Geometric Transformations
Drag a shape to translate, rotate and scale it and see the transformation matrix.
Drag the shape to move it. Drag the green handle to rotate and the orange handle to scale. Drag a corner to reshape it.
About the Geometric Transformations
Free transformations grapher. Drag a shape to translate it, rotate and scale it with handles, reflect or shear it, and see the coordinates and transformation matrix update. Built for math, the geometric transformations runs instantly in your browser: change a setting or drag an object and the result updates at once, so you learn by trying things out rather than only reading about them.
Drag a shape to translate, rotate and scale it and see the transformation matrix. Use it to explore math ideas at your own pace, then check what you found against the key ideas further down this page.
How to use the Geometric Transformations
- Use the controls to change Shape, Rotation, Scale factor. The simulation reacts instantly.
- Pick an option such as Triangle, Square, House, Arrow to switch modes or load an example.
- Press "Reflect in y-axis", "Reflect in x-axis", "Hide original", "Reset" to start, reset or change what is happening.
- Where you see a glowing handle, object, weight or atom, drag it with your mouse or finger. Everything responds in real time.
- Watch the readouts and graphs update as you experiment, and compare what you see with the key ideas below.
Things to try
- Drag the shape to translate it by (3, −2).
- Rotate by 90° and note the new coordinates.
- Scale by 2 and by 0.5.
- Reflect across the y-axis.
Key ideas you can learn
- A translation moves a shape without changing its size or orientation.
- A rotation turns a shape about a point, and a reflection flips it across a line.
- A scale (dilation) multiplies distances from a centre by a factor k.
Where this is used in the real world
Computer graphics, video games, robotics, animation and CAD all use transformations to move and orient objects in 2D and 3D.
Who is this simulation for?
Algebra, geometry, trigonometry and calculus students, teachers preparing demonstrations, and self-learners who want to see the maths move.
For teachers: project it on the board, let students predict what will happen, then run it together. For students: change one thing at a time and write down what changes.
Frequently asked questions
What is a transformation matrix?
A small grid of numbers that turns every point of a shape into its image. Different matrices rotate, scale, reflect or shear.
What is the difference between a reflection and a rotation?
A reflection flips the shape to its mirror image. A rotation turns it about a point but keeps its handedness.
Is the Geometric Transformations free to use?
Yes. It is completely free, with no signup, no download and no ads inside the simulation. It runs in your web browser.
Does the Geometric Transformations work on a phone or tablet?
Yes. It uses touch as well as the mouse, so you can drag objects with your finger. A larger screen makes the controls easier to see.