Rotation Calculator
Calculate point rotation around origin or arbitrary pivot with rotation matrices
Calculate Point Rotation
Positive: counterclockwise, Negative: clockwise
Original point: (0, 0)
Pivot: (0, 0) (Origin)
Rotation Results
Rotation Matrix
Distance Verification
Step-by-Step Solution
Rotation around origin formula:
x' = x·cos(θ) - y·sin(θ)
y' = x·sin(θ) + y·cos(θ)
Step 1: Apply rotation formulas
x' = 0·cos(0.0000°) - 0·sin(0.0000°)
x' = 0·1.0000 - 0·0.0000 = 0.0000
y' = 0·sin(0.0000°) + 0·cos(0.0000°)
y' = 0·0.0000 + 0·1.0000 = 0.0000
Visual Representation
Red: Pivot point, Blue: Original point, Green: Rotated point, Purple arc: Rotation direction
Example Calculation
Rotate point (3, 4) by 60° around origin
Given: Point (3, 4), Angle = 60°, Pivot = (0, 0)
Step 1: cos(60°) = 0.5, sin(60°) = 0.866
Step 2: x' = 3×0.5 - 4×0.866 = 1.5 - 3.464 = -1.964
Step 3: y' = 3×0.866 + 4×0.5 = 2.598 + 2.0 = 4.598
Result: Rotated point = (-1.964, 4.598)
Rotation Properties
Counterclockwise
Positive angles
Standard mathematical convention
Clockwise
Negative angles
Opposite to standard convention
Isometric
Preserves distances
Shapes remain congruent
Rotation Tips
Rotation preserves distances and angles
Use origin (0,0) for simple rotations
Matrix multiplication applies rotation
90° = π/2 radians
180° = π radians
Understanding Coordinate Rotation
What is Coordinate Rotation?
Coordinate rotation is a transformation that rotates points around a pivot point by a specified angle. It's an isometric transformation, meaning distances and angles are preserved.
Key Properties
- •Isometric: Preserves distances between points
- •Linear: Can be represented by matrix multiplication
- •Orientation: Preserves the orientation of shapes
Rotation Formulas
Around Origin:
x' = x·cos(θ) - y·sin(θ)
y' = x·sin(θ) + y·cos(θ)
Around Arbitrary Point (x₀, y₀):
x' = x₀ + (x-x₀)·cos(θ) - (y-y₀)·sin(θ)
y' = y₀ + (x-x₀)·sin(θ) + (y-y₀)·cos(θ)
Note: θ is positive for counterclockwise rotation, negative for clockwise.
Real-World Applications
Computer Graphics
Rotating objects, sprites, and 3D models in games and animations.
Engineering
Mechanical design, robotics, and coordinate system transformations.
Navigation
GPS coordinate transformations and map projections.