The library geometry-blender is designed to smoothly blend geometric objects using the following operations

• Union
• Intersection
• Subtraction
• Global Deformations

Solid objects in 3-dimensional are represented using smooth mathematical functions f = f(x,y,z), so that a point (x,y,z) belongs to the object if and only if it satisfies the inequality

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

The border of such an object is a surface that consists of set of points (x,y,z) that satisfy the equality

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

We can mesh this level set using algorithms like Marching Cubes or Marching Tetrahedra, and then render into the screen by using a graphing system (like plotly).

(See Jupyter notebook DoubleTorus.ipynb )

We can create a 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. 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 (see Figure below)

We now construct a new object obj2 by making two holes to obj1. We do this by subtracting two cylinders c1 and c2 from 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)

We can twist one of the of handles of the double torus 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 constructs 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 resulting object obj3 is shown in the figure below.

To avoid package conflicts it is recommended to create a Python Virtual Environment for installation. This can be done at a terminal a follows

• Anaconda : conda create --name geometry python=3

We then need to activate this environment as follows:

• Anaconda : conda activate geometry

The packages numpy, sympy and scikit-image need to be present in the virtual environment. This can be achieved by typing (at a terminal)

• Anaconda : conda install numpy sympy scikit-image

Clone the repository by typing

git clone https://github.com/valerocar/geometry-blender.git,

To be able to execute the Jupyter Notebooks in demos you will need to install the plotly graphing library as follows:

• Anaconda Python: conda install -c plotly plotly

![Animation]