Matrices as transformations

Multiplying a point (x, y) by a 2×2 matrix moves it to a new point. Doing this to every point transforms the whole plane.

Read the columns

The first column of the matrix is where (1, 0) lands; the second column is where (0, 1) lands. Every other point follows, because (x, y) = x·(1, 0) + y·(0, 1).

The example moves the corners of the unit square with each one.

from math import cos, sin, radians

def apply(m, p):
    x = m[0][0] * p[0] + m[0][1] * p[1]
    y = m[1][0] * p[0] + m[1][1] * p[1]
    return (round(x, 3) + 0.0, round(y, 3) + 0.0)

square = [(0, 0), (1, 0), (1, 1), (0, 1)]
t = radians(90)
moves = [("rotate 90°", [[cos(t), -sin(t)], [sin(t), cos(t)]]),
         ("scale", [[2, 0], [0, 3]]),
         ("shear", [[1, 1], [0, 1]])]
for name, m in moves:
    print(name, [apply(m, p) for p in square])

Combining moves

To rotate first and then scale, compute S·(R·p) = (S·R)·p. The matrix of the combined move is the product, written right to left, and order matters: rotating then scaling differs from scaling then rotating.

Where it is used

Every 3-D game and drawing program stores positions as vectors and moves them with matrices. Image processing, robotics arms and computer vision use the same idea.

Common mistakes

• Multiplying in reading order. The move applied first goes on the right.
• Forgetting that sin and cos in Python take radians: use math.radians(90), not 90.

Mark lesson complete