Rotation

A rotation turns the plane about the origin by an angle. The columns of the matrix are where i and j land, and the determinant stays 1.

Two folds

The determinant stays 1. The mark turns. It does not flip.

Angle θ 50°

Two folds make one turn. The turn is twice the angle between the creases. The first fold flips the mark. The second fold sets it right, as a rotation.

Rotation by θ. Read each row from left to right.
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Where this is used

Games turn a camera or a character with a rotation.
Many robot joints are rotations. A robotics engineer sets those angles.
An aerospace engineer describes the attitude of a satellite or a drone. Attitude is a set of rotations.
A phone turns its screen with a rotation. Motion sensors report how the phone has turned.
Animation tools store a turn as a rotation of a bone or an object.

Common mistakes

This matrix turns about the origin. A turn about some other point needs a slide, a rotation, and a slide back.
The matrix entries are the cosine and the sine of the angle, not the number of degrees.
A rotation does not change area. The determinant stays 1, and the mark does not flip.
Two folds make one turn. The turn is twice the angle between the creases.

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