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 +90^o 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