Calculator Inputs
Formula Used
The calculator models the forward affine transformation as:
It builds the combined matrix with this order:
The inverse transformation then becomes:
Here, S is scaling, F is reflection, H is shear, R is rotation, and t is translation. The inverse exists only when the determinant of A is not zero.
How to Use This Calculator
- Enter the transformed point coordinates you already know.
- Provide the original forward transformation settings: scale, reflection, shear, rotation, and translation.
- Choose degrees or radians for the rotation value.
- Set the decimal precision for displayed results.
- Click the calculate button to recover the original point.
- Review the matrix tables, verification output, and graph.
- Use the CSV or PDF buttons when you need a saved copy.
Example Data Table
| Case | Transformed Point | Scale | Reflection | Shear | Rotation | Translation | Recovered Original |
|---|---|---|---|---|---|---|---|
| Worked Example | (7, 10) | (2, 1) | None | (0, 0) | 90° | (5, 4) | (3, -2) |
| Reflection Example | (-6, 3) | (1, 1) | Y-axis | (0, 0) | 0° | (0, 0) | (6, 3) |
Frequently Asked Questions
1. What does this calculator reverse?
It reverses a 2D affine transformation built from scaling, reflection, shearing, rotation, and translation. You enter the transformed point and the forward settings, then the calculator recovers the original coordinates.
2. When does the inverse fail?
The inverse fails when the combined matrix has determinant zero. This usually happens with zero scaling on an axis or a shear combination that collapses the plane into a line.
3. Why is translation handled separately?
Translation is not part of the 2×2 linear matrix. The calculator first subtracts the translation vector from the transformed point, then applies the inverse of the combined linear matrix.
4. Which order of transformations is used?
The calculator applies scale, then reflection, then shear, then rotation, and finally translation. Because order matters, changing the sequence can change the recovered original point.
5. Can I use radians instead of degrees?
Yes. Select radians in the angle unit field, then enter the rotation directly in radians. The calculator also reports the angle in radians inside the result summary.
6. Why are the recovered and verification points almost identical?
That match confirms the inverse worked. Tiny differences can appear because decimal arithmetic in software may introduce extremely small rounding errors during matrix calculations.
7. What does the graph show?
The graph plots the transformed point you entered, the recovered original point, and a verification point created by applying the forward transformation to the recovered result.
8. What should I export in CSV or PDF?
Use CSV for spreadsheet work or batch checking. Use PDF when you want a clean report containing coordinates, matrix values, determinant, and verification details for documentation.