Page 79 - DCAP504_Computer Graphics

Computer Graphics

The figure 6.6 depicts the rotation of a point coordinate.

Figure 6.6: Rotation of a Point Coordinate

The rotation matrix to rotate the object by an angle θ counterclockwise is given by:
cos θ −sin  θ
R ( θ) =   

 sin θ cos θ 
Multiply all the point coordinates (point coordinate matrices) of the object with the rotation matrix to
rotate the object by an angle θ . The following is the matrix multiplication of rotation matrix R ( θ ) and
point coordinate matrix (P) to obtain the rotated point coordinate matrix (P’).
R
' P = ( θ) ⋅P
  ' x cos θ − sin   θ x 
    =        
 ' y   sin θ cos θ   y 
  ' x  ⋅ cosx θ − y ⋅sin  θ
    =     ………………………….. (Eq.6.2)
 ' y   x ⋅sin θ + y ⋅ cos θ 

The equation 6.2 is used to calculate the coordinates of the object that is to be rotated counterclockwise
by an angleθ . All the coordinates of the object have to be multiplied with the rotation matrix and the
rotated coordinates of the original object is calculated. Similarly, we can rotate the object clockwise. The
equation 6.3 is used to rotate the object by an angle θ clockwise.
  ' x  x ⋅ cos θ + y ⋅sin θ 
    =    ………………………….... (Eq.6.3)

 ' y   − x ⋅sin θ + y ⋅ cos θ 

A triangle object is to be rotated counterclockwise by an angle 90 degree, with
reference to the origin of the 2-D space. The triangle object has three point

coordinates (0, 0), (2, 0), and (1, 2). Using the equation 6.2, the point coordinate
values of the triangle are calculated. The point coordinate values of the triangle
after rotating it by an angle 90 degree are (0, 0), (0, 2), and (-2, 1). The object is
rotated clockwise by using the equation 6.3. The coordinates of the object
rotated clockwise are (0, 0), (0, -2), and (-1, 2).

72 LOVELY PROFESSIONAL UNIVERSITY