In geometry, rotation means turning a figure around a fixed point. This kind of transformation doesn’t change the shape or size of the figure, only its position.
A rotation calculator is a tool that helps you figure out the new location of a point after you’ve rotated it. In coordinate geometry, this involves some math, especially when you’re rotating around a point other than the origin (0,0).
We’ll go over the basics of rotation in coordinate geometry, how to rotate around the origin, how to rotate around other points, and how to use a rotation calculator to make things easier.
What is Rotation in Coordinate Geometry?
In coordinate geometry, rotation refers to how we change the coordinates of a point. Essentially, we’re spinning that point around a fixed location.
Here are the key things to remember about rotations:
- Rotations can go two ways: clockwise or counterclockwise.
- Clockwise rotations use negative angles, while counterclockwise rotations use positive angles.
- We can measure the angle of rotation in degrees or radians.
Two important terms to know are the center of rotation (the point we’re spinning around) and the angle of rotation (how far we’re spinning).
Calculating Point Rotation Around the Origin
If you want to know how to calculate the rotation of a point around the origin, there are some standard formulas you can use, and they rely on trigonometry (sine and cosine).
Formulas for Rotation Around the Origin
For counterclockwise rotation around the origin, you’d use these formulas:
- x’ = xcos(θ) – ysin(θ)
- y’ = xsin(θ) + ycos(θ)
In these formulas, (x, y) are the coordinates of the original point, and (x’, y’) are the coordinates of the new, rotated point. The symbol θ stands for the angle of rotation.
A clockwise rotation uses a negative angle, and a counterclockwise rotation uses a positive angle.
Example Calculation
Let’s rotate the point (1,0) 90 degrees counterclockwise around the origin. Here’s how that would work:
- x’ = 1cos(90) – 0sin(90) = 0
- y’ = 1sin(90) + 0cos(90) = 1
So, the new point is (0,1).
Rotation Around an Arbitrary Point
Our calculator can also rotate a point around any arbitrary point. Here’s how it works:
- Translation to the origin: First, we translate the point you want to rotate, as well as the center of rotation, to the origin (0,0). We do this by subtracting the coordinates of the center of rotation (which is also known as the pivot point) from the coordinates of the point to be rotated.
- Rotation: Then, we apply the standard rotation formulas to the translated point.
- Translation back: Finally, we translate the rotated point back to the original coordinate system by adding the coordinates of the original center of rotation to the rotated point’s coordinates.
This gives us the following formulas for rotation around an arbitrary point:
(x’, y’) = ((x-a) cos(θ) – (y-b) sin(θ) + a, (x-a) sin(θ) + (y-b) cos(θ) + b) where (a,b) is the arbitrary point.
Using Matrices for Rotation
The calculator uses matrix math to perform the rotation calculations. Rotation can be expressed by a 2×2 matrix, which looks like this:
To find the new coordinates after rotation, the calculator multiplies this rotation matrix by a column vector representing the original point’s coordinates. The result is a new column vector, which gives the coordinates of the rotated point.
How to use an online rotation calculator
Most online rotation calculators have similar input fields. You’ll typically need to enter:
- The angle of rotation (usually in degrees or radians)
- The x and y coordinates of the point you want to rotate
- The x and y coordinates of the point you want to rotate around (the pivot point)
The calculator then uses these values to apply the rotation formulas, whether around the origin or the pivot point you entered.
Final Thoughts
We’ve looked at coordinate rotation, rotation around the origin and other points, matrix representations, and using a rotation calculator.
Coordinate rotation calculators are efficient, practical tools for performing rotation calculations.