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Rotation Around A Point Calculator

Rotation Formula:

\[ \begin{cases} x' = x_c + (x - x_c) \cdot \cos(\theta) - (y - y_c) \cdot \sin(\theta) \\ y' = y_c + (x - x_c) \cdot \sin(\theta) + (y - y_c) \cdot \cos(\theta) \end{cases} \]

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1. What Is Rotation Around A Point?

Rotation around a point is a geometric transformation that moves a point around a fixed center by a specified angle while maintaining the distance from the center. This operation is fundamental in computer graphics, robotics, and physics.

2. How The Rotation Formula Works

The rotation formula consists of three steps:

\[ \begin{cases} x' = x_c + (x - x_c) \cdot \cos(\theta) - (y - y_c) \cdot \sin(\theta) \\ y' = y_c + (x - x_c) \cdot \sin(\theta) + (y - y_c) \cdot \cos(\theta) \end{cases} \]

Where:

Explanation: The formula translates the point to the origin, performs the rotation, then translates back to the original center.

3. Applications of Point Rotation

Details: Point rotation is used in computer graphics for object transformations, in robotics for joint movements, in physics for angular motion calculations, and in game development for character movements.

4. Using The Calculator

Tips: Enter the point coordinates, rotation center coordinates, and rotation angle in degrees. The calculator will automatically convert the angle to radians and compute the new position.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between rotation around origin and arbitrary point?
A: Rotation around origin is simpler (xc=0, yc=0). Rotation around arbitrary point requires the full translation-rotation-translation process.

Q2: Does the rotation angle direction matter?
A: Yes, positive angles typically represent counter-clockwise rotation, while negative angles represent clockwise rotation.

Q3: Can this be extended to 3D rotation?
A: Yes, but 3D rotation requires more complex mathematics using rotation matrices or quaternions.

Q4: How precise are the calculations?
A: The calculator provides results rounded to 4 decimal places, sufficient for most applications.

Q5: What if I need multiple rotations?
A: You can chain rotations by using the output of one rotation as input to the next rotation calculation.

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