Rotation Transforms in the Plane - Rotation Transforms for Computer Graphics - page 96

Pre-multiplying a vector p by the product nm creates p such that given:

p = p 1 e 1 + p 2 e 2

nmp =| n || m | ( cos β − sin β e 12 )(p 1 e 1 + p 2 e 2 )

=| n || m | ( cos βp 1 e 1 + cos βp 2 e 2 + sin βp 1 e 2 − sin βp 2 e 1 )

| ( cos βp 1 − cos βp 2 ) e 2

p =| n || m sin βp 2 ) e 1 + ( sin βp 1 +

p 1
p 2

cos β p 1

p 2

− sin β

=| n || m |

sin β cos β

and confirms that the vector p is rotated β and scaled by | n || m | .

Post-multiplying a vector p by the product nm creates p

such that given:

p = p 1 e 1 + p 2 e 2

pnm = (p 1 e 1 + p 2 e 2 ) | n || m | ( cos β − sin β e 12 )

=| n || m | (p 1 e 1 cos β − p 1 e 2 sin β + p 2 e 2 cos β + p 2 e 1 sin β)

| (p 1 cos β p 2 cos β) e 2 p =|

n || m + p 2 sin β) e 1 + ( − p 1 sin β +

p 1
p 2

p 1
p 2

cos β sin β

=| n || m |

− sin β cos β

and confirms that the vector p is rotated . By making n

and m unit vectors, the product nm rotates a vector without scaling, which is an
essential quality for a rotation transform.
Before proceeding, we should clarify the effect of reversing the product nm to
mn . Therefore, assuming that vectors n and m remain unchanged, the product mn
is given by:

− β and scaled by | n || m |

n = n 1 e 1 + n 2 e 2

m = m 1 e 1 + m 2 e 2

mn = m · n + m ∧ n

=| n || m | cos β +| m || n | sin β e 12

sin β e 12 ) .
Pre-multiplying a vector p by the product mn creates p such that given:

=| n || m | ( cos β +

p = p 1 e 1 + p 2 e 2

mnp =| n || m | ( cos β + sin β e 12 )(p 1 e 1 + p 2 e 2 )

=| n || m | ( cos βp 1 e 1 + cos βp 2 e 2 − sin βp 1 e 2 + sin βp 2 e 1 )

| ( cos βp 1 + cos βp 2 ) e 2 p =|

n || m sin βp 2 ) e 1 + ( − sin βp 1 +

p 1
p 2

p 1
p 2

cos β sin β

=| n || m |

−

sin β cos β

and confirms that the vector p is rotated − β and scaled by | n || m | .