[v0] < iso: triindex |= 1
 if g.val[v1] < iso: triindex |= 2
 if g.val[v2] < iso: triindex |= 4
 if g.val[v3] < iso: triindex |= 8

 return trianglefs[triindex](g, iso, v0, v1, v2, v3)

def VertexInterp(isolevel,p1,p2,valp1,valp2):
 if abs(isolevel-valp1) < 0.00001 :
  return(p1);
 if abs(isolevel-valp2) < 0.00001 :
  return(p2);
 if abs(valp1-valp2) < 0.00001 :
  return(p1);
 mu = (isolevel - valp1) / (valp2 - valp1)
 return Vector(p1.x + mu * (p2.x - p1.x), p1.y + mu * (p2.y - p1.y), p1.z + mu * (p2.z - p1.z))

if __name__ == "__main__":
 main()

Blender 2.6 add-on tutorial

3D Convex hull in Python

A convex hull in pure Python

Difference from the original implementation

Marching Tetrahedrons in Python

Marching Tetrahedrons

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