2D Rotations�

Deriving the rotation matrix

Say we have a point (x1,y1) and we want to find the 2×2 transformation matrix that will rotate it (anticlockwise) around the origin by an angle θ to a new point, (x2,y2).

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• We can now pull out the coefficients and express this as a matrix multiplied by the first point i.e (X1,Y1)
• That 2×2 matrix is the 2D rotation matrix. You can multiply it by any point (or series of points) to rotate them anticlockwise about the origin by the angle θ.

Properties of rotation matrices

• Inverse = Transpose

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Determinant = 1

Rotation X Rotation = Rotation

• Rotation matrices have the property that if you multiple two of them together, you always get another rotation matrix. 

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Experiment

3D Rotations�

• When we add a third dimension, we add it in the Z direction. According to the right-hand rule, the Z direction is "out of the page" compared to the typical X-Y plane.

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Right Hand Grip Rule�

•  The right-hand grip rule comes into play here. It tells us that if we look from above, a positive rotation around the Z axis is an anticlockwise rotation (i.e. moving from the positive X axis through the first quadrant to the positive Y axis). 

Rotating about the Z axis

3D Rotations�

• The z value could really be anything - it's just a height in space - and it shouldn't affect our rotated x and y positions at all, so we want the z contribution to be 0 multiplied by whatever the original z value is.

Rotating about the other axes��

Rotating anywhere in space

• we could rotate first around the Z axis, then around the Y axis, and then around the X axis. You can even use the same axis twice (for the first and last ones), e.g. Z-Y-Z. So we can multiply the rotation matrices for the individual dimensions to form a single matrix.