What is the usage of translations in the ScalingMatrix?

[Blatko1]

I was playing with the project's source code for better understanding, and when I replaced the tx and ty variables with 0, the results looked the same. What is the usage of translations in the ScalingMatrix? Calculating the tx and ty gives some tiny negligible numbers.
I am talking about this specific part of the code:

let (texture_width, texture_height) = texture_size;
let (screen_width, screen_height) = screen_size;

let width_ratio = (screen_width / texture_width).max(1.0);
let height_ratio = (screen_height / texture_height).max(1.0);

// Get smallest scale size
let scale = width_ratio.clamp(1.0, height_ratio).floor();

let scaled_width = texture_width * scale;
let scaled_height = texture_height * scale;

// Create a transformation matrix
let sw = scaled_width / screen_width;
let sh = scaled_height / screen_height;
let tx = (screen_width / 2.0).fract() / screen_width;
let ty = (screen_height / 2.0).fract() / screen_height;
#[rustfmt::skip]
let transform: [f32; 16] = [
    sw,  0.0, 0.0, 0.0,
    0.0, sh,  0.0, 0.0,
    0.0, 0.0, 1.0, 0.0,
    tx,  ty,  0.0, 1.0,
];

[parasyte]

The scaling matrix is for adding the border around the image when the surface is not an exact multiple of the image size. A good example is the screenshot used for the minimal-web demo:

A better example is the Android screenshot:

Without the border, the texture would be unnaturally stretched to fit the surface size.

[Blatko1]

I understand the usage of sw and sh variables. As you said, the whole image would be stretched without these variables. I know these variables scale or modify the width and height of the screen triangle in the shader code, but I am asking for the exact usage of the tx and ty variables. Their general use would be to move (translate) the image left or right and up or down, but their values are mostly below 0.001, which is negligible. I provided pictures of my findings in the following part.
With transformations removed (tx and ty variables set to 0):

With transformations applied (using actual tx and ty):

As you can see, there is no difference.
I tried to multiply the tx and ty variables by 100 to see how they impact the result, and all I can notice is that, when resizing the window, the image jiggles left and right by a small amount which is invisible whilst these variables have their typical values.

Maybe there is no need for the tx and ty variables?

[parasyte]

Specifically, tx and ty are for translation. If the surface is an odd number of pixels in either direction, these variables will nudge the triangle by half a pixel so that the pixels in the triangle align with pixels on the physical screen.

Without them, you would get an ugly half-pixel offset. See #231.

[Blatko1]

Thank you for the explanation, I now see that it fixes a very subtle bug.