2D Coordinate Transformation Calculator
This calculator helps you transform a 2D point (X, Y) by applying scaling, rotation, and translation operations. Understand how geometric transformations affect coordinates in a 2D plane, essential for computer graphics, game development, and robotics.
Transformed Coordinates:
Enter values and click 'Calculate' to see the transformed coordinates.
Understanding 2D Coordinate Transformations
2D coordinate transformations are fundamental operations in various fields, including computer graphics, game development, robotics, and image processing. They allow us to change the position, orientation, and size of objects or points within a two-dimensional space. This calculator helps you visualize the effect of combining three primary transformations: scaling, rotation, and translation.
1. Scaling
Scaling changes the size of an object or the distance of a point from the origin. It's applied by multiplying the original coordinates by respective scale factors. If Sx is the scale factor for the X-axis and Sy for the Y-axis, an original point (X, Y) becomes (X * Sx, Y * Sy) after scaling. A scale factor greater than 1 enlarges, less than 1 shrinks, and 1 leaves it unchanged.
2. Rotation
Rotation changes the orientation of an object or point around a fixed origin (or another specified pivot point, though this calculator uses the origin). The rotation angle is typically measured counter-clockwise from the positive X-axis. For a point (X, Y) rotated by an angle θ (in radians), the new coordinates (X', Y') are calculated as:
X' = X * cos(θ) - Y * sin(θ)Y' = X * sin(θ) + Y * cos(θ)
It's crucial to convert degrees to radians for trigonometric functions (radians = degrees * π / 180).
3. Translation
Translation simply moves an object or point from one location to another without changing its orientation or size. It's applied by adding offset values to the original coordinates. If Tx is the translation offset for the X-axis and Ty for the Y-axis, an original point (X, Y) becomes (X + Tx, Y + Ty) after translation.
Order of Operations
The order in which these transformations are applied is critical. In this calculator, the standard order is followed: Scaling first, then Rotation, and finally Translation. This sequence is common in many graphics pipelines and ensures consistent results.
How to Use the Calculator
- Original X/Y Coordinate: Enter the starting X and Y values of the point you wish to transform.
- Scale Factor X/Y: Input the desired scaling factor for the X and Y axes.
- Rotation Angle (Degrees): Enter the angle in degrees by which you want to rotate the point. Positive values rotate counter-clockwise.
- Translation X/Y Offset: Provide the values by which you want to shift the point along the X and Y axes.
- Click "Calculate Transformed Coordinates" to see the final position of your point after all operations.
Example Transformation
Let's say we have an original point at (10, 5).
- We want to scale it by
Sx=2,Sy=1.5. - Then rotate it by
45 degrees. - Finally, translate it by
Tx=3,Ty=-2.
Step 1: Scaling
X_scaled = 10 * 2 = 20Y_scaled = 5 * 1.5 = 7.5- Point after scaling:
(20, 7.5)
Step 2: Rotation (45 degrees = 0.7854 radians)
X_rotated = 20 * cos(0.7854) - 7.5 * sin(0.7854)X_rotated = 20 * 0.7071 - 7.5 * 0.7071 = 14.142 - 5.303 = 8.839Y_rotated = 20 * sin(0.7854) + 7.5 * cos(0.7854)Y_rotated = 20 * 0.7071 + 7.5 * 0.7071 = 14.142 + 5.303 = 19.445- Point after rotation:
(8.839, 19.445)
Step 3: Translation
X_final = 8.839 + 3 = 11.839Y_final = 19.445 + (-2) = 17.445- Final transformed point:
(11.839, 17.445)
Use the calculator above with these values to verify the results!