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Comment by kstenerud | Hacker News Reader

http://www-math.mit.edu/~djk/calculus_beginners/chapter03/se...

We can determine the linear function which takes value f(a) at a and f(b) at b by the following formula:

                x-b        x-a
    f(x) = f(a) --- + f(b) ---
                a-b        b-a

The first term is 0 when x is b and is f(a) when x is a, while the second term is 0 when x is a and is f(b) when x is b. The sum of the two is therefore f(a) when x is a and f(b) when x is b. And it is a linear function. Linear functions have a term that is x multiplied by some constant, and may also have a constant term as well.

Umm... whut?

This has me completely lost. I don't see how this can be geared towards beginners and artists.

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Try this idea:

  g(t) = (1-t) f(a) + t f(b)

where t goes from 0 to 1. At the end points, we get

  g(0) = (1) f(a) + (0) f(b) = f(a)
  g(1) = (0) f(a) + (1) f(b) = f(b).

This can be used to linearly blend any two things, as for t in between 0 and 1, we'll get part of f(a) and part of f(b).

Now, if we want to use x instead of t, where x goes from a to b, we need to convert:

  x = a  <=>  t = 0
  x = b  <=>  t = 1

The distance that x has travelled from a is (x-a). The total distance from a to b is (b-a). t is the proportion of that total distance, so t is the distance travelled over the total distance:

      x-a
  t = ---
      b-a

Now (renaming g to f as we're changing parameterisation* )

  f(x) = (1 - (x-a)/(b-a)) f(a) + ((x-a)/(b-a)) f(b)
       = (1 + (x-a)/(a-b)) f(a) + ((x-a)/(b-a)) f(b)             //since -(b-a) = a-b
       = ((a-b)/(a-b) + (x-a)/(a-b)) f(a) + ((x-a)/(b-a)) f(b)   //since 1 = (a-b)/(a-b)
       = ((a-b+x-a)/(a-b)) f(a) + ((x-a)/(b-a)) f(b)             //combining terms with common denominator
       = ((x-b)/(a-b)) f(a) + ((x-a)/(b-a)) f(b)

Does that explain it?

* g(x) is something different: g(x) = (1-x) f(a) + x f(b), just by substituting in.

1. X-b befomes 0 if x=b right?

2. This means anything multiplying this x-b becomes 0 as well when x=b right?

His definition of linear is also wrong in my book.

Linear: f(x) = a x

Affine: f(x) = a x + b

(For some a, b)

> His definition of linear is also wrong in my book.

There are two variations of "linear function". To quote https://en.wikipedia.org/wiki/Linear_function:

> In mathematics, the term linear function refers to two distinct but related notions:

> - In calculus and related areas, a linear function is [...] a polynomial function of degree one or zero.

> - In linear algebra, mathematical analysis, and functional analysis, a linear function is a linear map.

I personally prefer the terminology "linear function" for the former definition (polynomials of degree 1 or 0), and "linear transform" for the latter definition (a linear map).

What you are calling [“linear”, “affine”] are in some contexts called [“homogeneous linear”, “linear”] or the like.

This is not “wrong”, but just a different convention.

Is that comment for me or about the referenced page? If it's for me, I'm not sure what you're saying.

I'm trying to explain what he meant. Which step don't you understand specifically in my comment?

I'm trying to understand why there's f(a) and f(b), why is b being subtracted from x, why it's divided by a-b, why there's a similar thing for f(a), why the two are being added, why the zero values are important, and what the whole thing actually means.

It’s simpler if you use 0 and 1 instead of a and b. If we want a linear function such that

f(0) = A, f(1) = B,

then the unique solution is

f(t) = (1-t)A + tB

The more general version with a and b is just composing this with a transformation of [a, b] into [0, 1]

Relevant keywords: linear interpolation, affine combination, barycentric coordinates.

It's demonstrating how to find the line that passes through two points x=a, x=b on the function y=f(x).

Without having read the actual page, I assume he's going to move a and b close to eachother to approximate the tangent of the function at a point.

edit: In higher level math, you use circles/spheres or parabolas/paraboloids to approximate functions, but in high school level calculus you stick to using a straight line to approximate a function

Unfortunately, he doesn't do that. There's an even more complicated formula under that which is supposed to be related to the slope of some line and a ratio between b and a (I think?) and then I'm guessing he reduces it, but doesn't explain how he did it.

I've tried learning calculus 3-4 times during my life, using materials for an "absolute beginner", and this has always been my experience, as if one were teaching programming by going from "This is a variable. It can store data." directly to "A monad is a monoid in the category of endofunctors." To this day I have no idea what calculus even is, or what it's for.

You want a linear function f(x) such that f(a)=A and f(b)=B given a, b, A, B. A valid f(x) (actually it's the only one) is defined as

             x-b     x-a
    f(x) = A --- + B ---
             a-b     b-a

You can calculate f(a) and f(b) to confirm that it works.

However, I don't find this kind of guide any good for learning math...

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