4.3. Rotation

Rotates a point by angle $\theta$ about the origin.

Rotation in a 2D space involves turning a point around a fixed center point, which is usually the origin of the coordinate system (0, 0). The angle of rotation, denoted by $\theta$, represents how much we are turning the point. If $\theta$ is positive, we typically rotate the point counterclockwise; if it's negative, we rotate clockwise. This is a fundamental transformation in geometric modeling, as it allows us to change the orientation of shapes.

Think of the rotation like the minute hand on a clock. As the minute hand moves away from the 12 o'clock position, it is rotating counterclockwise. If you imagine a point on the tip of the minute hand, that point is rotating around the center of the clock (the origin). Depending on how many minutes have passed, the angle $\theta$ changes, and thus the position of the point changes.

Transformation Matrix:

Applied as:

The rotation matrix for a 2D point is a $2 \times 2$ matrix that contains trigonometric functions. This matrix, denoted as $R(\theta)$, is used to rotate a point characterized by its coordinates (x, y). When you multiply this matrix by the coordinates, it effectively computes the new position of the point after rotation. Inside the matrix, $\cos(\theta)$ and $\sin(\theta)$ determine how much of the point's original position contributes to its new position based on the angle of rotation.

Imagine spinning a wheel. Each point on the edge of the wheel represents a position (x, y). The angle $\theta$ represents how far the wheel has turned. The rotation matrix calculates where that point ends up after the wheel has turned. So if you have a point at (1, 0) on the edge and you spin the wheel $90$ degrees counterclockwise, the matrix tells you that this point will move to (0, 1) — it’s following the circular path around the center.

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