My math understanding is like a block of swiss cheese: full of holes. I see something like this & can't even imagine how it's done. Sometimes what I do is pick out something to poke at until I understand it, which is what I plan on doing this graph by @giohio . +
@giohio 2. Will keep revisiting until all the pieces makes sense. I hope. I will ask for help when I get stuck. Starting with what I know: I know that these are two representations of the same expression, but one is graphed on the Cartesian plane, the other using polar coordinates +
3. I know that 3 sin (2x) represents a cycle and a single cycle is 2 pi along the x axis, so sine 2x means something cycles twice as often as a single cycle, hence there are two complete sine waves between 0 and 2pi. I know that the 3 in the equation represents the magnitude+
4. or distance away from the x-axis that the wave goes. That's it for what I know already. I didn't remember how to graph the equation with polar coordinates, but I already looked under the hood, & I remember that I have to use theta instead of x. Trying that out to be sure now+
5. Okay, I have to do more than just change the x to theta. I also have to change f(x) to r.
6. Stuck already. Seems what I need to sort out is what m, M, and A are. Wondering if they have special meaning in Desmos or are they defined somewhere in the way Steve has written this. Will be trying to figure this out next. What I do see is that m is the lower boundary value+
7. and M is the upper boundary for the graph. Wondering why m, M are being used.
8. Okay, have m & M sorted out: min & Max for boundaries, values defined by Steve. Then he defines t being between m & M. Now Christmas is calling, so will return to this later. But this is has been a good leap forward, just being able to recognize lingo & provided definitions.
9. Now looking at t. t is likely thought about as time but it's also x, the values on the horizontal axis, which is, for this graph between m & M, thus between 0 and 2pi. And f(t) is the y coordinate. I have to do this translation to get comfortable. +
9. Now I'm seeing a capital T. Steve has a list that defines 37 different values of T. So I think the small t the continuous generalization of the x axis and the capital T is specific points along the axis. +
10. Line 12 on the graph defines a function A(t). From my grappling with M I know that A isn't some special Desmos thing, but a specific function that Steve is defining. But he's defining it in a way that I'm not comfortable with. I think this is a parametric thing, which +
11. ....I think is why he's using t. Either f(t) is what is chosen for parametric equations, like it's an agreement in math, or it's something that Desmos likes for parametric equations. Or I might have this all wrong. Time to look at some learn.demos videos.
12. Just looked at learn.desmos video for polar graphing. Wasn't really helpful for what I am looking for, but I did see that polar graphing isn't just for trig functions: I can polar graph any old function.
12. Okay. Well this parametric thing turned out to be easier than I thought. I've thought about parametrics before, they've always seemed elusive, but tonight they've made sense. I think. We'll see.
Now I know that when I see t in brackets like (t, f(t)) or even just (t,t) +
Now I know that when I see t in brackets like (t, f(t)) or even just (t,t) +
13. ...that it's a function being written in a parametric way. Desmos will only do a parametric with t. desmos.com/calculator/vud…
Notationally speaking, it's a nicer way of graphing. The first t is what I think of as the x coordinate, the second expression is the generalized y=
Notationally speaking, it's a nicer way of graphing. The first t is what I think of as the x coordinate, the second expression is the generalized y=
14. Still sorting out line 12. It says A(t)= (t, f(t)) 'which means that there is an f(t) somewhere. This is really strange. Seems that desmos has made f(t) kind of synonymous with f(x). Does that mean that line 12 could say A(x) = 3 sin 2(x)? I think so.
15. That's all for tonight. I think I will be able to get through lines 13 -16 tomorrow,. I can see there's notation going on in line 17 that I have no idea about, specifically the .x and .y . But I will put that aside until I get there.
16. Returning to deciphering parts of Steve's graph. Still on Line 12, been thinking about how t & T are introduced, & how desmos assumes both of them are referencing the initial function. Steve is using the t as the generalized equation, then using T to define specific points.
17. I've also been thinking about why the parametric form suddenly made sense to me. Here Steve has written it as (t, f(t)). I had previously seen the parametric as two functions within the parentheses, like the way the Lissajous curve is written, which just scrambled my brain.+
18. ...But seeing it written as t, f(t) somehow gave me something to scaffold on. It's not hard to then think (2t,f(t)) and have it make sense. Now it all makes sense. What a relief. If I get nothing else out of this exercise that finally feeling comfortable with the parametric +
19. ...form, that would still be a big win for me. Okay, enough gushing. Back to line 12. Looks like what he's done is to define A(t) so that he can more easily reference the line 2 equation later on. +
20. Line 11, T is defined so that points appear on A(t). Line 13 creates these points on the graph. I can't see the points that are made by this line unless I turn off the purple. Not sure why these points are not a solid color. Would like to figure that out before I move on.
21. Turns out that the open circle point is just a stylistic choice. Moving on. Line 14 says r(theta) = f(theta) AGain, this is Desmos knowing that f(theta) refers to the equation is in line 1. There are all these leaps of knowing here that are done behind a curtain. +
22. I think what's happening is in Line 14 is that Steve is creating a shortcut way of expressing line 1 in polar form. Line 15 a whole new letter is showing up! Now there's a B(t). It's being defined as something. Will take sometime to examine this, but I'm thinking maybe+
23. ... this is where he starts linking the two forms together?
24. now am seeing the line 15 is where the polar graph is defined as B(t). I have linger over this for awhile to understand how (r(t)cos(t), r(t) sin(t)) is the parametric polar form of f(x)=3 sin (2x) #BrainMayOverheat
25. This is wild to me. Looks like this begins with a function f(x) on the grid, then it's expressed using parametric notation (t, f(t)) , then it's expressed as a polar equation. But still perplexed about how to get to (r(t)cos(t), r(t) sin(t)) from f(x)=3 sin (2x).+
26. ... My conjecture is that the polar form needs a sin and a cos in it, but I'm not really convinced this is true. have to look up how to transform a function into polar form. Not even sure if I'm thinking about this with the right words.
26. Looked into converting the Cartesian equation, which the videos I've looked at call rectangular equations, into polar form. Not really clear on this, but I'm going to go with this: since rectangular equation can be see as y=3 sin(2x) I see that +
27. there will be cos theta and sin theta in the parametric. And since it's been established that r(t) is a stand in for y=3 sin(2x) I can just plop that right in front the cos & sin, thus getting (r(t) cos theta, r(t) sin theta) . This thinking seems kind of simplistic, +
28.... not really sure if it's a real way of working.
Found myself grumbling a bit about all these different forms, then I remembered that what attracted me to this graph is the connections that are being made between the different forms, so of course they have to show up here.+
Found myself grumbling a bit about all these different forms, then I remembered that what attracted me to this graph is the connections that are being made between the different forms, so of course they have to show up here.+
26. ...seems that the parametric bridges the two. And now my daughter has brought some tea up to visit with me so will get back to this tomorrow. Would welcome any insight on my take on Line 15.
30. My numbering got mixed up 26 above should've been 29. Daughter is drawing now, just likes company, so I am getting back to this. Thinking I need to recap, to remind myself of the big picture. First, function is defined. Some points are defined on the function. Function is
31. ... expressed in polar form. Same points are defined in the polar form. Then both equations are expressed in parametric form. +
32. I suspect that the parametric form is needed to find a common way to express both equations. Otherwise it's apples and oranges. The parametric gets them to be fruit, so we can talk about them simultaneously.
33. So the polar form and the rectangular form (a new word for me, feels odd to be using it) are both expressed parametrically, then expressed more simply at A(t) and B(t).
Twitter spell check does not like the word parametrically. Does it have grounds not to like this word?
Twitter spell check does not like the word parametrically. Does it have grounds not to like this word?
34. line 18. Now I need help. I have no idea what is going on with this notation. It's the .x and .y in the expression that is completely unfamiliar. I don't even know where to begin to figure this out. closing down for the night. Hoping when I come back help has shown up. :)
35. Line 18. This is where all those pieces that have been defined come together. What I want to do it make sense of it, then understand it enough so that if I wanted to make this graph from scratch, I could, though that may be too far a reach.
36. First thing to notice is that I have to contend with n. Just ignored n earlier. From line 6, n=1. It also travels between 0 and one. How complicated could that be? Now it's completely mysterious. It seems to be the tool that transforms the Cartesian points to the polar. +
37. I'm not too worried about the .x and .y any more, having read this and feeling better about my understanding of line 15 because of
38. Back to thinking about n. n appears to be the tool that transforms the line/curve from the Cartesian curve to the Polar curve. There's not really any information embedded in the steps of the transformation, other than being able to say something like, hey, it's 64% there.
39. This n is also like the comic relief, as it's a funny squiggle when it's in mid-transformation. But how does one come up with thinking of how to make this?
40. My daughter is here, sees me burying my head in my hands, worries she is bothering me, which I assure her she is not. I'm beginning to unravel line 18 & it's just so lovely. But the dog is drinking my coffee, so my bliss is seriously interrupted.
41. Line 18, it's a parametric equation, in which the x is the sum of the Cartesian and the Polar, and the y is also the sum of the Cartesian and the Polar. n scales the A(t) while the inverse of n scales the B(t) in both the x and y. Like water flowing from one vessel to another
42. Line 17 and Line 18 are pretty much the same, but one references t, which is the generalized curve, and the other reference T which are the points. +
43. Lines 19 and 20 are hidden, so I'm not going to try to figure these out now. But I did hide everything but the moving n then turned on lines 19 and 20 and it was quite fun. But moving on. +
44. Line 21 is blank, line 22 is a duplicate of line 18, so there is only one line left to decipher. It uses s, which is defined in line 8. Here in line 23 it is a multiplier of T, which are points along the curve. s controls the numbers of points on the curves.
45. Line 23 is the most extraordinary, in that it pairs the points on the two graphs, and even tell me which of the 37points on each graph are being paired. Husband wants me to go the hardware store, so I am stopping here, but, again, the notation will be stopping me soon anyhow.
46. Continuing to look at Line 23 of desmos.com/calculator/6t6… The first thing I wondered about was, why the curly brackets? Took them away & desmos objected. Without the curly brackets desmos told me that the colon mark made no sense. So I took away the part in front of the +
47. ...colon mark. I mean, why do we have to know that s>0? After all, it's defined that way on line 8. But desmos has no patience for my logic, simply said told me, by clicking the triangle with the ! in it, that this piecewise function needed at least one condition.
48. I had to pause to consider why this function is piecewise. OK, because we're using T here, which is the set of points, each point being a piece. Don't know why desmos want a condition. Just wants to assert that I better realize this is a piecewise function? If that;s the case
49, Is it important what the condition is? Turns out desmos is happy with many different conditions, as long they are less than the number of points, in this case 37. I tried s>-55, t>0, T > -15 . There were lots of conditions that didn't work, but it appears that the fact that+
50. ..there is a condition is more important than what that condition is exactly. +
51. I'm trying to put into words what's happening on Line 23, but whatever words I put together seem to sound like mumbo jumbo. Instead, I'm going to try to put into words what I would be wanting line 23 to do if I were the creator of the graph.
52. On the two different ways of graphing the expression, want to connect the corresponding pairs of points. This means that as I move from one point to another on one graph I move from one point to another on the other graph. t represents the continuous, generalized function
53. ...for both graphs, so t needs to be involved in scaling all expressions simultaneously. Like in line 17 (or 18), the connections between the two graphs happen within a parametric equation in which the x and the y are describing both graphs. #pauseInAwe +
54. so the x of the parametric equation scales A(T) by t, and B(T) by the inverse, same for y. But there is something else going on, since we're matching up pairs of points. We want to see these pairs, one set at a time+
55. ...so we need to tell our equation to show us pair 1, then pair 2, etc through pair 37 one at a time. Doing this is the power of s, though I will be going back to figure out how s got this power. so be it, everything has to be expressed as a function of s, so is that called+
56. a function within a function? One thing that I found fun was removing s's from line 23. I left s>0 since, as I said, Desmos seems to want a constraint but isn't particularly fussy about what that constraint is. So s has to be a function of the functions & all is well. End :)
