Geometry Blender

GeoBlend is a library to smoothly blend geometric objects using the following operations

• Union
• Intersection
• Subtraction
• Global Deformations

The resulting object is an analytic expression f = f(x,y,z) (i.e a closed mathematical formula) such that the resulting geometric object will be defined by the inequality

f(x,y,z) <= 1/2

The surface of this object is the level set

f(x,y,z) == 1/2

We can mesh this level set using algorithms like Marching Cubes or Marching Tetrahedra, and then render it using a graphing system.

Example: Twisted Double Torus in 3-dimensional space

(See Jupyter notebook TwistedDoubleTorus.ipynb for the complete code)

We can create a "twisted double torus" as follows. First, we join two lenticular shaped objects using the code

b1 = Ball3D()
b1.scale(1/2,1/2,1/3)
b1.translate(-1/2,0,0)

b2 = Ball3D()
b2.scale(1/2,1/2,1/3)
b2.translate(1/2,0,0)

obj1 = b1 + b2

The ball b1 represent a ball radius 1 centered at the origin. Then we scale b1 by a factor 1/2 along the x and y plane, and by a factor of 1/3 along the z axis; This produces a lenticular shaped object which is thinner in the z-direction. Next, we translate b1 by an amount of 1/2 in the x-direction. The object b2 is constructed in a similar way, but is translated by an amount of -1/2 in the x-direction. Finally, we construct obj1 as the blended sum (union) of b1 and b2.

We now make two holes to the above object by subtracting two cylinders c1 and c2 to obj1

c1 = Cylinder3D()
c1.scale(1/8,1/8,1)
c1.translate(-1/2,0,0)

c2 = Cylinder3D()
c2.scale(1/8,1/8,1)
c2.translate(1/2,0,0)

obj2 = obj1-(c1+c2)

Finally, we twist one of the holes as follows

theta = pi/2
b2r = b2.rotated(1,0,0,theta)
c2r = c2.rotated(1,0,0,theta)
obj3 = (b1+b2r)-(c1+c2r)

The above code produces b2r and c2r by rotating b2 and c2 an angle of pi/2; using the vector (1,0,0) as the axis of rotation. The object obj3 is shown in the figure below.

Requirements

• Numpy - For the main library
• Sympy - For the main library
• Scikit-image - For plotting implicit surfaces. Installation:
  • Anaconda Python: conda install scikit-image
  • Pip: pip install scikit-image

Installation

Download the folder and open the Jupyter notebooks.