How do you define a computer screen?

1) Physically it is a set of pixels (picture elements), which can be on or off, and display colors (to be defined, how many).

That leads to a set of discrete positions (x,y), and associated visual vector (to keep is simple) v(x,y).

2) At logic level, you can generalise with continuous [say in real plan R**2] positions (x,y), and continous vector values v(x,y)

Digital topology relevant to computer screens?

Topology is an area of mathematics coming partially from geometry, and partially from the reduction to a thin minimum of what is often described using distance. Neighborhoods are defined, without the need for a metrics or distance.

Graph theory is an interesting way to approach discrete problems or to discretise them efficiently, for computability. The inventor of graph theory seems to be Leonhard Euler, when solving an opeartions research problem: the 7 bridges of Koenigsberg (today Kaliningrad; to be noted that this city was the city of leading philosopher Immanuel Kant).

Historically computer screens and tv screens have been made using electron guns, shooting towards the screen.

For analogue TV there was the notion of going from left to right on a succession of lines, to display an image. This led to the awkward representation of the rectangle of the screen through line 1, 2... n.

Hence the dimensionality was heavily disturbed, using 1 dimensional objects to describe a two dimensional one in (x,y).

You may match that early age with implicit ordinal ranking of lines: 1, 2, 3, etc, combined with distance along the lines (say just the x coordinate, to make it simple).

Point P(x,y) on screen was then described by:

-P belongs to line D(i) for a certain i

-P has coordinate x from the origin (left, say O(i)) of D(i).

You find a lot of engineering methods for detection in TV signals using this awkward representation closest to the process of producing images in analog TV.

For instance the early scrambling of PayTV "Canal +" in France would segment the picture in lines, and fragment each line in segments, and perform a permutation of these segments to display them again "scrambled". This was meant to let people get a vague view of what the programme was about, and if they wanted to see it in clear, they would have to pay a subscription.

Now you may use other inherently digital representations: (x,y) on a discrete set say (x(i), y(j)) with i in I and j in J, subset of natural numbers set N...

You may also use an origin on the screen O (say intersection of the two diagonals of the rectangle of the screen), and turn around it, along discrete shapes, like --, ¦, going right horizontal, down vertical, left horizontal, up vertical, with a discrete square or rectangular spira (use an homothetic shape of the rectangle defined by the screen, if it's 16:9, keep the L:l ratio).

What metric do you want to use? Taxicab called Manhattan or better L1 is a good candidate, very efficient.

You may use a more streamlined topology, with neighborhoods defined as you wish (shapes suitable for the problem addressed)

A classical picture processing in computer imaging divides the rectangle of the screen in smaller rectangles and within each, tasks of smaller scales are performed. This allows for parallel processing, with the classical issue that an object may be across several of such zones, and glueing back everything together is not easy...

Projective geometry and the computer screen?

Projective geometry may also be interesting.

Each point of the screen can be seen as the intersection of a line and the rectangle surface of the screen. behind the screen think of a cone constructed on any shape, then beyond the shape seen on the screen, there may be a cone...

Etc...

If this is to represent what we see from the world as in painting or photography, a lot has been done long ago.

Canaletto the famous Italian painter of Venice invented a great method, for systematic painting of views of his city, which are close to photography (focal distance, etc). This would be worth revisiting, iof you're interested, maybe this time with different mathematical tools, from your topology background...

Ref:

https://www.amazon.it/Canaletto-quaderno-veneziano-Catalogo-illustrata/dp/8831713000

You may also look for publications of Frank Nielsen on digital imaging. Frank being a topologist, will have addressed your concenrns in much more depth than my summary above.

https://www.lix.polytechnique.fr/~nielsen/

Thank you dear professor Renaud.

Thank You Dear Professor.

Sharing link,

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Regards!