Examples for

Geometric Transformations

Geometric transformations are bijections preserving certain geometric properties, usually from the xy-plane to itself but can also be of higher dimension. In particular for each linear geometric transformation, there is one unique real matrix representation. Wolfram|Alpha has the ability to compute the transformation matrix for a specific 2D or 3D transformation activity or to return a general transformation calculator for rotations, reflections and shears.

Rotations

Compute the matrix of a rotation transformation and visualize it.

Visualize a rotation and compute its matrix:

Rotate a point:

Rotate the graph of a function:

Visualize a rotation in 3D:

Reflections

Compute the matrix of a reflection transformation and visualize it.

Visualize a reflection and compute its matrix:

Reflect a point:

Reflect the graph of an implicitly defined function through a line:

Visualize a reflection in 3D:

Euler Angles

Compute the matrix of a rotation transformation given by a sequence of rolls, yaws and pitches and visualize it.

Specify a rotation in 3D using the angles of rolls, pitches and yaws and visualize it:

Specify a rotation in 3D using Euler angles:

Shears

Compute the matrix of a shear transformation and visualize it.

Visualize a shear and compute its matrix: