Concept units

Rotating points with a matrix

Basic

Read these first if they are new to you

The idea in one paragraph

To turn a shape around the origin, multiply each of its points by a rotation matrix. The matrix for a turn by an angle θ is built from the cosine and sine of that angle. Its columns are where the two axis arrows land after the turn. A rotation keeps every distance and every angle, so the shape keeps its size and form. In 3D the same idea works around an axis, and the axis stays fixed.

The columns are the turned axis arrows

The matrix unit showed that a matrix's columns tell you where (1, 0) and (0, 1) land. So to build a turn, ask where a turn sends those two arrows.

Turn the arrow (1, 0) counterclockwise by an angle θ (the Greek letter theta). Its tip moves along a circle of radius 1 and stops at (cos θ, sin θ). That is what cosine and sine mean: the x and y of a point on that circle. The sine-waves unit explains them. The arrow (0, 1) starts a quarter turn further on than (1, 0), so it ends a quarter turn further on than (cos θ, sin θ). A quarter turn counterclockwise sends any arrow (a, b) to (−b, a): multiply it by the quarter-turn matrix [[0, −1], [1, 0]] to see this. So (0, 1) lands at (−sin θ, cos θ). Put the two landing spots in as columns:

R(θ) = [[cos θ, −sin θ], [sin θ, cos θ]]

The inner brackets are the rows. This is the rotation matrix. Multiply any point by it and the point turns by θ around the origin. A positive angle turns counterclockwise.

Every point turns by the same angle and keeps its distance from the origin
R = [[0.866, −0.500], [0.500, 0.866]]
The orange tip (0, 2) lands at (−1.000, 1.732).
Try this. The orange dot is the top of the flag's pole, (0, 2). It moves along the dashed circle, so its distance from the origin stays 2. Predict the angle at which the pole points straight left, and what the matrix is then. Slide to check.
Worked example

Turn the point (3, 4) by 90°.

  1. cos 90° = 0 and sin 90° = 1, so R = [[0, −1], [1, 0]].
  2. Row 1 dotted with (3, 4): 0 × 3 + (−1) × 4 = −4.
  3. Row 2 dotted with (3, 4): 1 × 3 + 0 × 4 = 3.
  4. The point lands at (−4, 3). Its distance from the origin was √(9 + 16) = 5, and it is still √(16 + 9) = 5.

Your turn. Turn (2, 0) by 60°. Use cos 60° = 0.5 and sin 60° ≈ 0.866.

Answer

Row 1: 0.5 × 2 + (−0.866) × 0 = 1. Row 2: 0.866 × 2 + 0.5 × 0 = 1.732. The point lands at (1, 1.732), still 2 from the origin.

In 3D, a rotation turns around an axis

In 3D, a turn happens around a line called the axis. Points on the axis stay put. To turn around the z-axis, which points straight up, use the 2D matrix for x and y and leave z alone:

Rz(θ) = [[cos θ, −sin θ, 0], [sin θ, cos θ, 0], [0, 0, 1]]

A point's height does not change; only its position around the axis does. Matrices for turns around the x- and y-axes have the same shape, with the 1 in a different place. For the y-axis the two sines also swap signs:

Ry(θ) = [[cos θ, 0, sin θ], [0, 1, 0], [−sin θ, 0, cos θ]]

Programs that need a shape in many random orientations make one of these matrices from a random angle and multiply every point by it.

This is a simplification

Three details are hidden here. First, the sign convention: some fields measure angles clockwise, which swaps the signs of the sines. Second, in 3D the order of turns matters: turning around x then z gives a different result from z then x. Third, code that stores points as rows of a table writes points @ R.T rather than R @ point.

rot.py: NumPy's sin and cos take radians, so convert degrees first

import numpy as np

def rotation(degrees):
    t = np.radians(degrees)
    return np.array([[np.cos(t), -np.sin(t)],
                     [np.sin(t),  np.cos(t)]])

R = rotation(90)
p = np.array([3.0, 4.0])
q = R @ p
print(np.round(q, 6))
print(np.linalg.norm(p), np.linalg.norm(q))   # same length

print(np.round(rotation(30) @ np.array([1.0, 0.0]), 3))

[-4.  3.]
5.0 5.0
[0.866 0.5  ]

Check yourself

  1. Write the rotation matrix for 180°. What does it do to (2, 5)?

    Answer

    cos 180° = −1 and sin 180° = 0, so R = [[−1, 0], [0, −1]]. It sends (2, 5) to (−2, −5): every point goes to the opposite side of the origin.

  2. You turn a shape by 40° and then by −40°. Where does it end up?

    Answer

    Where it started. The second turn undoes the first.

  3. You turn the 3D point (1, 0, 7) by 90° around the z-axis. Where does it land?

    Answer

    (0, 1, 7). The x and y turn as in 2D, and the height 7 stays the same.

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