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 30 degrees
rotation transform calculator

Rotate a point:

rotate {1, 1} pi/3 radians

Rotate the graph of a function:

rotate y=x^2 by 30 degrees around {0.2, -0.04}

Visualize a rotation in 3D:

rotate (3 pi)/4 radians around the z-axis
rotate {cos(t), sin(t), sin(2t)} by 30 degrees about (1,0,0)
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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:

yaw=30 degrees, roll=22 degrees
yaw=30 degrees, pitch=40 degrees, yaw=33 degrees

Specify a rotation in 3D using Euler angles:

Euler angles: 30 degrees, 40 degrees, 33 degrees
Euler angles calculator
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Reflections

Compute the matrix of a reflection transformation and visualize it.

Visualize a reflection and compute its matrix:

reflect across y=2x
mirror transformation matrix

Reflect a point:

reflect {2, 1} over y = -2x

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

reflect x^2+y^2=1 about y=x+1

Visualize a reflection in 3D:

reflect across x+y+z=1
reflect {3 cos(t), 3 sin(t), 0} across x + y + z = 1
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Shears

Compute the matrix of a shear transformation and visualize it.

Visualize a shear and compute its matrix:

vertical shear 45 degrees
shear transform
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