Implement a 2D rotation transformation using a rotation matrix to rotate a list of 2D points by a given angle $\theta$ in degrees. The goal is to apply this transformation to each point and return the list of rotated points.

In geometric transformations, rotation is a fundamental concept that involves changing the orientation of an object in a 2D or 3D space. The rotation matrix is a mathematical representation of this transformation, which can be used to rotate points, vectors, and other geometric entities. The rotation matrix $R$ for a 2D rotation is given by:

\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}

To apply this transformation to a point $[x, y]$ , we can use the following steps:

Convert the angle from degrees to radians.
Compute the cosine and sine of the angle.
Apply the rotation matrix to the point using the formulas: $x' = x\cos\theta - y\sin\theta$ and $y' = x\sin\theta + y\cos\theta$ . The key formulas are:

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

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

This technique is widely used in computer vision and image processing applications.

The rotation angle $\theta$ is given as 90 degrees, which in radians is $\frac{\pi}{2}$ , so we calculate $\cos\theta = \cos(\frac{\pi}{2}) = 0$ and $\sin\theta = \sin(\frac{\pi}{2}) = 1$ .
For the point $[1, 0]$ , we apply the rotation transformation: $x' = 1\cos(\frac{\pi}{2}) - 0\sin(\frac{\pi}{2}) = 0$ and $y' = 1\sin(\frac{\pi}{2}) + 0\cos(\frac{\pi}{2}) = 1$ , resulting in $[0.0, 1.0]$ .
For the point $[0, 1]$ , we apply the rotation transformation: $x' = 0\cos(\frac{\pi}{2}) - 1\sin(\frac{\pi}{2}) = -1$ and $y' = 0\sin(\frac{\pi}{2}) + 1\cos(\frac{\pi}{2}) = 0$ , resulting in $[-1.0, 0.0]$ .
The final output is the list of rotated points, each coordinate rounded to 4 decimal places: $[[0.0, 1.0], [-1.0, 0.0]]$ .

theta is in degrees
points is a list of [x, y] pairs
Return list of [x', y'] pairs rounded to 4 decimal places
Use math.cos, math.sin, math.radians

Editor

Python 3.13.1

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📘

2D Rotation Matrix

EasyGeometric Transformations

\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}

To apply this transformation to a point $[x, y]$ , we can use the following steps:

Convert the angle from degrees to radians.
Compute the cosine and sine of the angle.
Apply the rotation matrix to the point using the formulas: $x' = x\cos\theta - y\sin\theta$ and $y' = x\sin\theta + y\cos\theta$ . The key formulas are:

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

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

This technique is widely used in computer vision and image processing applications.

Example:

Input:

theta = 90
points = [[1, 0], [0, 1]]

Output:

[[0.0, 1.0], [-1.0, 0.0]]

Reasoning:

The rotation angle $\theta$ is given as 90 degrees, which in radians is $\frac{\pi}{2}$ , so we calculate $\cos\theta = \cos(\frac{\pi}{2}) = 0$ and $\sin\theta = \sin(\frac{\pi}{2}) = 1$ .
For the point $[1, 0]$ , we apply the rotation transformation: $x' = 1\cos(\frac{\pi}{2}) - 0\sin(\frac{\pi}{2}) = 0$ and $y' = 1\sin(\frac{\pi}{2}) + 0\cos(\frac{\pi}{2}) = 1$ , resulting in $[0.0, 1.0]$ .
For the point $[0, 1]$ , we apply the rotation transformation: $x' = 0\cos(\frac{\pi}{2}) - 1\sin(\frac{\pi}{2}) = -1$ and $y' = 0\sin(\frac{\pi}{2}) + 1\cos(\frac{\pi}{2}) = 0$ , resulting in $[-1.0, 0.0]$ .
The final output is the list of rotated points, each coordinate rounded to 4 decimal places: $[[0.0, 1.0], [-1.0, 0.0]]$ .

Constraints:

theta is in degrees
points is a list of [x, y] pairs
Return list of [x', y'] pairs rounded to 4 decimal places
Use math.cos, math.sin, math.radians

Editor

Python 3.13.1

Test Results

0/0

Run code to see test results.