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Stream: learning: questions

Topic: Categories for Drawing

Alex Kreitzberg (Apr 17 2025 at 17:02):

So, I spent a lot of time thinking about what the "right" category for drawing is supposed to be. I don't think that's the right way to think about it.

I want everything Durer describes in Vnderweysung der Messung to count as "a drawing".

He has a bunch of drawing algorithms, I'm partial to this one for example:

image.png

It's fun, because it's a beautiful construction with a real basis in modern technical art, but it is a misunderstanding of how Conchoids are constructed.

The point of the machine is to draw curves. I want to note, that using Euclidean geometry it isn't too hard to take any segment of this curve, and rotate it and translate it. So any similar segment to such a drawing machine exists between any two points of my "category of drawings".

Alex Kreitzberg (Apr 17 2025 at 17:02):

Going through the first chapter of the book introduces many different sorts of curves

Alex Kreitzberg (Apr 17 2025 at 17:03):

For example, here is Durer constructing a sine curve:

image.png

Alex Kreitzberg (Apr 17 2025 at 17:04):

And here Durer specifies a logarithmic curve via a relationship.

image.png

I think this is the only time in the book he doesn't give an imperative description

Alex Kreitzberg (Apr 17 2025 at 17:06):

Here Durer is constructing a conic section:

image.png

Alex Kreitzberg (Apr 17 2025 at 17:06):

And here Durer is constructing a cardioid:

image.png

Alex Kreitzberg (Apr 17 2025 at 17:10):

So, at one point Durer explains, takes for granted, or foreshadows, how to build by hand certain linear functions, quadratic functions, arbitrary algebraic functions, trig functions, and analytic functions. And further, you're allowed to do this with the aid of machines or tools.

In view of these constructions, I don't think it's outside the spirit of Durer's approach, to describe a curve via a differential equation, even though this isn't usually seen as drawing. If you can maintain how to track what it would take to draw a solution to a differential equation by hand, Durer would've used it.

Alex Kreitzberg (Apr 17 2025 at 17:12):

In the modern era artists ended up exploring ideas related to homotopy, like in Escher's "Metamorphosis"

Alex Kreitzberg (Apr 17 2025 at 17:13):

If a given category has a well defined notion of "path", then it could be used in a drawing.

Alex Kreitzberg (Apr 17 2025 at 17:15):

So, an idea I want to explore, is in order to have curves constructed with very different constraints, in the same drawing, there needs to be forgetful functors from the specific drawing categories, to the "canvas" category.

Alex Kreitzberg (Apr 17 2025 at 17:18):

So I could make a category of piecewise analytic curves, or a category of lazy paths, and both of these have a forgetful functor into a Moore path category of continuous paths.

Later I'm going to see if that lets me make interesting adjunctions with these forgetful functors.

Alex Kreitzberg (Apr 17 2025 at 20:25):

I guess to have adjoint functors, you need at least a bijection between sets,

So $C(F(a), b) \cong A(a, U(b))$ implies there should be the same number of paths, but in general you're going to have less algebraic paths than you do continuous paths, so you can't get adjoint functors in the way I'm going about this.

I guess I'm thinking about adjoint functors lately and got excited/ahead of myself.

I think everything I said above about embedding paths from special categories into a canvas category is useful somehow.

Alex Kreitzberg (Apr 19 2025 at 18:06):

I believe these drawing categories should be internal categories.

It's useful to have a euclidean metric on the objects, or points, and it's useful to have an $L^2$ metric on the morphisms, or paths.

Now if I approximate a curve, with curves of some rigid drawing category, like a linear piecewise category. I have the language to say how good my approximation is.