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Lab No 10
Question No 01:
Show that the composition of two transformations is additive by
connecting the matrix presentation for R(θ1) and R(θ2) to
obtain:
R(θ1). R(θ2)= R(θ1+θ2)
Question No 02:
Prove that the multiplication of transformation matrix for each
of the following sequence of operations is commutative.
a) Two successive Rotations
b) Two successive Translations
c) Two successive Scaling
Hint:
a) R(θ1). R(θ2)= R(θ2). R(θ1)
b) T1(tx1,ty1).T2(tx2,ty2)= T2(tx2,ty2). T1(tx1,ty1)
c) S1(Sx1,Sy1).S2 (Sx2,Sy2)= S2 (Sx2,Sy2). S1(Sx1,Sy1)
Question No 03:
Draw a point P(1,1) which is rotated 90 degrees along an
arbitrary point A(3,3)
Question No 04:
Rotate triangle about its third vertex P3(4,1); consider
triangle is formed with the following points:
P1(1,1), P2(1,4) and P3(4,1)
Hint:
Question No 03
1. First calculate the Translation of point P = Tp(x,y) of
point A, i-e Tp(x,y)= P – A
2. Then calculate rotation coordinate points P’(x’,y’)from
equations 3 and 4.
3. Now again calculate the Translation point P = Tp’(x,y) of
point A, i-e Tp’(x’,y’)=P’+ A
4. This last translation point is called the rotation of point
P.
Hint:
Question No 04
1. Repeat above steps for P1 and P2 and find the rotation of
triangle with respect to P3.
Given:
For Translation (x,y) = (2,3)
Translation Factor tx = 5
Translation Factor ty = 3
Angle for rotation is 60 degrees
Solution
>> tx = 5
tx = 5
>> ty = 3
ty = 3
>> A = [2;3];
>> A
A=
2
3
>> B = [tx;ty];
>> B
B=
5
3
>> T = A + B;
>> T
T=
7
6
>> theta = 60;
>> angle = 60;
>> a = cos(60);
>> a
a = -0.9524
>> b = -sin(60);
>> c = sin(60);
>> d = cos(60);
>> R1 = [a b 0;c d 0; 0 0 1];
>> R1
R1 =
-0.9524 0.3048
0
-0.3048 -0.9524
0
0
0 1.0000
>> A = [3 3 1];
>> A
A= 3 3 1
>> A =[3;3;1]
A=
3
3
1
>> R = R1 * A;
>> R
R=
-1.9428
-3.7717
1.0000
>> theta = 60 * 3.142/180;
>> a = cos(theta);
>> b = -sin(theta);
>> c = sin(theta);
>> d = cos(theta);
>> R1 = [a b 0;c d 0; 0 0 1];
>> R1
R1 =
0.4999 -0.8661
0
0.8661 0.4999
0
0
0 1.0000
>> R = R1 * A;
>> r
??? Undefined function or variable 'r'.
>> R
R=
-1.0986
4.0979
1.0000