Rotation Coordinates Calculator
The Rotation Coordinates Calculator is an essential tool designed to compute the new coordinates of a point after it has been rotated around the origin. This calculation is a fundamental aspect of various applications ranging from designing animations in computer graphics to solving geometric problems in engineering.
Formula of Rotation Coordinates Calculator
At the heart of this calculator is a simple yet powerful formula:
x' = x * cos(θ) - y * sin(θ) y' = x * sin(θ) + y * cos(θ)
Here, x’ and y’ represent the new coordinates after rotation. The original coordinates of the point are denoted by x and y, while θ stands for the angle of rotation expressed in radians.
Table for General Terms
To facilitate easier calculations, here is a table that includes common angles with their corresponding sine and cosine values:
| Angle (Degrees) | Angle (Radians) | Cosine | Sine |
|---|---|---|---|
| 0 | 0 | 1 | 0 |
| 30 | π/6 | √3/2 | 1/2 |
| 45 | π/4 | √2/2 | √2/2 |
| 60 | π/3 | 1/2 | √3/2 |
| 90 | π/2 | 0 | 1 |
| …and so on. |
Example of Rotation Coordinates Calculator
Let’s consider an example: rotating a point (2,3) by 45 degrees (π/4 radians). Using the formula, we calculate the new coordinates:
x' = 2 * cos(π/4) - 3 * sin(π/4) y' = 2 * sin(π/4) + 3 * cos(π/4)
This results in x’ ≈ 0.707 and y’ ≈ 3.535.
Most Common FAQs
A1: Radians provide a direct correlation between the angle and the arc length, making calculations more straightforward in many geometric and trigonometric contexts.
A2: It helps in determining the new positions of graphic elements after rotation, which is crucial for animation and modeling.
A3: While the provided formula is for 2D rotation, similar principles apply in 3D, though with more complex formulas involving the z-axis.