Learning Objectives

Key learning outcome Improve the understanding of the mathematical descriptions of 3D Motion, by testing implementations in Python.

Secondary outcomes if you have time, it is worth browsing different libraries and frameworks to build, visualise, and animate scenes using 3D objects.

Briefing

Recap: a simple object

Remember, last week we worked with this data structure in Python:

In [1]: print(vertices)
[[-1.   0.5  0.5]
 [ 1.   0.5  0.5]
 [ 0.  -0.5  0.5]
 [-1.   0.5  0.5]
 [ 1.   0.5  0.5]
 [ 0.   0.5 -0.5]
 [-1.   0.5  0.5]
 [ 0.  -0.5  0.5]
 [ 0.   0.5 -0.5]
 [ 1.   0.5  0.5]
 [ 0.  -0.5  0.5]
 [ 0.   0.5 -0.5]]

The rows are points in 3D. Note that there are only four distinct points. If we divide the matrix into sets of three rows, each triplet defines a triangle. These four triangles form the faces of an irregular tetrahedron.

This is a standard way to define a 3D object. More complex objects need more triangles.

Note that the textbook have vertices as column vectors, while we here use row vectors. This means that we need to transpose matrices when we translate textbook formulæ to python formulæ.

Consider, for instance a rotation matrix

\[ \begin{align} R = \begin{bmatrix} \cos(\pi/6) & -\sin(\pi/6) & 0 \\ \sin(\pi/6) & \cos(\pi/6) & 0 \\ 0 & 0 & 1 \end{bmatrix} \end{align} \]

from numpy import *
R = array( [[ cos(pi/6), -sin(pi/6), 0], [ sin(pi/6), cos(pi/6), 0], [0,0,1]] )

This matrix is orthonormal, which we can check:

matmul( R, transpose(R) )

To rotate a vector \(\vec{v}\), we would calculate \(\vec{u}= R\cdot \vec{v}\), where \(\vec{u}\) and \(\vec{v}\) are column vectors. If you have a shape \(V\) where the columns are points, it could be rotated as \(U=R\cdot V\). In python this has to become

V = transpose(vertices)
U = matmul(R, V)
newvertices = transpose(U)

Homogeneous co-ordinates

STL files and the STL library

Exercise

Rotations and translations

Homogenous Coordinates

Motion defined by homogenous matrix

STL files and the STL library

• Load Model
• View Model
• Change Model
• Save Model

Additional Materials

• Other relevant python libraries