You can't divide a point into 2. Say you have an interval between -1 and 1, and you want to decide where 0 belongs. Like you want to cut it in equal halves. You have [-1,0) and (0,1] with 0 left out. The set (0,1] is not the same as [0,1]. But, as far as I know, the integral over each set would be the same, as long as the integrand is well behaved at 0. No singularities. For exmple, integtrate 1 dx.

Maybe it depends on what question you are asking, as to whether it makes a difference.

With an endless unlimited resolution, it would certainly be possible to divide a square into two equal in size triangles. In concept at least.

One triangle will be greater than the other, as the resolution decreases. There is also a very scary thing happening at the diagonal line of both triangles, if one triangle must never overlap the other triangle, and there must be no gap between the triangles. As the 4 corner points do stay 90 degrees from each other as the resolution decreases, the diagonal ones do not stay 45 degrees, and if one triangle cannot share the same points (overlapping), they will end up with different diagonal angles.

Triangle rasterization is tricky. If not done right, it simply doesn't look right.

@Brony Number 42

The problem is that you can't cut a [-1, 1] interval in equal halves. They will have the same measure, but they will be conceptually different. You must include 0 in one interval or the other, so this creates an asymmetry. My proposal is to get rid of the asymmetry by dividing a point (in this case 0) into two halves. Of course it would require redefining the foundations of mathematics, by using a different set of axioms than the ones normally used. I'm not thinking about integration or other advanced stuff, this is all about very basic ideas.

@Splashee

Even with unlimited resolution, it creates a conceptual problem as to where the points at the diagonal belong, my idea aims to get rid of this problem.

lim x->0⁺ vs lim x->0^- but 0 itself you can't split. It's like splitting an electron.

@Brony Number 42 My idea is that there is no such thing as "zero itself". "Zero" is a set consisting of lim x->0⁺ and lim x->0^-. By the way, if we take reciprocals of these two "zeros", we get positive and negative infinity, which are also used when taking limits. So if "zero itself" doesn't exist, "infinity itself" (or "actual infinity") also doesn't exist, and we only have "potential infinity".

Two different kinds of zero that behave the same way for all our normal math purposes. But then how many integers are there in the set [-1,1]? 2 or 3?

@Brony Number 42 If you mean it as an interval, then under my rules the answer is 2. Half of -1, 0, and half of 1.

.....HA nerds...... 

No, I mean how many integers in the interval [-1, 1]. Or the set {-1, 0-, 0+, 1}. That looks like 4, which might cause problems.

@Brony Number 42 If we define an integer (or any other real number) as the set of both "parts" of the number, then it's not a problem. Also keep in mind that other numbers behave the same way as zero, so they have two parts as well. (If we introduce complex numbers, then the situation complicates, because we have 2 dimensions, so we need to use the "pie chart" model.)

in what ways can we apply this in reality?🤔

@Kujamih None, I suppose.

1 minute ago, PawelS said:

@Kujamih None, I suppose.

maybe...who knows right?

1 hour ago, Kujamih said:

in what ways can we apply this in reality?🤔

We can save a lot of money in the Anime industry, if we can solve this problem.

I once tried to find the most optimal "next step" line drawing tool that would fill perfectly 1 pixel on the screen, not leak when filling a shape, and not do any overlapping. It never happened, as no one had found a way, and I am not good enough to solve that problem by myself. So instead, people have to draw with digital tools, and still do extra clean-up, which is taking extra time, and leaves outline overlapping filled regions. I am sure you understand this problem @Kujamih, as you do draw and fill shapes manually, and know how long it takes to do it in a clean way.

By being able to find a way to divide two shapes equally (like this blog is talking about), and somehow get a perfect interpolation between those shapes, it would certainly save a lot of time for us artists.

I laughed at Microsoft's interpretation of the shape problem as they had to give up and use the words "It is deemed that there is no single -best- way to perform antialiased line rendering". Nowadays I'm always saying "it has been deemed" with a smile since it is clearly a way to give up trying

34 minutes ago, Splashee said:

We can save a lot of money in the Anime industry, if we can solve this problem.

I once tried to find the most optimal "next step" line drawing tool that would fill perfectly 1 pixel on the screen, not leak when filling a shape, and not do any overlapping. It never happened, as no one had found a way, and I am not good enough to solve that problem by myself. So instead, people have to draw with digital tools, and still do extra clean-up, which is taking extra time, and leaves outline overlapping filled regions. I am sure you understand this problem @Kujamih, as you do draw and fill shapes manually, and know how long it takes to do it in a clean way.

By being able to find a way to divide two shapes equally (like this blog is talking about), and somehow get a perfect interpolation between those shapes, it would certainly save a lot of time for us artists.

I laughed at Microsoft's interpretation of the shape problem as they had to give up and use the words "It is deemed that there is no single -best- way to perform antialiased line rendering". Nowadays I'm always saying "it has been deemed" with a smile since it is clearly a way to give up trying

you heard @Splashee@PawelS! get to work! we need that right now! 

1 hour ago, Kujamih said:

you heard @Splashee@PawelS! get to work! we need that right now! 

Actually, I want to be the first in the world to do that, since it is a huge market for it. So no need to look into that

....then what are you waiting for!? Collab with @ pawels now!!

lol

@Splashee @Kujamih I don't really understand the problem, and I'm not sure if my version of geometry would help solve it...