Matrix Shear Simulator

Visualize horizontal and vertical shear transformations in 2D

Shear type

Shear factor k

Vector x

Vector y

Shear Matrix Formula

Horizontal: [[1, k], [0, 1]], Vertical: [[1, 0], [k, 1]]

A shear transformation slants a shape in one direction while keeping parallel lines parallel. Basic shear matrices have determinant 1.

⚠Shear preserves area but does not preserve angles or lengths in general.

How to Read the Simulator

Horizontal Shear

x changes by k times y while y stays fixed.

Vertical Shear

y changes by k times x while x stays fixed.

Basis Arrows

Blue and green show transformed basis directions.

Determinant

The determinant remains 1 for these shear matrices.

Example: Horizontal shear with k = 1 maps (x, y) to (x+y, y). The point (2, 2) becomes (4, 2).

Applications

Linear AlgebraComputer GraphicsShape TransformsMechanics

Frequently Asked Questions

What is a shear matrix?▼

A shear matrix slants coordinates while preserving parallel lines. Horizontal shear uses [[1, k], [0, 1]] and vertical shear uses [[1, 0], [k, 1]].

Does shear preserve area?▼

Basic horizontal and vertical shear matrices have determinant 1, so they preserve area but change angles and shape.

Where is shear used?▼

Shear transformations are used in computer graphics, geometry, mechanics, and linear algebra visualization.

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