Calculator Input
Use the responsive calculator grid below. It displays in three columns on large screens, two on smaller screens, and one on mobile.
Example Data Table
Sample case: rotate 90 degrees, then translate by (2, 1). Matrix used:
[ 0 -1 2 ]
[ 1 0 1 ]
[ 0 0 1 ]
| Point | Original x | Original y | Mapped x | Mapped y |
|---|---|---|---|---|
| P1 | 0 | 0 | 2 | 1 |
| P2 | 2 | 0 | 2 | 3 |
| P3 | 2 | 1 | 1 | 3 |
| P4 | 0 | 1 | 1 | 1 |
Formula Used
p' = M p
p = [x, y, 1]^T
M = T · P · L · P^-1
L may include scaling, shearing, rotation, and reflection.
Rotation: R(θ) = [[cosθ, -sinθ], [sinθ, cosθ]]
Scaling: S = [[sx, 0], [0, sy]]
Shear: H = [[1, shx], [shy, 1]]
Determinant = ad - bc
Area scale factor = |determinant|
Trace = a + d
Eigenvalues = (trace ± √(trace² - 4det)) / 2
This page uses homogeneous coordinates, so translation fits inside the same 3×3 matrix as rotation, scaling, and shear.
The determinant of the 2×2 linear part controls orientation and area scaling. Translation shifts location only.
How to Use This Calculator
- Enter scaling, shear, rotation, translation, and pivot values.
- Select any reflection preset and choose the operation order.
- Pick a matrix mode to use the builder matrix, custom matrix, or both.
- Enter points as x,y, one point per line.
- Click Transform Matrix to calculate the final matrix.
- Review determinant, rank, eigenvalues, inverse, mapped coordinates, and the plot.
- Use the export buttons to save the result table as CSV or PDF.
FAQs
1. What does the determinant tell me?
It shows area scaling and orientation behavior. A positive determinant preserves orientation, a negative one reverses orientation, and zero means the mapping collapses geometry.
2. Why are homogeneous coordinates used?
Homogeneous coordinates let translation join rotation, scaling, and shear inside one matrix. That makes composite transformations easier to compute and apply.
3. Does translation affect determinant or eigenvalues?
No. Translation changes only position. Determinant, trace, rank, and eigenvalues come from the linear part of the transformation matrix.
4. What is the pivot used for?
The pivot changes the center of rotation, scaling, shear, and reflection. It is useful when transformations should happen around a chosen point instead of the origin.
5. Why does operation order matter?
Matrix multiplication is not commutative. Scaling then rotating usually gives a different result than rotating then scaling, even with identical values.
6. When is the inverse unavailable?
The inverse does not exist when the determinant of the linear part is zero. In that case, the transformation loses information and cannot be undone uniquely.
7. Can I use the point list as a polygon?
Yes. If you provide at least three points, the calculator estimates polygon area by connecting points in the given order. Repeating the first point closes the shape clearly.
8. What does the graph show?
The graph plots the original and transformed point sets together. It helps you compare direction, scale, rotation, shear, and translation visually.