Example Two Dimension Rotation
Example 1 / rotate the line P1(1,6) and P2(5,1) anticlockwise 90 degrees.
sol//
Example 2 /Rotate the square (3,2),(5,2),(3,4),(5,4) counterclockwise with Ǿ =45 around the point (3,2).
sol //
TX=3 , TY=2 Translation
Point(3,2)
X1new =X1-TX=3-3=0
Y1new Y1-TY=2-2=0 (0,0)
Point(5,2)
X2new =X1-TX=5-3=2
Y2new Y1-TY=2-2=0 (2,0)
Point(5,4)
X3new =X1-TX=5-3=2
Y3new Y1-TY=4-2=2 (2,2)
Point(3,4)
X4new =X1-TX=3-3=0
Y4new Y1-TY=4-2=2 (0,2)
Rotation : Ǿ =45
sin(45)=0.85 , cos(45)=0.52
XP2=XP1*cos Ǿ -YP1* sin Ǿ
YP2=YP1*cos Ǿ +XP1* sin Ǿ
point(0,0)
X1new=0*0.52-0*0.85=0
Y1new=0*0.52+0*0.85=0 (0,0)
point(2,0)
X2new=2*0.52-0*0.85=1.04
Y2new=0*0.52+2*0.85=1.7 (1.04,1.7)
point(2,2)
X3new=2*0.52-2*0.85=-0.66
Y3new=2*0.52+2*0.85=2.74 (-0.66,2.74)
point(0,2)
X4new=0*0.52-2*0.85=-1.7
Y4new=2*0.52+0*0.85=1.04 (-1.7,1.04)
Back Translation
TX=3 ,TY=2
Point(0,0)
X1new =X1+TX=0+3=3
Y1new Y1+TY=0+2=2 (3,2)
Point(1.04,1.7)
X2new =X1+TX=1.04+3=4.04
Y2new Y1+TY=1.7+2=3.7 (4.04,3.7)
Point(-0.66,2.74)
X3new =X1+TX=-0.66+3=2.34
Y3new Y1+TY=2.74+2=4.74 (2.34,4.74)
Point(-1.7,1.04)
X4new =X1+TX=-1.7+3=1.3
Y4new Y1+TY=1.04+2=3.04 (1.3,3.04)
Example 3/: Performa counterclockwise 45 rotation of triangle A (2, 3), B (5,5) and C(4,3) about point (2,3).
sol //
TX=2 , TY=3 Translation
Point(2,3)
X1new =X1-TX=2-2=0
Y1new Y1-TY=3-3=0 (0,0)
Point(4,3)
X2new =X1-TX=4-2=2
Y2new Y1-TY=3-3=0 (2,0)
Point(5,5)
X3new =X1-TX=5-2=3
Y3new Y1-TY=5-3=2 (3,2)
Rotation : Ǿ =45
sin(45)=0.85 , cos(45)=0.52
XP2=XP1*cos Ǿ -YP1* sin Ǿ
YP2=YP1*cos Ǿ +XP1* sin Ǿ
point(0,0)
X1new=0*0.52-0*0.85=0
Y1new=0*0.52+0*0.85=0 (0,0)
point(3,2)
X2new=3*0.52-2*0.85=-0.14
Y2new=2*0.52+3*0.85=3.59 (-0.14,3.59)
point(2,0)
X3new=2*0.52-0*0.85=-1.04
Y3new=0*0.52+2*0.85=1.7 (1.04,1.7)
Back Translation
TX=2 ,TY=3
Point(0,0)
X1new =X1+TX=0+2=2
Y1new Y1+TY=0+3=3 (2,3)
Point(-0.14,3.59)
X2new =X1+TX=-0.14+2=1.86
Y2new Y1+TY=3.59+3=6.59 (1.86,6.59)
Point(1.04,1.7)
X2new =X1+TX=1.04+2=3.04
Y2new Y1+TY=1.7+3=4.7 (3.04,4.7)
Example 4 /- Rotate the object defined by (43,88), (84,50),(66,72) in rotating at anticlockwise by angle 53 then scale it with scaling factor Sx=1, Sy=2 where (43,88) is the fixed point?
sol//
Rotation : Ǿ =53
sin(53)=0.8 , cos(53)=0.6
XP2=XP1*cos Ǿ -YP1* sin Ǿ
YP2=YP1*cos Ǿ +XP1* sin Ǿ
point(43,88)
X1new=43*0.6-88*0.8=-44.6
Y1new=88*0.6+43*0.8=87.2 (-44.6,87.2)
point(84,50)
X2new=84*0.6-50*0.8=10.4
Y2new=50*0.6+84*0.8=97.2 (10.4,97.2)
point(66,72)
X3new=66*0.6-72*0.8=-18
Y3new=72*0.6+66*0.8=96 (-18,96)
TX=43 , TY=88 Translation
Point(-44.6,87.2)
X1new =X1-TX=10.4-43=-32.6
Y1new Y1-TY=97.2-88=9.2 (-32.6,9.2)
Point(10.4,97.2)
X2new =X1-TX=-44.6-43=-87.6
Y2new Y1-TY=87.2-88=-0.8 (-87.6,-0.8)
Point(-44.6,87.2)
X1new =X1-TX=-44.6-43=-87.6
Y1new Y1-TY=87.2-88=-0.8 (-87.6,-0.8)
Point(-18,96)
X1new =X1-TX=-18-43=-61
Y1new Y1-TY=96-88=8 (-61,8)
Scaling SX=1 , SY=2
Point(-87.6,-0,8)
X1new=-87.6*1=-87.6
Y1new=-0.8*2=-1.6 (-87.6,-1.6)
Point(-32.6,9.2)
X2new=-32.6*1=-32.6
Y2new=9.2*2=18.4 (-32.6,18.4)
Point(-61,8)
X3new=-61*1=-61
Y3new=8*2=-16 (-61,16)
Back Translation TX=43 , TY=88
Point (-87.6,-1.6)
X1new=-87.6+43=-44.6
Y1new=-1.6+88=86.4 (-44.6,86.4)
Point (-32.6,18.4)
X2new=-32.6+43=10.4
Y2new=18.4+88=106.4 (10.4,106.4)
Point (-61,16)
X3new=-61+43=-18
Y3new=16+88=104 (-18,104)
Example 5/ Rotate the object defined by (44,68), (104,66),(70,102) in rotating at counter-clockwise by angle 37 degrees after scaling with scale factor Sx=3, Sy=2 where (44,68) is the fixed point?
sol//
TX=44 , TY=68 Translation
Point(44,68)
X1new =X1-TX=44-44=0
Y1new Y1-TY=68-68=0 (0,0)
Point(104,66)
X2new =X1-TX=104-44=60
Y2new Y1-TY=66-68=-2 (60,-2)
Point(70,102)
X3new =X1-TX=70-44=26
Y3new Y1-TY=102-68=34 (26,34)
Scaling SX=3 , SY=2
Point(0.0)
X1new=0*3=0
Y1new =0*2=0 (0,0)
Point(60.-2)
X2new=60*3=180
Y2new =-2*2=-4 (180,-4)
Point(26.34)
X3new=26*3=78
Y3new =34*2=68 (78,68)
Back Translation TX=44, TY=68
point(0,0)
X1new=0+44=44
Y1new=0+68=68 (44,68)
point(180,-4)
X2new=180+44=224
Y2new=-4+68=64 (224,64)
point(78,68)
X1new=78+44=122
Y1new=68+68=136 (122,136)
Rotation : Ǿ =37
sin(37)=0.6 , cos(37)=0.8
XP2=XP1*cos Ǿ -YP1* sin Ǿ
YP2=YP1*cos Ǿ +XP1* sin Ǿ
Point( 44,68)
X1new=44*0.8-68+0.6=-5.6
Y1new=68*0.8+44*0.6=80.8 (-5.6,80.8)
Point( 224,64)
X2new=224*0.8-64+0.6=140.8
Y2new=64*0.8+224*0.6=185.6 (140.8,185.6)
Point( 122,136)
X1new=122*0.8-136+0.6=16
Y1new=136*0.8+122*0.6=182 (16,182)
