In geometry, a transformation maps every point of a figure to an image point. Depending on the transformation, the figure's position, orientation, size, or shape may change. Applying several transformations in sequence creates 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 (x,y) is reflected across the x-axis, its image can be found by multiplying this matrix by the position vector of the point: (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).
Suppose the matrices related to transformations T1 and T2 are M1=(prqs) and M2=(tvuw) respectively.
Then, the matrix related to the composition of transformations T2∘T1 (Transformation T1 followed by T2) is M2M1=(tvuw)(prqs).
The order of matrix multiplication determines the order of the transformations. The matrix for the transformation performed first (M1) is written on the right.
Apply three transformations to point P(2,5). First reflect it across the x-axis, then rotate it 90∘ about the origin, and finally rotate it 180∘ about the origin. Determine its image.
Apply three transformations to point P(2,5). First reflect it across the y-axis, then rotate it 180∘ about the origin, and finally reflect it across the line y=x. Determine its image.
