Enter transformation inputs
Example data table
| Scenario | Shape | Operation order | Input values | Expected insight |
|---|---|---|---|---|
| Lecture demo | Triangle | T, R, S | tx=3, ty=2, rotation=30, sx=1.2, sy=0.8 | Shows translation plus nonuniform scaling after rotation. |
| Symmetry check | Square | F, T | reflection=y, tx=-2, ty=1 | Reverses orientation and preserves side lengths before translation. |
| Affine stretch | Rectangle | H, S, T | shx=0.4, shy=0.2, sx=1.5, sy=1.1 | Illustrates skewing, area amplification, and centroid movement. |
Formula used
Each point is stored as a homogeneous coordinate column vector. The calculator multiplies affine matrices in the selected order.
[x′ y′ 1]^T = M × [x y 1]^T
The core matrices are:
Translation: [[1, 0, tx], [0, 1, ty], [0, 0, 1]]
Rotation: [[cosθ, -sinθ, 0], [sinθ, cosθ, 0], [0, 0, 1]]
Scaling: [[sx, 0, 0], [0, sy, 0], [0, 0, 1]]
Shear: [[1, shx, 0], [shy, 1, 0], [0, 0, 1]]
Reflection: depends on the selected axis or line
Composite matrix: M = M_n × ... × M_2 × M_1
Determinant: det = a·d - b·c for the 2×2 linear part
Area factor: |det|
A positive determinant preserves orientation. A negative determinant reverses orientation. A determinant near zero collapses the figure toward a line or point.
How to use this calculator
- Choose a preset shape or switch to the custom polygon option.
- Enter base dimensions for preset shapes, or provide custom coordinates.
- Fill in translation, rotation, scaling, shear, reflection, and pivot values.
- Set the operation sequence using T, R, S, H, and F codes.
- Press the submit button to generate the graph, matrix, and tables.
- Use the export buttons to download a CSV or PDF summary.
Frequently asked questions
1. What does the determinant tell me?
It measures the signed area scaling of the linear part. Positive values keep orientation, negative values flip orientation, and zero indicates collapse into a lower-dimensional shape.
2. Why does operation order matter?
Affine transformations do not generally commute. Rotating then translating usually gives a different result from translating then rotating, so the chosen order changes the final coordinates.
3. What is the pivot used for?
The pivot defines the anchor point for rotation, scaling, shear, and reflection. Using a nonzero pivot moves the local center of the operation away from the origin.
4. Can I test classroom coordinate sets?
Yes. Select Custom polygon and enter one x,y pair per line. The calculator will transform every listed point and visualize the resulting figure automatically.
5. Why can transformed area change?
Area changes with the absolute determinant of the linear transformation. Translation alone keeps area unchanged, while scaling and some shear combinations can enlarge or compress the region.
6. Does reflection always reverse orientation?
A single reflection gives a negative determinant and reverses orientation. Two reflections can combine into a rotation or translation-like effect and restore positive orientation.
7. Why are perimeter changes less predictable?
Perimeter depends on transformed edge lengths. Uniform scaling changes all lengths equally, but nonuniform scaling, shear, and rotation combinations alter each edge differently.
8. When should I use custom order codes?
Use custom order codes when you want to compare composition behavior. They help illustrate matrix multiplication order, geometric interpretation, and sensitivity in advanced algebra exercises.