Shear Matrix Calculator Form
Horizontal mode forces shy to zero. Vertical mode forces shx to zero. Combined mode keeps both factors active.
Formula Used
A 2D shear transformation uses the matrix shown below. It slants coordinates by adding a proportion of one axis to the other.
Shear matrix: S = [[1, shx], [shy, 1]]
Point transformation: [x′, y′]ᵀ = S[x, y]ᵀ
Expanded form: x′ = x + shx·y, y′ = shy·x + y
Determinant: det(S) = 1 − shx·shy
Inverse: S⁻¹ = (1 / det(S)) · [[1, −shx], [−shy, 1]], when det(S) ≠ 0
The determinant controls area scaling and invertibility. A zero determinant means the transformation collapses the plane into a lower-dimensional result.
How to Use This Calculator
- Choose horizontal, vertical, or combined shear mode.
- Enter the input point coordinates you want to transform.
- Enter the shear factor values shx and shy.
- Set plot width and height for the comparison rectangle.
- Choose decimal precision for clean output formatting.
- Click the calculate button to show results above the form.
- Review the matrix, inverse, determinant, transformed point, and graph.
- Use the CSV and PDF buttons to export the computed results.
Example Data Table
| Mode | shx | shy | Input Point | Determinant | Transformed Point |
|---|---|---|---|---|---|
| Horizontal | 1.50 | 0.00 | (2, 3) | 1.00 | (6.50, 3.00) |
| Vertical | 0.00 | 0.75 | (4, 2) | 1.00 | (4.00, 5.00) |
| Combined | 0.40 | 0.30 | (5, 1) | 0.88 | (5.40, 2.50) |
| Combined | 1.20 | 0.90 | (2, -1) | -0.08 | (0.80, 0.80) |
FAQs
1. What does a shear matrix do?
A shear matrix slants a shape by shifting one coordinate in proportion to the other. It changes angles and orientation behavior while keeping lines parallel.
2. What is the difference between shear and rotation?
Rotation turns a figure around an origin and preserves angles. Shear tilts the figure instead, so right angles usually become non-right angles.
3. Why is the determinant important?
The determinant tells you whether the transformation is invertible and how it scales signed area. Positive values preserve orientation, negative values reverse it, and zero makes the matrix singular.
4. When does the inverse exist?
The inverse exists whenever the determinant is not zero. For this calculator, that means 1 − shx·shy must be nonzero.
5. What happens in horizontal shear only?
Horizontal shear sets shy to zero. The x-coordinate changes by shx times y, while the y-coordinate stays unchanged.
6. What happens in vertical shear only?
Vertical shear sets shx to zero. The y-coordinate changes by shy times x, while the x-coordinate stays unchanged.
7. Why does the graph use a rectangle?
A rectangle makes the slant effect easy to see. You can compare the original shape with the transformed shape and visually inspect area and orientation changes.
8. Can this help with linear algebra study?
Yes. It connects matrix multiplication, determinants, inverses, basis vectors, and geometric interpretation in one place, which is useful for algebra, graphics, and transformation practice.