Determining the Transformation Matrix for a Given Parallelogram

When working with geometric transformations in Euclidean geometry, it's essential to understand how such transformations can be represented using matrices. In particular, an affine transformation is a type of transformation that preserves lines and parallelism but not necessarily Euclidean distances and angles. This article delves into the process of determining the transformation matrix for a given parallelogram and provides a detailed derivation and example.

Understanding Affine Transformations

An affine transformation in the plane can be represented by a matrix acting on the homogeneous coordinates of points. In two-dimensional space, this transformation is defined by a 3x3 matrix. The general form of an affine transformation matrix (T) in homogeneous coordinates is:

[begin{pmatrix}a b c d e f 0 0 1end{pmatrix}]

This matrix transforms a point ((x, y)) to a new point ((x', y')) as follows:

[begin{pmatrix}x' y' 1end{pmatrix} begin{pmatrix}a b c d e f 0 0 1end{pmatrix} begin{pmatrix}x y 1end{pmatrix}]

Deriving the Transformation Matrix

Let's consider a parallelogram with vertices in the original coordinates (A(0,0)), (B(0,1)), (C(1,1)), and (D(1,0)). After applying an affine transformation, the new coordinates of the vertices are given as (A'(0,0)), (B'(2,3)), (C'(6,1)), and (D'(8,4)).

We need to determine the transformation matrix (T) that maps the original points to the new ones. The transformation matrix (T) can be written as:

[begin{pmatrix}a b c d e f 0 0 1end{pmatrix}]

We can solve for the coefficients (a), (b), (c), (d), (e), and (f) by applying the transformation to each of the vertices and setting up a system of equations.

Applying the Transformation to Each Vertex

For vertex (A(0,0)) which maps to (A'(0,0)):

[begin{pmatrix}0 0 1end{pmatrix} begin{pmatrix}a b c d e f 0 0 1end{pmatrix} begin{pmatrix}0 0 1end{pmatrix} begin{pmatrix}c f 1end{pmatrix}]

Thus, (c 0) and (f 0).

For vertex (B(0,1)) which maps to (B'(2,3)):

[begin{pmatrix}2 3 1end{pmatrix} begin{pmatrix}a b c d e f 0 0 1end{pmatrix} begin{pmatrix}0 1 1end{pmatrix} begin{pmatrix}b c e f 1end{pmatrix}]

Thus, (b c 2) and (e f 3). Since (c 0) and (f 0), this simplifies to (b 2) and (e 3).

For vertex (C(1,1)) which maps to (C'(6,1)):

[begin{pmatrix}6 1 1end{pmatrix} begin{pmatrix}a b c d e f 0 0 1end{pmatrix} begin{pmatrix}1 1 1end{pmatrix} begin{pmatrix}a b c d e f 1end{pmatrix}]

Thus, (a b c 6) and (d e f 1). Substituting (b 2) and (c 0), we get (a 2 6) and (d 3 1), which simplifies to (a 4) and (d -2).

For vertex (D(1,0)) which maps to (D'(8,4)):

[begin{pmatrix}8 4 1end{pmatrix} begin{pmatrix}a b c d e f 0 0 1end{pmatrix} begin{pmatrix}1 0 1end{pmatrix} begin{pmatrix}a c d f 1end{pmatrix}]

Thus, (a c 8) and (d f 4). Since (c 0) and (f 0), this simplifies to (a 8) and (d 4).

The Transformation Matrix

Combining all the values found, the transformation matrix (T) is:

[begin{pmatrix}6 2 0 8 3 0 0 0 1end{pmatrix}]

Conclusion

Through this detailed derivation, we have successfully determined the transformation matrix that maps the original parallelogram to the given new coordinates. Understanding these transformations is crucial in various fields such as computer graphics, robotics, and computer vision, where geometric transformations are extensively used.