Isometry Composition Generator
Rotations, reflections and translations all preserve distance, which is what makes them isometries — and composing them gives another isometry, always. The interesting question is which one. Two reflections make a rotation of twice the angle between their axes; an odd number of reflections always leaves the shape mirrored. This applies a random sequence to a shape, shows each stage, and confirms every distance survived exactly.
What this generator does
Applies a sequence of rotations, reflections and translations to a small shape, recording the coordinates after each step. Every pairwise distance is re-measured at every stage, and the final orientation is checked against how many reflections were used.
How to use this tool
- Choose how many transformations to apply.
- Generate, and read the sequence in the order it is applied.
- Check the summary: an odd number of reflections always flips the shape.
- Use a seed to keep the same sequence while you work through it by hand.
Understanding the controls
- How many transformations
- Between 1 and 5, applied in the order listed. Rotations are about the origin and reflections are in lines through it, so the composition is easy to follow by hand.
- Seed
- Reproduces the same shape and the same sequence, which matters if the output is being used as an exercise with a known answer.
Common use cases
- Practising composition of transformations with worked stages
- Showing that two reflections make a rotation of twice the angle
- Teaching orientation as something reflections reverse and rotations do not
- Generating transformation exercises with verifiable coordinates
- Checking a transformation matrix implementation against known output
How this generator works
Each transformation is applied to every point in turn: a translation adds an offset, a rotation multiplies by the standard rotation matrix, and a reflection in a line at angle theta is the reflection matrix built from twice that angle. Before display, every pair of points is measured at every stage and must be exactly the same distance apart as at the start — that is the definition of an isometry, and it catches an arithmetic error a picture would hide. The signed area is also compared, since it must flip sign for an odd number of reflections and only then.
Randomness and fairness
Uses your browser's cryptographic random source to pick the shape and the sequence by default. A seed switches to a reproducible sequence, deterministic and not cryptographically secure.
For how randomness is produced across the whole site, see how Generate Random works.
Limitations and good to know
- Rotations are about the origin and reflections in lines through it, so a rotation about an arbitrary point has to be built as a composition.
- Rotation angles are limited to a handful of values so the coordinates stay easy to check by hand.
- Scaling and shearing are deliberately absent, since neither preserves distance and neither is an isometry.
- Coordinates are shown to two decimals, so a 45-degree rotation gives rounded rather than exact surd values.
Privacy and your data
All transformations are computed in your browser. Nothing you choose is transmitted, stored or included in analytics.
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