Pick dimension, set shear factors, and preview matrices. Apply to vectors, see steps, and properties. Download clean reports and keep calculations organized always securely.
A shear is a linear transformation that shifts one coordinate by a multiple of another.
Sample inputs and expected outputs for quick verification.
| Case | Dimension | Factors | Matrix | Example point | Transformed |
|---|---|---|---|---|---|
| 1 | 2D X-shear | k = 2 | [[1,2],[0,1]] | (1,3) | (7,3) |
| 2 | 2D General | shx = 1.5, shy = 0.2 | [[1,1.5],[0.2,1]] | (2,4) | (8,4.4) |
| 3 | 3D Single | x′ = x + 0.5·z | [[1,0,0.5],[0,1,0],[0,0,1]] | (2,1,6) | (5,1,6) |
A shear matrix is a linear operator that keeps the origin fixed and slides points parallel to an axis. In 2D, the off‑diagonal factors shx and shy describe how x changes with y and how y changes with x. With ones on the diagonal, the matrix introduces skew without direct scaling. This calculator builds the matrix from your chosen mode so you can verify coefficients before transforming vectors.
In 2D, a single factor k produces a classic x‑shear or y‑shear, such as x′ = x + k·y. In 3D, the idea extends to dependencies between any pair of coordinates. The general 3D option exposes six factors, letting you model compound shears like x′ = x + sh_xy·y + sh_xz·z and compare results across configurations and test cases.
Properties summarize the geometry quickly. The trace is fixed at 2 in 2D and 3 in 3D here, so it mainly checks that the diagonal stayed at one. The determinant is crucial: it signals whether area/volume is preserved and whether the mapping is reversible. Many single‑factor shears have det(A)=1, meaning they preserve area/volume even while shapes skew and lengths change. A negative determinant would indicate orientation reversal.
When det(A) is near zero, inversion is unstable or impossible. The calculator reports invertibility and provides the inverse only when the determinant is safely away from zero. In 2D, the inverse uses a simple 1/det(A) formula. In 3D, cofactor expansion is used; it works well for typical inputs but can magnify rounding error for very large shear factors. Use moderate values when validating homework or pipelines.
Shear matrices appear in graphics, CAD drafting, simulation preprocessing, and decomposing affine transforms into basic steps. A practical workflow is to enter factors, test a few points, and export the result for documentation. CSV is convenient for spreadsheets and further computation, while PDF gives a compact record of the matrix, determinant, and transformed points. Keeping exports with project notes supports repeatable reviews. It simplifies future troubleshooting for teams.
X-shear changes x using y (x′ = x + k·y). Y-shear changes y using x (y′ = y + k·x). Both keep the diagonal at one and skew shapes without translating the origin.
Often yes. A single-factor shear typically has det(A)=1, so area/volume is preserved even though angles and lengths may change. In general mode, det(A) depends on the product of opposing shear factors.
If det(A) is zero or extremely close to zero, the matrix is not invertible. The calculator will show “not invertible” and omit the inverse table to avoid misleading results.
Yes. Negative factors shear in the opposite direction, which is useful for reversing a skew or modeling mirrored offsets. Watch det(A): very large magnitudes can make inversion numerically unstable.
Enter one point per line, comma-separated. Use “x,y” for 2D and “x,y,z” for 3D. Extra values after the required coordinates are ignored, and non-numeric lines are skipped.
It controls rounding in the displayed matrix, properties, transformed points, and exports. It does not change the internal calculation precision, but higher decimals can make small numerical differences easier to spot.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.