The concept of coordinate rotation involves rotating points in a Cartesian plane by specific angles. These angles include 90°, 180°, 270°, and 360° around the origin (0,0). Each of these rotations has distinct effects on the coordinates of a point. Below is a breakdown of what happens during each of these rotations.
| Rotation around Origin (0,0) | Rule | Example |
| 90° Counterclockwise Rotation | (x , y) → (-y , x) | (3, 4) → (-4, 3) |
| 90° Clockwise Rotation | (x , y) → (y , -x) | (3, 4) → (4, -3) |
| 180° Rotation | (x, y) → (-x, -y) | (3, 4) → (-3, -4) |
| 270° Counterclockwise Rotation | (x, y) → (y, -x) | (3, 4) → (4, -3) |
| 270° Clockwise Rotation | (x, y) → (-y, x) | (3, 4) → (-4, 3) |
| 360° Rotation | (x, y) → (x, y) | (3, 4) → (3, 4) |
Do you want to learn more? 🙋 The concept of coordinate rotation can be extended beyond just rotating around the origin (0,0). It also includes rotating around any arbitrary point (h, k) in the Cartesian plane. To rotate a point (x, y) by an angle (θ) around the point (h, k), follows 3 steps.
Step 1. Translate the point (x , y) → (x’ , y’)
Step 2. Rotate the point (x’ , y’) → (x” , y”)
Step 3. Translate back (x” , y”) → (x”’ , y”’)
| 3 Step Process | Example |
| 1. Translate the point (x’, y’) = (x – h, y – k) 2. Rotate the point a. Counterclockwise x” → x’cos(θ) – y’ sin (θ) y” → y’cos(θ) + x’ sin (θ) b. Clockwise x” → x’cos(θ) + y’ sin (θ) y” → y’cos(θ) – x’ sin (θ) 3.Translate back (x”’, y”’) = (x” + h, y” + k) | Rotate (5, 1) around the point (2, 3) by 90° counterclockwise Step 1 : (5, 1) → (5 – 2, 1 – 3) (x’ , y’) = (3, -2) Step 2 : x” → 3 * cos (90) – (-2) * sin (90) y” → (-2) * cos (90) + 3 * sin (90) (x” , y”) = (2 , 3) Step 3 : (2 , 3) → (2 + 2, 3 + 3) (x”’, y”’) = (4, 6) |
Applications
- Robotics: For path planning where rotations around a robot’s center must be calculated.
- Computer Graphics: When objects need to be rotated around specific points for animations and rendering.
- Geometric Transformations: In computer-aided design (CAD) and modeling, where elements may need to be rotated around different pivot points.
Understanding these specific rotations allows for better navigation of points and figures within a Cartesian coordinate system. This understanding is essential in practical applications across many scientific and engineering fields.
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