Graphics Reference
In-Depth Information
7.3.4 Rotate and Translate a Point
There are two ways we can combine the rotate and translate transforms into a single
transform. The first method starts by translating a point P(x,y) using T (t x ,t y ) to an
intermediate point, which is then rotated using R β . The problem with this strategy
is that the radius of rotation becomes large and subjects the point to a large circular
motion. The normal way is to first subject the point to a rotation about the origin
and then translate it:
P =
T t x ,t y R β P
x
y
1
10 t x
01 t y
00 1
cos β
sin β
0
x
y
1
=
sin β
cos β
0
0
0
1
=
.
x
y 1
cos β
sin β x
x
y
1
sin β
cos β y
0
0
1
For example, consider rotating the point P( 1 , 0 ) , 90° and then translating it by
( 1 , 0 ) . The rotation moves P to ( 0 , 1 ) and the translation moves it to ( 1 , 1 ) .Thisis
confirmed by the above transform:
1
1
1
0
11
100
001
1
0
1
=
.
7.3.5 Composite Rotations
It is worth confirming that if we rotate a point β about the origin, and follow this
by a rotation of θ , this is equivalent to a single rotation of θ
+
β ,so R θ R β =
R θ + β .
Let's start with the transforms R β and R θ :
cos β
sin β
0
R β =
sin β
cos β
0
0
0
1
cos θ
sin θ
0
.
R θ =
sin θ
cos θ
0
0
0
1
We can represent the double rotation by the product R θ R β :
cos θ
sin θ
0
cos β
sin β
0
R θ R β =
sin θ
cos θ
0
sin β
cos β
0
0
0
1
0
0
1
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