Transformations - Transformations and Projections in Computer Graphics - page 12

rotate
about
origin

translate back

translate

rotate

(a) (b) (c)

Figure 1.6: Rotation About a Point.

(because 2 α 2 =sin 2 45 ◦ +cos 2 45 ◦ = 1). Note that det T =

− 1 (i.e., pure reflection).

Exercise 1.13: Show that the result in the example is correct.

Example: Reflection about an arbitrary line . Given the line y = ax + b ,
it is possible to reflect a point about this line by transforming the line to the x axis,
reflecting about that axis, and transforming the line back. Since a is the slope (i.e., the
tangent of the angle α between the line and the x axis) and b is the y intercept, the
individual transformations needed are (1) a translation of b units in the y direction, (2)

a clockwise rotation of α degrees about the origin, (3) a reflection about the x axis, (4) a
counterclockwise rotation, and (5) a reverse translation. The combined transformation
matrix is therefore

−

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

100
010
0

cos α − sin α 0 100

0

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

T reflect = sin α cos α 0 10

001

−

− b 1 0 0 1

⎛ ⎞ ⎛ ⎞

cos α sin α 0 100

010
0

⎝ ⎠ ⎝ ⎠

× − sin α cos α 0

0 0 1 b 1

⎛ ⎞

cos(2 α ) si (2 α )0
sin(2 α ) − cos(2 α )0
−b sin(2 α ) b cos 2 α

⎝ ⎠ .

= (1.9)

1

The determinant of this transformation matrix equals 1, as should be for pure reflec-

tion. For the two special cases α = b =0and α =45 ◦

−

and b = 0, Equation (1.9)

becomes