Rotate 3D Point Around Z-Axis

3D Geometry

Medium

Problem

Theory

Rotation around the Z-axis spins a point or vector in the XY-plane while keeping the Z-coordinate unchanged. Looking down the positive Z-axis (toward negative Z), a positive angle rotates counterclockwise.

This is one of the three fundamental rotations in 3D space, along with rotations around X and Y axes.

The Rotation Matrix

The 3x3 rotation matrix for angle \theta around the Z-axis is:

R_z(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix}

To rotate a point \mathbf{p} = (x, y, z)^T:

\mathbf{p}' = R_z(\theta) \mathbf{p}

Deriving the Formula

Consider a point (x, y) in the XY-plane. In polar coordinates:

x = r\cos\phi, \quad y = r\sin\phi

where r is the distance from origin and \phi is the current angle.

After rotating by \theta, the new angle is \phi + \theta:

x' = r\cos(\phi + \theta) = r(\cos\phi\cos\theta - \sin\phi\sin\theta) = x\cos\theta - y\sin\theta

y' = r\sin(\phi + \theta) = r(\sin\phi\cos\theta + \cos\phi\sin\theta) = x\sin\theta + y\cos\theta

The Z-coordinate is unchanged: z' = z

The Expanded Equations

For a point (x, y, z) rotated by angle \theta around Z:

x' = x\cos\theta - y\sin\theta

y' = x\sin\theta + y\cos\theta

z' = z

Or in matrix form:

\begin{pmatrix} x' \\ y' \\ z' \end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix}

Worked Example: 90 Degree Rotation

Point: (1, 0, 5)

Angle: \theta = 90° = \frac{\pi}{2}

Trigonometric values:

• \cos(90°) = 0
• \sin(90°) = 1

Calculation:

x' = 1 \cdot 0 - 0 \cdot 1 = 0

y' = 1 \cdot 1 + 0 \cdot 0 = 1

z' = 5

Result: (0, 1, 5)

The point moved from the positive X-axis to the positive Y-axis.

Worked Example: 45 Degree Rotation

Point: (2, 0, 3)

Angle: \theta = 45° = \frac{\pi}{4}

Trigonometric values:

• \cos(45°) = \frac{\sqrt{2}}{2} \approx 0.707
• \sin(45°) = \frac{\sqrt{2}}{2} \approx 0.707

Calculation:

x' = 2 \cdot 0.707 - 0 \cdot 0.707 = 1.414

y' = 2 \cdot 0.707 + 0 \cdot 0.707 = 1.414

z' = 3

Result: (1.414, 1.414, 3)

Worked Example: General Point

Point: (3, 4, 2)

Angle: \theta = 30° = \frac{\pi}{6}

Trigonometric values:

• \cos(30°) = \frac{\sqrt{3}}{2} \approx 0.866
• \sin(30°) = 0.5

Calculation:

x' = 3 \cdot 0.866 - 4 \cdot 0.5 = 2.598 - 2 = 0.598

y' = 3 \cdot 0.5 + 4 \cdot 0.866 = 1.5 + 3.464 = 4.964

z' = 2

Result: (0.598, 4.964, 2)

Sign Convention

Positive angle (counterclockwise):

Looking down the Z-axis (from positive Z toward origin):

• Positive X rotates toward positive Y
• Positive Y rotates toward negative X

Negative angle (clockwise):

The opposite direction. Equivalent to rotating by -\theta.

Special Angles

\theta = 0°: No rotation, identity matrix

\theta = 90°: (x, y, z) \to (-y, x, z)

\theta = 180°: (x, y, z) \to (-x, -y, z)

\theta = 270° or -90°: (x, y, z) \to (y, -x, z)

\theta = 360°: Back to original, identity

Properties of Rotation Matrices

Orthogonal:

R_z^T R_z = I

Determinant equals 1:

\det(R_z) = \cos^2\theta + \sin^2\theta = 1

Inverse is transpose:

R_z^{-1}(\theta) = R_z^T(\theta) = R_z(-\theta)

Preserves length:

||R_z \mathbf{v}|| = ||\mathbf{v}||

Composing Rotations

Rotating by \theta_1 then \theta_2 around Z:

R_z(\theta_1 + \theta_2) = R_z(\theta_2) R_z(\theta_1)

Note: Matrix multiplication is right to left, but angles add.

For rotation around Z, order does not matter since all rotations share the same axis.

Rotation Around Other Axes

Around X-axis:

R_x(\theta) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{pmatrix}

Around Y-axis:

R_y(\theta) = \begin{pmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{pmatrix}

Note: R_y has opposite sign pattern due to the cyclic nature of cross products.

Euler Angles

Any 3D rotation can be decomposed into three rotations around coordinate axes.

ZYX (aerospace):

R = R_z(\psi) R_y(\theta) R_x(\phi)

XYZ (common in graphics):

R = R_x(\phi) R_y(\theta) R_z(\psi)

The order matters! Different orders give different rotations.

Axis-Angle to Matrix

For rotation of angle \theta around Z-axis specifically, we have the matrix above.

For arbitrary axis \hat{\mathbf{k}}, use Rodrigues' rotation formula:

R = I + (\sin\theta)K + (1 - \cos\theta)K^2

where K is the skew-symmetric matrix of \hat{\mathbf{k}}.

Rotating Multiple Points

To rotate many points efficiently:

1. Precompute \cos\theta and \sin\theta once
2. Apply the formulas to each point
3. Or use matrix-vector multiplication with batched operations

For n points, this is O(n) after constant-time setup.

Homogeneous Coordinates

In 4x4 homogeneous form (for combining with translations):

R_z^{4\times4}(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta & 0 & 0 \\ \sin\theta & \cos\theta & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}

This can be composed with translation matrices.

Numerical Stability

Near-zero angles:

When \theta \approx 0:

• \cos\theta \approx 1
• \sin\theta \approx \theta

Use Taylor series for better precision if needed.

Accumulated rotations:

After many small rotations, the matrix may drift from being perfectly orthogonal. Re-orthogonalize periodically.

Applications

3D Graphics:

• Camera rotation
• Object transformation
• Animation systems

Robotics:

• Joint rotations
• End-effector orientation
• Path planning

Computer Vision:

• Image rotation
• Camera calibration
• 3D reconstruction

Physics Simulation:

• Rigid body dynamics
• Angular momentum

Quaternion Alternative

Rotations can also be represented as quaternions:

q = \cos\frac{\theta}{2} + \sin\frac{\theta}{2}(0i + 0j + 1k)

For Z-axis rotation, only the scalar and k components are non-zero.

Quaternions avoid gimbal lock and interpolate smoothly.

Examples

Hints

1. Read coordinates with values[..., 0], values[..., 1], and values[..., 2].
2. Stack the transformed coordinates with np.stack(..., axis=-1).

Requirements

• Apply the rotation to every point without looping over batch rows
• Leave each z-coordinate unchanged
• Accept shape (3,) or (N, 3)
• Return a NumPy array with the input shape

Constraints

• theta is measured in radians
• Use NumPy only

Starter Code

import numpy as np

def rotate_around_z(points: list, theta: float) -> np.ndarray:
    """
    Returns a NumPy array with the same shape as points.
    """
    # Write code here
    pass

Test Cases