Combination of Transformations

The transformations discussed above are the examples of single transformation. When an object has been transformed, its image can again be transformed to form a new image. Such transformation is called a combination of transformations. After a combination of transformations, the change from the single object to the final image can be described by a single transformation.

Let R₁ be a transformation which maps a point P to the point P' and R₂ be another transformation which maps P' to the point p''. Then, the transformation which maps a point P to P'' is said to be the combination of R₁ and R₂ or it is said to be the composite transformation of R₁ and R₂. It is denoted by R₂ or R_1.The composite transformation R₂ R₁ is also called as the transformation R₁ followed byR₂.

The transformations R₁R₂ and R₂R₁ have different meanings. R₁R₂ gives a combination of transformations R₂ followed by R₁ whereas, R₂R₁ gives the combination of transformations R₁ followed by R₂.

If T and W are two different transformations, then the product TW gives the combination of the transformations W followed by T (this means to work out W first and then apply T), which can be denoted by single transformation say Z and we write it as Z = TW.

The composite transformation of R with itself is denoted by R.R = R²

Combination of Two Translations

A translation followed by a translation is equivalent to a single translation.

Let T₁ = (\()\frac{a}{b}\) be a translation vector. It describes the translation of units in the X - direction and units in the Y - direction.Let T₁ translates a point P to P'.

Let T₂ =(\(\frac{c}{d}\)) be another translation vector. It describes the translation ofc units in the X-direction and units in the Y-direction. Let T₂ translates P' to the point P''.

Now, we want to find out the translation vector which translates the point P to P'. Then,

x - component of T = PB = PA + AB = PA + P'C = a + c

Y - component of T = BP'' = BC + CP'' = AP' + CP'' = b + d

∴ T = T₂oT₁ = (\(\frac{a + c}{b + d}\))

Hence, if the translation T₁ = (\(\frac{a}{b}\)) is followed by the translation T₂ = (\(\frac{c}{d}\)), then the composite translation T₂oT₁ is described by (\(\frac{a + c}{b + d}\)).

Combination of Two Rotations

A rotation followed by a rotation is equivalent to a single rotation

1. When two rotations are about the same centre

A point or an object once rotated about the centre through a given angle can further be rotated about the same centre through another given angle.

Let P (x, y) be any point in the plane. If Q₁ be the rotation about origin through 90^o, then

Q₁: P (x, y)→ P' (-y, x).

Again, if Q₂ be the rotation about the origin through 90^o, then

Q₂: P'(-y, x)→ P'' (-x, -y).

Here, Q₂.Q₁: P (x, y)→ P''(-x, -y).

But (-x, -y) is the image of the point (x, y) under the rotation about the origin through 180^o.

Hence, a rotation of x^o followed by a rotation of y^o about the same centre is equivalent to the rotation of (x^o + y^o) about the same centre.

2. When two rotations are about the different centre

A point or an object once rotated about a centre through a given angle can further be rotated about the different centre through another given angle.

Let R₁ be the rotation through an angle x^o about the centre and R₂ be another rotation through an angle y⁰ about the different centre. Then, the equivalent transformation (R₂oR₁) is again a new rotation through the angle (x^o + y^o) about the third centre. This centre is the meeting point of the perpendicular bisectors of the line segments joining corresponding vertices of the object and its final image. In other words, perpendicular bisectors of the line segment joining corresponding vertices of the object and its final image are concurrent and the point of concurrence is the centre of new rotation ( i.e. combined rotation).

Let ABC be the given triangle (object). It is rotated to Δ A'B'C' through 90^o about the centre P. The triangle A'B'C' is again rotated to ΔA''B''C'' through 90^o about the centre Q. Then ΔA"B"C" is the image of ΔABC under the rotation through 180^o about the third centre R. Perpendicular bisectors of AA", BB", CC" intersect each other at the point R.

Combination of Two Reflection

A point or object once reflected can further be reflected to form a new image. The axes of these reflections may be parallel to each other or they intersect each other at a point. So, there are two cases for the combination of two reflections.

1.  When the axes of reflections are parallel

Let Ab and CD be parallel to each other. Let P' be the image of P under the reflection in the line AB. Then,

PM = MP' and so, PP' = 2 MP'.

Let P' be the image of P" under the reflection of the line CD. Then,

P'N = NP" and so, P'P" = 2P'N.

Here, P" is the image of P under the reflection in the line AB followed by the reflection in the line CD.

Now, \(\overrightarrow{P}{P"}\) = \(\overrightarrow{P}{P'}\) + \(\overrightarrow{P'}{P"}\) = \(\overrightarrow{M}{P'}\) + \(\overrightarrow{P'}{N}\) = 2( \(\overrightarrow{M}{P'} + \overrightarrow{P}'{N})\) = 2 \(\overrightarrow{M}{N}\)

Hence, if the axis of reflection is parallel, a reflection followed by another reflection is equivalent to the translation. The distance of the translation is twice the distance between the axes of reflections and the direction is perpendicular to the axis of reflections.

2. When the axes of reflection intersect at a point

Let OA and OB intersect each other at the point O. Let P" be the image of P' under the reflection in the line OB. Then, OB is the perpendicular bisector of PP'.

∴∠POM =∠POM =θ (say)

Let P" is the image of P' under the reflection in the line OB. Then, OB is the perpendicular bisector of P'P".

∴∠P'ON =∠P"ON = Φ(say)

Here, P" be the image under the reflection in the line OA followed by the reflection in the line OB.

Now,

∠POP" =∠POM +∠P'OM +∠P'ON +∠P"ON = θ+θ +Φ +Φ = 2(θ +Φ)

= 2(∠P'OM + ∠P'ON) = 2∠MON.

Hence, if the axes of reflections intersect at a point O, then a reflection followed by another reflection is equivalent to a rotation about the centre O through the angle twice the angle between the axes of reflection. The direction of rotation is the direction from OA to OB (i.e. the direction from P to P").

A reflection followed by a reflection is equivalent to either a translation or a rotation.

Fig:- combination of two reflections over intersecting lines

In the following figure, O is the point of intersection of two straight lines OP and OQ. ΔA'B'C' is the image of ΔABC under the reflection in the line OP and ΔA"B"C" is the image of ΔA'B'C' under the reflection in the line OQ. In fact, ΔA"B"C" is the image of ΔABC under the rotation about O through an angle 2∠POQ.

Fig:- combination of two reflections over two parallel lines  

In the following figure, AB and CD are two parallel straight lines. ΔP'Q'R' is the image of Δpqr under the reflection in the line AB and ΔP"Q"R" is the image of ΔP'Q'R'. In fact ΔP"Q"R" is the image of ΔPQR under a translation whose magnitude is twice the distance between AB and CD.

Combination of Two Enlargements

An object once enlarged can further be enlarged. Similarly, an object once reduced can further be reduced.

There are two cases in the combination of two enlargements:

1. When the centre of enlargements are same

Let E₁(O, k) be the enlargement with centre O and the scale factor k.

Let A'B' be the image of the given line AB under the enlargement E₁ with centre o and the scale factor k.

Then, A'B' =kAB.

Let E₂[O, k'] be the enlargement with centre O and scale factor k'.

Let A"B" be the image of A'B' under the enlargement E₂ with centre O and scale factor k'. Then A"B" = k'A'B'.

Now,

A"B" = k'A'B' = k'k AB.

∴ A"B" is the image of AB under an enlargement E(O, kk') with centre O and scale factor kk'.

Hence, an enlargement with centre O and scale factor k followed by another enlargement with the same centre O and scale factor k' is equivalent to an enlargement with the same centre and scale factor kk'.

2. When the centres of enlargements are different

Let E₁ [O, k] be the enlargement with centre O and scale factor k. Let AB be the image of AB under enlargement E₁ with centre O and scale factor k.

Then, A'B' = kAB.

Let E₂ [O', k'] be the enlargement with centre O' and scale factor k'.

Let A"B" be the image of A'B' under enlargement E₂ with centre O' and scale factor k'.

Then, A"B" = k'A'B'.

Now, A"B" = k'A'B' = k'kAB.

Join A"A' and B"B'. The intersect each other at the point O".

So, A"B" is the image of AB under an enlargement E[O, k'k] with the third centre O'' and scale factor kk'.

Hence, An enlargement with centre O and scale factor K followed by an enlargement with different centre O' and scale factor k' is equivalent to enlargement with the O" and scale factor KKK'. The centre of combined enlargement can be found by drawing.

An enlargement followed by another enlargement is equivalent to a new enlargement.

Combination of Reflection and Rotation

An object once reflected can further be rotated to get a new image. A reflection followed by a rotation is also a type of combination of two transformations.

1. Reflection in X-axis followed by a rotation about origin through 90^o

Let R₀ be the reflection in X-axis.

Then, R₀: P' (x, -y)→ P"(y, x)

∴ R_0.R_e: P(x, y) → P"(y, x) ...............................(i)

Again, let R be the reflection in the line y = x.

Then, R:P (x, y) → P"(y, x) ......................................................(ii)

So, reflection in the X-axis followed by a rotation about origin through 90^o is equivalent to the reflection in the line y = x.

2. Reflection in Y-axis followed by a rotation about origin through 90^o

Let R_e be the reflection in Y-axis.

Then, R_e: P(x, y)→ P'(-x, y).

Let R₀ be the rotation about origin through 90⁰.

Then, R₀: P (x, y) → P" (-y, -x)

∴ R₀. R_e: p (x, y) = P"(-y, -x) .........(i)

Again, let R be the reflection in the line y = -x.

Then, R: P(x, y) → P"(-y, -x) ............................(ii)

So, reflection in Y-axis followed by a rotation about origin through 90^o is equivalent to the reflection in the line y = -x.

3. Reflection in the line y = x followed by a rotation through 90^o about the origin.

Let R_e be the reflection in te line y = x.

Then, R_e: P (x, y)→ P'(y, x)

Let R₀: P' (y, x)→ P" (-x, y) ....................(i)

Again, Let R be the reflection in the Y-axis.

Then, R:P (x, y) → P"(-x, y) .........................(ii)

Hence, reflection in the line y = x followed by a rotation about the origin through 90^o is equivalent to the reflection in Y-axis.

Similarly, we ca establish the following results:

4. Reflection in X-axis followed by a rotation through - 90^o about the origin is equivalent to the reflection in the line y = -x.

5. Reflection in Y-axis followed by a rotation through -90^o about the origin is equivalent to the reflection in the line y = x.

6. Reflection in the line y = x followed by a rotation through -90^o about the origin is equivalent to the reflection in X-axis.

Combination of Rotation and Reflection

A point or an object once rotated can further be reflected to get a new image. A rotation followed by a reflection is also a type of combination of transformation.

1. Rotation about origin through 90^o followed by a reflection in the x-axis

Let R_o be the rotation about origin through 90^o. Then,

R_o: P(x, y)→ P'(-y, x).

Let R_e be the reflection in the x-axis.

Then, R_e: P' (-y, x) → P"(-y, -x)

∴ R_e.R₀: P (x, y) → P" (-y, -x) .....................(i)

Again, Let R be the reflection in the line y = -x.

Then, R: P (x, y) →P"(-y, -x) .............................(ii)

Hence, rotation about origin through 90^o followed by a reflection in X-axis is equivalent to the reflection in the line y = -x.

2. Rotation about origin through 90⁰ followed by reflection in Y-axis

Let R_o be the rotation about origin through 90^o.

Then, R_o: P (x, y) → P'(-y, x).

Let R_e be the reflection in the y-axis.

Then, R_e: P' (-y, x) → P"(y, x).

∴ R_e.R_o: P (x, y) → P"(y, x) ........................(i)

Again, let R be the reflection in the line y = x.

Then, R: P(x, y)→ P" (y, x) .................(ii)

So, rotation about origin through 90^o followed by the reflection in Y-axis is equivalent to a reflection in the line y = x.

3. Rotation about origin through 90^o followed by reflection in the lie y = x

Let R_o be the rotation about origin through 90⁰.

Then, R_o:P (x, y)→ P' (-y, x).

Let R_e be the reflection in the line y = x.

Then,R_e: P'(-y, x)→ P"(x, -y)

∴ R_e.R_o: P(x,y)→ P"(x, -y) ............................................(i)

Again, let R be the reflection in X-axis.

Then, R:P (x, y)→ P"(x, -y) ...............................(ii)

So, rotation about origin through 90^o followed by a reflection in the line y = x is equivalent to a reflection in X-axis.

As above, we can establish the following results:

4. A rotation about origin through - 90^o followed by a reflection in X-axis is equivalent to a reflection in the line y = x.

5. A rotation about origin through -90^o followed by a reflection in Y-axis is equivalent to a reflection in the line y = -x.

6. A rotation about origin through -90^o followed by a reflection in the line y = x is equivalent to a reflection in the y-axis.

Combination of Reflection and translation

A point or an object once reflected can further be translated. Similarly, a point or an object once translated can further be reflected.

Let r be the reflection in X-axis and T =(\(\frac{a}{b}\)) be a translation.

Then, R:P (x, y)→ P'(x, -y) and

T: P' (x, -y)→ P"(x + a, -y + b).

T_oR: P (x, y)→ P" (x + a, -y + b)

So, P"(x + a, -y + b) is the image of P (x, y) under the reflection in X-axis followed by translation T.

Again,T:P (x, y)→ P'(x + a, y + b) and

R:P' (x + a, y + b)→ P' (x + a, -y - b)

R_oT: P (x, y) → P"(x + a, -y -b)

So, P" (x + a, -y - b) is image of P(x, y) under translation T followed by reflection R.

Combination of Rotation and Translation

A point or object once rotated can further be translated. Similarly, a point or an object once translated can further be rotated.

Let R be the rotation through 90^o about the origin and T = \(\begin{bmatrix}a\\b\\ \end{bmatrix}\) be a translation.

Then, R: P (x, y)→ P' (-y, x).

∴ T: P' (-y, x)→ P" (-y + a, x + b)

ToR: P(x, y) → P" (-y -b, x + a)

So, P' (-y -b, x + a) is image of P(x, y) under reflection R followed by translation T.

Again, T: P (x, y)→P' (x + a, y + b) and

R: P' (x + a, y + b)→ P" (-y -b, x + a)

∴ RoT: P (x, y)→ P"(-y -b, x + a).

So, P" (-y - b, x + a) is the image of P (x, y) under translation T followed by rotation R.

Combination of Reflection and Enlargement

Let R be the reflection in X-axis and E [O, k] be the enlargement with centre O and scale factor k.

Then, R: P (x, y) → P'(x, -y) and

E:P' (x, -y) → P' (kx, -ky).

∴ EoR: P (x, y) → P'' (kx, -ky)

So, P" (kx, -ky) is image of p (x, y) under reflection r followed by enlargement E.

Again, E: P(x, y) → P'(kx, ky) and

R: P'(kx, ky) → P" (kx, -ky)

∴ RoE: P(x, y) → P"(kx, - ky).

So, P" (kx, -ky) is an image of p (x, y) under enlargement E followed by reflection R.

Combination of Rotation and Enlargement

Let R be the rotation through 90^o about origin and E[O, k] be the enlargement with centre O and scale factor k.

Then, R:P(x, y) → P'(-y, x) and

E:P'(-y, x)→ P"(-ky, kx)

∴EoR:P(x, y)→ P"(-ky, kx)

So, P" (-ky, kx) is an image of P (x, y) under rotation R followed by enlargement E.

Again, e:P(x, y)→ P' (kx, ky) and

R:P'(kx, ky)→ P"(-ky, kx)

RoE: p(x, y)→ p"(-ky, kx)

So, P"(-ky, kx)is an image of P(x, y) under enlargement E followed by rotation R.

Combination of Translation and Enlargement

Let T =\(\begin{bmatrix}a\\b\\ \end{bmatrix}\) be a translation and E [o, k] be an enlargement with centre O and scale factor k.

Then, T:P(x, y)→ P'(x + a, y + b)

E:P'(x + a, y + b)→ P"(kx + ka, ky + kb)

∴ EoT: P(x, y)→ P"(kx + ka, ky + kb)

So, P" (kx + ka, Ky + kb) is image of P (x, y) under translation T followed by enlargement E.

Again, E:P(x, y)→ P'(kx, ky) and

T:P'(kx, ky) → P"(kx + a, ky + b)

∴ToE: P(x, y)→ P" (kx + a, ky + b)

So, P"(kx + a, ky + b) is image of P(x, y) under enlargement e followed by translation T.