Ask HN: What to learn for math for modeling?

  > I have done math classes

What classes? It's a bit hard to suggest what is appropriate without understanding where you currently are.

  > to brush up.

But I think this might be the wrong approach. Especially for these topics. Instead of learning the minimum you need to just understand a specific algorithm, you should understand what the symbols mean, what they are doing, and how they are interacting with one another. Math is a language. But it is a language designed for precision.

A big part of why I'm saying this is because at their core, these are not difficult equations from a computational perspective. You really just need to understand addition, subtraction, multiplication, and division. My point is that the hard part about math often isn't the calculation part.

Let's take B-Splines for an example and look at what Wiki says

  > A B-spline is defined as a piecewise polynomial of order n, meaning a degree of n − 1.

Either this is simple or incomprehensible, so let's break it down.

What does piecewise mean? Here's an example piecewise function.

          a     if x > 0
  f(x) = {0     x = 0
          b     if x < 0

Given you're on HN and looking at computer graphics, I take it you're familiar with if statements? That's the same thing here. Let's translate. Looking at things from a different perspective can really help.

  if x > 0:
    return a
  else if x < 0:
    return b
  else: # x == 0
    return 0

We should think of x as a location and we're just describing something about that location. Here's another piecewise function, but now in everyday English: "Across the street all the houses end in even numbers, but on my side of the street all the houses are odd." There's always more ways to look at something. If you can't see what something means from this angle, try another. Be that another book teaching the same thing or another framework to describe the same thing. Get multiple perspectives. One will make more sense than another but which one that is tends to be personal.

"polynomial of order n"

A polynomial is an equation like a_0 + a_1 x + a_2 x^2 + ... + a_n x^{n}. Sorry, let me rewrite that: a_0 x^0 + a_1 x^1 + a_2 x^2 + ... + a_n x^{n}

Now in code

  for i in range(n):
      sum += a[i] * math.pow(x, i)

But we should break this down more, just like before. What is a_0? It is just a number. What does this represent? A point. What is a_1*x? That's a line. x is an arbitrary location. The a_1 is a scaling factor. So if a_1 is the number 2, then what our x is representing is all numbers (..., -4, -2, 0, 2, 4, ...} (is that all my neighbors across the street?) The a_2 x^2 is a curve, specifically a parabola. We can do the same thing. But also remember we're summing them together. So you have a point, a line, and a parabola. When you add them together you get something that is none of those things.

What is really important here is that these functions you are interested in have a really critical property. They can be used to approximate almost anything. That almost part is really important, but you're going to have to dig deeper.

My point here is that take time to slow down. Don't rush this stuff. I promise you that if you take time to slow down then the speed will come. But if you try to go too fast you'll just end up getting stuck. This is tricky stuff. When learning it doesn't always make a ton of sense and unfortunately(?) when you do understand it it is almost obvious. Think of the slowing down part like making a plan. In a short race you should just take off running without thinking. But if it is longer then you will go faster if you first plan and strategize. It is the same thing here and I promise you this isn't a short race. You don't win a marathon by winning a bunch of consecutive sprints. The only thing you win by trying that is a trip to the hospital.

So slow down. Break problems apart. Find one part you think is confusing and focus on that. If it all seems confusing then try to walk through and force yourself to say why it is confusing. Keep doing this until you have something you understand. It is an iterative process. It'll feel slow, but I promise it pays dividends. Any complex problem can be broken down into a bunch of small problems[0]

You got this!

[0] This sentence doesn't just apply to how to figure things out, it applies directly to what you're trying to do and why you want these functions. If it doesn't make sense now, revisit, it will later.

Very nice explanation!

In particular; the idea of breaking down a polynomial as a sum of terms, mapping each term to a graphical view (i.e. analytic to coordinate geometry) and then realizing such a sum of terms can be a complex curve (i.e. a complex graphical view) which can be an approximation of almost any function.

The teaching of mathematics has become so abstract that students are not taught how to map it to geometry which is THE way to build intuition. You can understand a lot by just imagining 2D/3D mappings before generalizing to n-dimensional vector spaces. Every student/beginner should study I.M.Gelfand's Functions and Graphs so that they can train themselves to imagine the graphs corresponding to a string of symbols.