2D Rotation Calculator

Calculate the new coordinates of a point after a 2D rotation around the origin.

Enter the initial coordinates of the point and the rotation angle to find its new position. This tool is essential for geometry, computer graphics, and physics.

Examples

Click on an example to load its values into the calculator.

90° Counter-Clockwise Rotation

degrees

Rotating a point (10, 5) by 90 degrees counter-clockwise.

Point: (10, 5)

Angle: 90 degrees

180° Rotation

degrees

Rotating a point (3, 4) by 180 degrees.

Point: (3, 4)

Angle: 180 degrees

45° Clockwise Rotation

degrees

Rotating a point (8, 0) by -45 degrees clockwise.

Point: (8, 0)

Angle: -45 degrees

Rotation by π/2 Radians

radians

Rotating a point (2, -2) by π/2 radians (equivalent to 90°).

Point: (2, -2)

Angle: 1.570796 radians

Other Titles
Understanding 2D Rotation: A Comprehensive Guide
Explore the principles of 2D rotation, its mathematical foundation, and its wide-ranging applications in technology and science.

What is 2D Rotation? Core Concepts

  • Understanding the transformation of points in a 2D plane
  • The role of the origin as the center of rotation
  • Differentiating between clockwise and counter-clockwise rotation
2D rotation is a fundamental geometric transformation that moves a point or an object around a fixed center, known as the center of rotation. In most standard cases, this center is the origin (0,0) of the Cartesian coordinate system. The transformation is defined by an angle, which specifies the amount of rotation. A positive angle typically corresponds to a counter-clockwise rotation, while a negative angle indicates a clockwise rotation. This process changes the coordinates of the point but preserves its distance from the center of rotation.
The Rotation Formulas
To find the new coordinates (x', y') of a point (x, y) after a counter-clockwise rotation by an angle θ around the origin, we use the following trigonometric formulas:
x' = x cos(θ) - y sin(θ)
y' = x sin(θ) + y cos(θ)
These equations are the bedrock of 2D rotation and are derived from applying trigonometry to the geometry of the rotation.

Fundamental Rotation Examples

  • Rotating point (1, 0) by 90° results in (0, 1)
  • Rotating point (2, 3) by 180° results in (-2, -3)
  • The distance from the origin remains constant after rotation.

Step-by-Step Guide to Using the Rotation Calculator

  • Entering the initial point's coordinates accurately
  • Specifying the rotation angle and its unit
  • Interpreting the calculated new coordinates
Our Rotation Calculator simplifies the process of finding a point's new location after rotation. Follow these steps for an accurate calculation.
Input Guidelines:
  • Initial Point Coordinates (X, Y): Enter the starting x and y values of your point in their respective fields. These can be positive, negative, or zero.
  • Rotation Angle (θ): Input the angle of rotation. Use a positive value for counter-clockwise rotation and a negative value for clockwise rotation.
  • Angle Unit: Select whether the angle you entered is in 'Degrees' or 'Radians'. This is a critical step as the trigonometric functions depend on this unit. The calculator handles the conversion internally.
Performing the Calculation:
Once all inputs are provided, click the 'Calculate Rotation' button. The tool will apply the rotation formulas and instantly display the new coordinates (x', y').
If you need to start over, the 'Reset' button will clear all input fields and results.

Practical Usage Examples

  • Input: X=5, Y=2, Angle=90, Unit=Degrees → Output: X'=-2, Y'=5
  • Input: X=1, Y=1, Angle=-45, Unit=Degrees → Output: X'≈1.414, Y'≈0
  • Input: X=4, Y=3, Angle=3.14159 (π), Unit=Radians → Output: X'=-4, Y'=-3

Real-World Applications of 2D Rotation

  • Computer graphics and video game development
  • Robotics and mechanical engineering
  • Physics simulations and data visualization
2D rotation is not just an abstract mathematical concept; it's a practical tool used in numerous fields.
Computer Graphics and Animation:
  • Object Manipulation: In design software like Adobe Illustrator or CAD programs, users rotate objects using these exact mathematical principles.
  • Game Development: Characters, cameras, and projectiles in 2D games are constantly rotated to face different directions or follow paths.
  • UI/UX Design: Animating icons, loading spinners, and other interface elements often involves rotation to create dynamic and engaging user experiences.
Engineering and Physics:
  • Robotics: The movement of a robotic arm's joints is calculated using rotation matrices, which are an extension of these 2D formulas.
  • Physics Engines: Simulating the motion of celestial bodies, spinning objects, or analyzing forces on a rotated reference frame requires rotation calculations.
  • Navigation Systems: On-board computers in airplanes and ships use rotation to adjust their orientation based on sensor readings.

Industry Applications

  • A sprite in a video game rotating to aim at a target.
  • A weather map being rotated to align with a different geographical orientation.
  • Simulating the orbit of a planet around its star.

Common Misconceptions and Key Insights

  • Rotation vs. Translation: Understanding the difference
  • The importance of the center of rotation
  • Correctly handling angle units (degrees vs. radians)
While rotation seems straightforward, several common misunderstandings can lead to errors. Clarifying these points is key to mastering the concept.
Rotation is Not Translation
A frequent error is confusing rotation with translation. Translation moves every point by the same distance in the same direction (a simple shift). Rotation, however, moves points along a circular path, and the distance and direction of movement depend on the point's distance from the center.
The Center of Rotation Matters
This calculator assumes rotation around the origin (0,0). Rotating a point around a different center (cx, cy) is a multi-step process: 1. Translate the system so the center of rotation is at the origin. 2. Perform the rotation. 3. Translate the system back. Failing to account for a non-origin center point will yield incorrect results.
Angle Units are Crucial
Most programming and mathematical libraries' trigonometric functions (sin, cos) expect angles in radians. Forgetting to convert from degrees to radians (by multiplying by π/180) is one of the most common sources of bugs in rotation implementations. Our calculator handles this for you, but it's essential to be aware of when implementing it yourself.

Clarification Examples

  • Rotating (2,0) by 90° gives (0,2). Translating it by (0,2) gives (2,2).
  • Rotating point (3,3) 90° around origin gives (-3,3). Rotating it 90° around (2,2) gives (1,3).

Mathematical Derivation and Matrix Form

  • Deriving the rotation formulas using polar coordinates
  • Representing rotation with a 2x2 matrix
  • Advantages of using the matrix representation
The rotation formulas can be elegantly derived and represented using different mathematical tools, including polar coordinates and matrices.
Derivation via Polar Coordinates
Any point (x, y) can be represented in polar coordinates (r, α), where r is the distance from the origin and α is the angle from the positive x-axis. So, x = r cos(α) and y = r sin(α). When we rotate the point by an angle θ, the new angle becomes (α + θ), while the radius r remains unchanged. The new Cartesian coordinates (x', y') are:
x' = r * cos(α + θ)
y' = r * sin(α + θ)
Using the trigonometric sum identities, cos(A+B) and sin(A+B), we can expand these: x' = r (cos(α)cos(θ) - sin(α)sin(θ)) = (rcos(α))cos(θ) - (rsin(α))sin(θ) = xcos(θ) - y*sin(θ). A similar expansion for y' yields the second formula.
The Rotation Matrix
The transformation can be expressed concisely using matrix multiplication. The rotation matrix R(θ) for a counter-clockwise rotation is:
[ cos(θ) -sin(θ) ]
[ sin(θ) cos(θ) ]
To find the new coordinates, we multiply this matrix by the column vector of the original point: [x'; y'] = R(θ) * [x; y]. This matrix form is powerful because multiple transformations (like rotation, scaling, and shearing) can be combined by simply multiplying their matrices.

Mathematical Representation Examples

  • The matrix for a 90° rotation is [[0, -1], [1, 0]]
  • The matrix for a 180° rotation is [[-1, 0], [0, -1]]