In geometry, a transformation is an operation that moves or changes the shape of an object. When multiple transformations are applied sequentially to an object, this is called a composition of transformations.
We can use matrices to represent many geometric transformations and also to find the result of the composition of these transformations.
We will focus on transformations that can be represented by 2×2 matrices. For example, reflection across the x-axis can be represented by the matrix (100−1). If the point is reflected across the -axis, its image can be found by multiplying this matrix by the position vector of the point: .
(x,y)
x
(100−1)(xy)
Here are some basic transformations along with their matrices that are often used in the composition of transformations:
Reflection across the x-axis: (100−1)
Reflection across the y-axis: (−1001)
Reflection across the line y=x: (0110)
Reflection across the line y=−x: (0−1−10)
Reflection across the origin O(0,0) (equivalent to a 180∘ rotation): (−100−1)
Rotation about the origin (0,0) by an angle θ: (cosθsinθ−sinθcosθ)
Dilation about the origin (0,0) with a scale factor k: (k00k)
Composition of transformations means performing several transformations in sequence. If transformation T1 is followed by transformation T2, we denote it as T2∘T1. This means T1 is applied first, then its result is transformed by T2.
Suppose the matrix corresponding to T1 is M1, and the matrix corresponding to T2 is M2. To find the image of point P(x,y) under the composition T2∘T1, there are two equivalent methods:
Applying Transformations Sequentially to the Point:
Calculate the image P′(x′,y′) of P(x,y) under T1: (x′y′)=M1(xy).
Then, calculate the image P′′(x′′,y′′) of P′(x′,y′) under : .
If we substitute step (a) into (b), we get: (x′′y′′)=M2(M1(xy)).
Finding the Composite Matrix First:
Determine the matrix M that represents the composite transformation T2∘T1. This matrix is the product M2M1.
Both methods yield the same final image due to the associative property of matrix multiplication, i.e., M2(M1P)=(M2M1)P, where P is the column vector (xy).
Illustrative Example:
Suppose T1 is a reflection across the y-axis, and T2 is a rotation about the origin O by 21π radians (90∘). We want to find the image of point P(x,y) under T2∘T1.
The matrix for T1 (reflection across the y-axis) is M1=(−1001).
The matrix for T2 (rotation 90∘) is M2=(cos90∘sin90∘−sin90∘cos90∘)=(01−10).
Both methods give the same result. Using the composite matrix (M2M1) is often more efficient if we need to transform many points with the same composition.
Suppose you want to perform three transformations on a point P(2,5), namely reflection across the x-axis, rotation 90∘ about the origin, and a half turn (180∘ rotation about the origin). Determine its image!
Suppose we want to perform three transformations on a point P(2,5), namely reflection across the y-axis, rotation 180∘ about the origin, and reflection across the line y=x. Determine its image!
