Matrix Vector Multiplication in MATLAB: Understanding Transformations

When working with computer graphics and transformations, it is essential to understand how matrices can be used to represent these operations. In this article, we will explore how matrix vector multiplication can be applied in MATLAB to perform various types of transformations.

Scaling Process

The scaling process is shown in the following figure:

As seen from the above equations, we can also represent them using column vectors. The pair (t x, t y) is called the translation vector or shift vector. By translating a point in 2D by adding the translation coordinates (t x, t y) to the original coordinate (X, Y), we get the new coordinate (X', Y'). From this figure, it is clear that:

* Translation: Moves an object to a different position on the screen.

Homogenous Coordinate System

Any Cartesian point P(X, Y) can be converted to homogenous coordinates by P' (X h, Y h, h). In this system, we can represent all transformation equations using matrix multiplication. This way, we can represent points with 3 numbers instead of 2 numbers, which is known as the Homogenous Coordinate system.

Converting 2x2 Matrix to 3x3 Matrix

To convert a 2x2 matrix to a 3x3 matrix, we need to add an extra dummy coordinate W. To shorten this process, we can use a 3x3 transformation matrix instead of a 2x2 transformation matrix.

Sequential Process for Transformations

When performing a sequence of transformations like translation followed by rotation and scaling, we need to follow a sequential process:

1. Scale the rotated coordinates.

2. Rotate the translated coordinates.

3. Perform any additional transformations as needed.

Transformations in Computer Graphics

Transformations play an important role in computer graphics to reposition graphics on the screen and change their size or orientation. When a transformation occurs on a 2D plane, it is called a 2D transformation. We can have various types of transformations like translation, scaling up or down, rotation, shearing, etc.

In conclusion, matrix vector multiplication in MATLAB is an essential tool for performing transformations in computer graphics. By understanding how matrices are used to represent these operations, we can create complex graphics with ease and precision.

Matrix vector multiplication matlab