Matrix reflection and shear

Description of reflection and shear of matrices with examples

Geometric operations, like rotating a position vector by a certain angle around some axis, can be achieved by multiplying the vector by an appropriate matrix. This page descript how such matrices are constructed.

Matrix Reflection

We will consider a point \(P\) with the coordinates \((x, y)\) in a two-dimensional space.

In the two dimensional space we draw the vector as

The matrix below produces a reflection of the vector across the X-axis

this results in the formula

A 3-dimensional reflection of the \(Y\) position is achieved by the following formula

To produce reflections in the \(X\) or \(Z\) planes, place a negative sign on the corresponding diagonal elements of the unit matrix.

Matrix Shear

The insertion of a non-zero element into a position outside the diagonal of the unit matrix produces a shear distortion of the position vector.

corresponds to

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