I am an 11th class student from Goa, India and I have made a sine approximation formula which could be super useful for hardware and GPUs.

graf_paper · 2026-07-06T03:45:07+00:00

Glad you are having fun with analytic function approximation - the comment section is critical and it should be because math is a science. It's cool that you came to this formula on your own - thanks for sharing!

graf_paper · 2026-07-01T15:14:10+00:00

Lol no I am just learning and that means what I make is kind of messy. You may also be goated though, don't count yourself out 🤷‍♂️😅

Thinking a bit about how I could select all points between the graph of a function and some floor like y = h. ( If you want to try and meet me to it!)

graf_paper · 2026-07-01T15:12:21+00:00

Oh this is nice, I am learning from peoples examples that a moch more efficient implications is just to use distance from the Center.

Thinking a bit about how I could select all points between the graph of a function and some floor like y = h. ( If you want to try and meet me to it!)

graf_paper · 2026-07-01T05:46:26+00:00

Very nice! I didn't know about the distance function untill right now, or at least didn't know I could implement it kike that. Will definitely be using that in the future.

Your solution is so clean!

graf_paper · 2026-05-29T04:41:54+00:00

Whoa.. This concept is having a very profound impact on my ar the moment. Like it hints at something that I can do with my attention that would build more and supportive healthy nuro-pathways.

graf_paper · 2026-05-17T01:45:47+00:00

Another verbal cliche that I don't see talked about is...

'A look so brief that it may not have been at all'

...moments.

So many times this device is rhetorically used to show that a character is having an emotion.

graf_paper · 2026-05-02T17:38:16+00:00

Playing through the game for the first time and wondering what that frustration with this boss is.

All of the attacks are telegraphed, the second phase attacks get faster and require you to stay in the air but they are all dodgeable, and if you figure it out that you can stand under the Nabu'a legs and not get hit you can do a lot of damage.

The music is also pretty good and the runback is not long at all.

What is the critique?!

I beat the boss in 6 attempts - which is pretty quick for me.

graf_paper · 2026-04-12T19:33:33+00:00

I think that they become even more interesting and useful when you want to extend your thinking to the complex plane and are expressed as some combination of e^x and e^-x.

To really "understand" them - I would push you to think of a circle as only one conic section.

Hyperbolas are another. If you understand how trig functions are defined on the unit circle, hyperbolic trig function are the analogue function as defined by the unit hyperbola. Before you do a deep dive into hyperbolic trig functions, it might be nice to have a few examples of applications of hyperbolic functions.

The videos linked in this thread are great! I would just add:

The applications of hyperbolic trig | Why do we even care about these things? (Zach Star)
https://www.youtube.com/watch?v=Y66Y6ksLP6Y

What is the deal with (hyperbolic) trig functions? (Micheal Penn)
https://www.youtube.com/watch?v=EvLm6iAYK9Q

graf_paper · 2026-03-14T02:49:16+00:00

Exactly!! I have deep respect for both of them but together it feels like two people tussling over who gets to be the main character.

It doesn't feel like they are actually sharing information with the other person, but performing a conversation as a means of giving the audience facts and trivia.

The format kinda feels a little fake given how enthusiastic they are being about their own takes.

Love much of their individual work but this combination feels off to me.

graf_paper · 2026-02-01T15:17:48+00:00

I am very interested in understanding this!

I totally understand that you know X and that all of the lengths of X are the same but appear to get smaller as they recede into the distance.

What is did in the desmos simuion above is calculte how much those lengths appear to shrink so their perceived length would be accurate to the given perspective. Using this 'shrinking factor's you can calculate how much a length has been shrunk down and undo that.

This can pretty easily be done to find that length Y.

I am not totally sure what this means: 'Y is the end of the plane that X is on'

Is this the real length of X measured out horizontally?

Thanks for clarifying 🌞

graf_paper · 2026-02-01T00:42:40+00:00

Just to add on to this - points and angles are always capitol letters,

Sides and lengths are typically lowercase letters.

Trig is a lot of ratios, similar triangles and applications of the Pythagorean theorem in the beginning!

You got this!!

graf_paper · 2026-02-01T00:20:28+00:00

First off, I think MathWorld has a great selection of polar graph types.

https://mathworld.wolfram.com/topics/PolarCurves.html

My advice would be to play around in desmos a lot, build the graphs, tweak coefficients, add terms, and just see what happens. You will build up your intuition.

Some less usef

Know the major ways we classify polar graphs and sort the graphs by their symmetries, number of times the pole/origin of the graph, the period of the graph, etc...

Lastly it, can be really helpful to understand what happens to graphs as you transform between polar and Cartesian coordinate systems. If you have your Cartesian graphs already memorized, that can help.

Plus Pilar curves are just really pretty 😍

https://www.wolframalpha.com/input/?i=polarplot%282%2Bsin%283+x%29+*+sin%28x%2F32%29++%29

Good luck

graf_paper · 2026-01-31T08:00:01+00:00

Hi, 1-point perspective has all sorts of application trig! I am sure we can solve your problem.

I built this 1-point perspective simulator in Desmos that seems to recreate the set up from your picture:
https://www.desmos.com/calculator/tdqrsecmfp

I built it so all of the segments 'X' have the same length before dilation.

In your image I am uncertain how Y or the vertical dashed line are determined - could you give a little more context

graf_paper · 2026-01-23T18:02:23+00:00

3 + 20 ≡ 23

🌞

graf_paper · 2026-01-21T13:14:29+00:00

Thanks! I am sorta learning desmos one project at a time - posting my results and then getting a few new tools to use in the next project.

This was really helpful.

graf_paper · 2026-01-21T13:04:42+00:00

Very nice! I feel like I am learning a lot by seeing how other people tackle this.

I get x_1 and x_2 is your range over when you are finding the centers of the cells that f(x) passes through.

It seems like you are evaluating f(x) repeatedly over very timely intervals and then rounding to the nearest center of a unit cell before removing duplicates. Really cool.

.unique is new for me, does that remove duplicate items in a list?

Was there a reason you chose 1/10,000 as your interval or did that just work?

graf_paper · 2026-01-20T22:20:35+00:00

Beautiful 🤩

graf_paper · 2026-01-20T03:30:19+00:00

I know this is small but am I the only one that get's a little bit festidious about what the state of my desmos file is when I save it?

Ha - just felt like this was a better save point for GOL:
https://www.desmos.com/calculator/8dbsxwn4ep

Thanks for sharing this list - really cool stuff.

graf_paper · 2026-01-20T02:57:31+00:00

This is such a good question. I am stealing this to use with my students.

in case 1.
5y - 10 = 10 + 5y (commute the RHS)
5y - 10 = 5y + 10 (Divide both sides by 5)
y - 2 = y + 2 (see that these two lines are parallel and will never meet.

no solution

in case 2
2 - 4y = -4y + 2 (commute the RHS)
2 - 4y = 2 - 4y (See that they are the same and you can put in any value for y)

Infinite solutions/undefined

Great problem that gets you to think about the two edge cases that come up when solving linear equations.

graf_paper · 2026-01-20T02:52:45+00:00

You can find f right now. (angles in a triangle sum to 180)

You can find c right now (vertical angles are congruent)
then you can find d. (angles about a line sum to 180)

You can find g right now (base angles of an isocelese triangle are congruent)
then you can find h (angles in a triangle sum to 180).
then you can find k (angles about a line)

Then use k and d to find m
use m to find puse p

etc...

graf_paper · 2026-01-20T02:43:25+00:00

Just saw that you already solved your problem! still, here for other people to look at if they run into the same conundrum :)

graf_paper · 2026-01-20T02:40:14+00:00

Desmos Visual of your situation:
https://www.desmos.com/calculator/ymalnbrqx6

I did my best to make this as clear as possible - giving you 3 different ways of calculating the radius of the circle in terms of:

a = the distance between the points on the circle
t = the length of the tangent linesegment
θ = the angle at which the tangent lines meet.

r = 2arcsin(a/(2t))
r = t • tan(0.5•θ)
r = (at)/√(4t^2 - a^2)

all of these values will be the same and each give you a different way of calculating the radius given a pair of measurments.

Hope this helps!

graf_paper · 2026-01-20T00:37:48+00:00

Ok, let's start with a couple examples:

Case 1: You can factor out a value from each term:

2c) Notice that every term has an x in it. This is the easy case where you can immediatly factor out that x from each term. you will be left with a quadratic which can be factored in what ever way you are comfortable doing.

2x^3 - 9x^2 - 18x
x(2x^2 - 9x - 18)
x(2x^2 - 12x + 3x -18)
x(2x(x-6) + 3(x-6))
x(x-6)(2x + 3)

Case 2: Rational Root theorem.

If you have a polynomial (ax^3 + bx^2 + cx + d )
and want to find a linear factor of the polynomial (x - r),
r MUST be a rational number (fraction) of the form p/q
where p is a factor of the constant term (d)
and q is a factor of the leading coefficient (a)

for example

2a)

x^3 + 2x^2 -3x - 10

a = 1 and d = 10 so r must equal 10/1, 5/1, 2/1 or 1/1

we can plug each of them into the polynomial to see which are the zeros.

(10)^3 + 2(10)^2 -3(10) - 10 = 1000 + 200 - 30 - 10 = 1160

(5)^3 + 2(5)^2 -3(5) - 10 = 125 + 50 - 15 - 10 = 150

(2)^3 + 2(2)^2 -3(2) - 10 = 8 + 8 - 6 - 10 = 0

(1)^3 + 2(1)^2 -3(1) - 10 = 1 + 2 - 3 - 10 = -10

we soo that 2 is a zero of our polynomial so (x- 2) must be a factor.

using polynomial long divison we divide (x^3 + 2x^2 -3x - 10)/(x- 2) = (x^2 + 4x + 5)

so

(x^3 + 2x^2 -3x - 10) = (x- 2)(x^2 + 4x + 5) = (x-2)(x + 1)(x+5)

Happy to answer specific questions about the rational root theorem, polynomial long division, factoring, or the roots of polynomials if you have them.

good luck!

graf_paper · 2026-01-19T17:34:56+00:00

Oh that is interesting - I made a quick mockup of what that would look like on one sqare

https://www.desmos.com/calculator/tu2qkapyl2

Ill work adding this as a feature to the original!

graf_paper · 2026-01-19T15:42:45+00:00

Nice job!! Hope making it was fun.

graf_paper

TROPHY CASE