linear-algebra-testcase

A small utility program in Python (pygame + numpy) to visualize concepts of linear transformations. Watch 3Blue1Brown for a nice introduction.

There is now an online version of this program using pyodide.

Installation

# clone repository
git clone https://github.com/Bluemi/linear-algebra-testcase.git
cd linear-algebra-testcase

# install requirements (numpy, pygame)
pip3 install -r requirements.txt

# run the programm
python3 ./linear_algebra_testcase/main.py

Consider using a virtual python-environment (eg virtualenvwrapper).

Usage

You can move around in the plane with the mouse. Zoom is controlled with the mouse wheel. If you click on the button on the left top, a menu opens.

The menu shows three sections:

• Object: You can have two kinds of objects: A single vector or a collection of vectors ordered in a circle (or ellipse, if modified). Add these objects by clicking on the plus buttons. You can change the values of a vector by dragging them in the coordinate system.
• Transforms: Here you can add Transformation matrices. The first plus-button creates a two-dimensional matrix. If you want to utilize translation as well you have to use the second plus-button to create a three-dimensional transformation matrix. These matrices are not rendered directly, but can be later used for transformations. To change the values of a matrix, hold an element of the matrix and move the mouse up or down. The bottom row of a 3-dimensional matrix does not change anything (as long as you don't use it for later transformations).
• Transformed: If you want to see the effect of your Transforms use this section. The first button creates a transformed version of one of your objects. To select an object click on the created transform and then on the left side on one of the objects. Then click on the transformed again and click on the transformation you want to use. This only works for 2-dimensional transformation matrices (WIP). The last add button creates a custom-transformation. If you click on the custom transformation a window will pop up, that enables you to write python code. See Custom Transformed for more information. To close the window press Esc. You can remove the last sign with Backspace and everything with Del. As you can see, this editor is very rudimentary (no removal/edit of signs that are not the last sign).

Controls

• To remove any object, transform or transformed hover over the element in the menu on the left side and press Del or Backspace.
• You can toggle between render mode LINE and POINT by hovering over a rendered element on the left side and pressing r.
• You can toggle visibility by hovering over a rendered element on the left side and pressing v.

Custom Transformed

If you create a custom transformed and click on it, a text window appears. Here you can write a python expression. You can use all object names listed above and numpy functions (accessible via np). Objects from above will be presented as numpy-ndarray.

Example: Lets say you have a vector v1 and a 2d transformation matrix T1. You could write T1 @ v1 to apply the transformation matrix on your vector and render the result. The same works for unit circles (replace v1 with u1).

As applying a 3d transformation matrix on a 2d vector is a bit complicated, you can use the special function mm()(matrix-multiplication): mm(T1, v1).

Limitations / Risks

• To evaluate custom-transformations the python builtin eval() is used, which allows arbitrary code execution. For example, you could use exit() as formula to end the program. So be a bit careful.