Linear Perspective Puns

Currently in a regression course; and a linear algebra course. My regression course is part one of a two semester regression course, where the first semester course is simple linear and multiple linear regression, and second semester is glms and introducing linear algebra with regression

However, right now, as I’m learning simple linear regression I’d like something which explains why certain formulas are the way they are form a linear algebra perspective. Why are the parameter estimates of B1 the cross product of X and Y divided by cross product of X and X? What does that look Like visually? From a linear algebra and vector perspective?

I would like a text which goes through regression with a linear algebra explanation, as I feel it would help me understand why the formulas look the way they look rather than just memorizing them.

Might be a dumb question. So I know that with Aerial perspective the further away it is the lighter it is.

But I am getting confused with one point perspective. I'm looking at images of hallways and maybe it's the the building is built or the fact that spaces in-between get smaller but it looks darker. Which goes against what I thought it would be.

Can one point perspective end up darker the further you go since it all ends up in a single point or is it a general rule that perspective becomes lighter the further you go?

During quarantine, one of my purposes was to improve my knowledge about perpective.

While I was working on it, I found something, a method that is able to convert information in 2D to linear perspective to perfection, but the most important, to viceversa too.

I think that it's like if I had found how both languages are connected between them, the key to translate from one to another.

And I am not sure why it works or what it represents, I just know it works.

I called this method The Principle of Infinity.

What I am going to expose next is the proof that it works, not how it does. You will have to wait for that for now.

Let's get started:

In the step number 1 we draw a random scalene triangle and the mediatrixes of the three edges that conform the triangle.

In the step number 2, using my method, we draw exactly the same triangle and the mediatrixes in linear perspective.

But, how can we know now if the triangles in both steps are the same at 100% and not only similar?

To know the answer we continue in the step number 3. Using Thales's theorem we divide the red-coloured edge in 3 parts. Now, from one of those parts, we draw a line until one of the vertex of the triangle, de violet line, the one that unites the green circumference and the red one.

Where the 3 mediatrixes cut, we have the blue circumference. From there, we are going to draw a line until the green circumference, the yellow line.

And now we are going to measure the angle of the vertex between the yellow and violet line.

The number in red is the value obtained.

As you can see, we established a simple pattern with the intention to compare both triangles.

So in the step number 4, we are going to repeat the same pattern but this time in the triangle in linear perspective. It is possible to use the Thales's theorem in perspective, and the rest of the lines are easy to draw.

As I said, if It is possible to convert 2D into linear pespective using my method, it is possible to convert linear perspective into 2D too.

To finish the step number 4, we convert the yellow and violet lines into 2D using the method, moment when we can measure the angle between them without any kind of distortion.

The value in blue is the answer the method gives.

The red value and the blue one are completely equals with such accuracy that I cannot think it is only a coincidence, it must mean that the two triangles are exactly the same with no marg

I know some people might crucify me for saying this (cause kings fall and wraith good. Everything else bad.) but deep stone crypt is no joke one of the best raids in terms of design that I’ve seen in a LONG time. Originally, raids had one specific way of doing encounters and there wasn’t much wiggle room to improvise or come up with strats. Garden limited you to tethering with a specific amount of people, Crown didn’t limit you as much but only provided little freedom with blessing strats. Scourge for the most part limited you in every regard apart from the last encounter. And every Leviathan raid was basically the same apart from final bosses.

Deep Stone Crypt sets a new bar for raid design and makes the entire raid flexible with strats. I can’t tell you how many groups I’ve joined where somebody shows me their strat that I’ve never seen before. There are endless possibilities to do all encounters in this raid whereas in others, you would only get some freedom in limited encounters. The sky is the limit. Do you want to solo operator for first encounter and exit manually? You can. Do you want to just switch buffs via terminal in first encounter instead? You can. Do you want to pair scanner with op in the same room at the start? You can. Do you want to gather all 6 fireteam members for tube damage? You can.

And that’s just one encounter. I only laid out a couple strats. I could make a whole page list of all the possible strats and ways of beating encounters but it would take too long. My point is that this raid sets the new benchmark for raid design that past raids haven’t quite been able to do. I hope future raids continue with this level of freedom from now on. This is amazing work from Bungie.

Was going through the organic forms and it seems all we need to do is add contour lines to organic forms to convey volume and form. I am excluding the textures for now.

Did I miss if linear perspective is applied? So that whole vanishing points, converging lines, horizon line all go out the door?

Since the direction an organic form is all based on its contour lines? I mean that’s much easier if it’s all I have to worry about.

It seems most people draw organics in an axonometric projection anyways, since convergence is negligible.

From what I understand Filippo Brunelleschi figured it out in the 15th century, but why exactly did it take such a long time? When we look at the world you can tell that two parallel lines will appear to converge into the distance

I understand that 3+ points might be harder to solve, but 1/2 is just something you can clearly observe

I come from a mathematical background, and I've been thinking about the mathematics of Factorio quite a bit. Recently I realized that there exists a formulation of Factorio factories within a context of a high-dimensional vector space.

For those without a background in linear algebra, I'll do my best to explain (although keep in mind I can't exactly give you an entire course on linear algebra in a single post). A vector space is, in a nutshell, a space where vectors can live in. A vector is a collection of numbers, where each number represents a coordinate for a dimension of the vector space. You may have seen vectors in the context of 2D space (or 3D space) before, where for each vector, the 1st number represents the distance along the x axis from the origin; the 2nd number, distance along the y axis; and the 3rd number (in 3D space), distance along the z axis. 2D and 3D space are 2 and 3 dimensional respectively, since it takes 2 and 3 numbers to uniquely describe each point in the respective space.

Vectors of the same number of dimensions can also be added together. This is done component-wise. For example, in 3D space, we can add (3, 4, 2) and (1, 7, 5) to get (3+1, 7+4, 5+2) = (4, 11, 7). We can also multiply vectors by numbers, which is also done component-wise. If we multiply (4, 11, 7) by 3, we get (12, 33, 21). (I'll explain why these operations are relevant below)

However, these numbers can represent different things as well, and you can add on as many dimensions as you like, albeit once you go beyond 3 dimensions, there's no intuitive way to visualize the vector space by graphing. That doesn't mean that the mathematics break down though, and this is where things get interesting.

Let's say there are n total products in the Factorio world. (There are a lot of products and I don't know the exact number of how many there are.) Then we can define a vector space, which I will call the Factorio vector space, as an n-dimensional vector space, where each dimension represents a different product. A vector in this vector space can then be considered a factory, where the dimension in the factory vector that corresponds to the product p denotes the net-production of p per second. Negative values indicates net-consumption of that product.

I'll explain with an example. Let's only consider 4 products: iron ore, coal, iron plates, and iron gears. Each dimension in this 4D vector space corresponds to the net production per second of the respective produc