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Combined Transformation

Check the notes of single transformation before going through this note!!



When an object is transformed under two successive transformations i.e., the initial image is re-transformed to obtain final image is called combination of transformation.



Combination of two translations:

Let T1=(a, b) be a translation vector and T2=(c, d) be another translation vector. If T1 maps P→P’ and T2 maps P’→P”, the translation vector(T) which transforms P→P’’ is given by;
                        T=T2oT1=(a+c, b+d)



#Transform P(x, y) under combined translation of T1oT2 where T1=(1, 2) and T2=(-3, 1).
Under combined transformation T1oT2,
T1oT2= T1+T2=(-2,3)
The image under combined translation of T1oT2 is given by translation vector T:


                            P(x, y)→P’(x-2, y+3), is the final image


Combination of two Rotation

Let R1 be a rotation and R2 be another rotation such that R1 transforms P→P’ and R2 transforms P’→P”, the combined rotation(R) which transforms P→P’’ is given by;

                            R=R2oR­­1=R2+R1


#Transform P(4, 5) through +90o with center O and again P’ is rotated through +90 with center O. Find the final image of P under combined transformation R2oR1.
→Under combined transformation R1oR2,
                    R1oR2= R1+R2=+90+(+90)=180
So, the image under combined transformation of R1oR2 is given by +180.
                    P(4,5 )→P’(-4,-5), is the final image.


Combination of two Reflection

i. When the axes are parallel.

    The combination of two reflections when the axes are parallel is given by the translation and the distance of the translation is twice the distance between the axes of reflection.

#If P(4, 5) is reflected through R1:x=2 and R2:x=-2, find the image under combined transformation.

→distance between the axes=-2-2=-4
Required translation vector=(2*-4, 0)   {as it is reflection under x axes}
Now, the combined transformation is given by translation vector
(-8, 0)

                     P(4,5 )→P’(-4,5), is the final image.


ii. When the axes of reflection intersect at a point.
    The combined transformation when the axes of reflection intersect at a point is given by the rotation about the point of intersection of axes of reflection and twice the angle between them. (angle between them is determined with the help of graph.)

#If P(4, -5) is reflected through x axis and then again through y axis, find the combined transformation.
Here,
Reflection under x-axis
   P(4, -5)→P’(4, 5)
Reflection under y-axis
   P’(4, 5)→P”(-4,5)
Here, the axes intersects at (0,0) and the angle between them is 90.
So, this is combined transformation through (90*2=180)o about origin