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http://www-math.mit.edu/~djk/calculus_beginners/chapter03/section02.html http://www-math.mit.edu/~djk/calculus_beginners/chapter03/se... We can determine the l

http://www-math.mit.edu/~djk/calculus_beginners/chapter03/section02.html 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.

baby 7y ago
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)
- kstenerud 7y ago
  Is that comment for me or about the referenced page? If it's for me, I'm not sure what you're saying.
  - baby 7y ago
    I'm trying to explain what he meant. Which step don't you understand specifically in my comment?
    - kstenerud 7y ago
      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.
      - Xeanort 7y ago
        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...
      - llamaz 7y ago
        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
        kstenerud 7y ago
        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.
      - jacobolus 7y ago
        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.
- cornstalks 7y ago
  > 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 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).
- jacobolus 7y ago
  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.
mkl 7y ago
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.

baby 7y ago

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)

kstenerud 7y ago
Is that comment for me or about the referenced page? If it's for me, I'm not sure what you're saying.
- baby 7y ago
  I'm trying to explain what he meant. Which step don't you understand specifically in my comment?
  - kstenerud 7y ago
    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.
    - Xeanort 7y ago
      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...
    - llamaz 7y ago
      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
      - kstenerud 7y ago
        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.
    - jacobolus 7y ago
      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.
cornstalks 7y ago
> 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 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).
jacobolus 7y ago
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.

mkl 7y ago

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.