This video is really cool, you have a super simple derivation for the transformation too. One thing I think you should have included was why using a rule to rotate the point counterclockwise ended up rotating the graph clockwise. It was because plugging the new expressions in for x and y was basically saying "the counterclockwise rotations of these points satisfy the equation." So the said points would be a rotation in the clockwise direction from the original equation (very similar to how replacing x with x+2 in a function actually moves the graph to left by 2, instead of the right). To go in the standard counterclockwise direction, you can plug in negative theta and simplify with sin and cos rules. Also, it explains why the parametric equations still rotated counterclockwise, because you replaced the functions with the expressions rather than x and y, so the values that were equal to the new x and y rotated counterclockwise, instead of the counterclockwise rotations of x and y satisfying an equation. Again, great video. You definitely deserve more subs for this quality of video and explanation.
@fahrenheit2101
Жыл бұрын
I knew some subtlety had to have been glossed over - thanks for this. Still a little weird to wrap my head around though.
@uchinanchuu58
Жыл бұрын
You answered the main question I had about this video. Thanks!
@may21136
Жыл бұрын
Yep. Given that positive theta = anticlockwise, we should ideally start with the transformed coordinate point (x',y'), rotate the point (x',y') back to our original coordinate system (x,y) in a clockwise direction (wherein negative theta comes), and then use the equation y = f(x). The final equation should be consisting of x' and y' terms. What this guy did, is that, instead of finding the locus P' (x',y'), he ended up finding the locus of P' conjugate (x',-y'), entirely going against the initial purpose.
@zackattack9228
Жыл бұрын
Thanks uncle Ben🙏
@rolosilver3256
Жыл бұрын
UNCLE BEN?!
@OrangeC7
Жыл бұрын
"All I ask for is infinite precision, is that so much?" Every mathematician ever
@omargoodman2999
Жыл бұрын
Heisenberg Uncertainty Principle: Yeah, about that...
@someoneonyoutube8622
Жыл бұрын
@@omargoodman2999 Gödel’s incompleteness theorems & Turing’s halting problem… we need to talk
@FireyDeath4
Жыл бұрын
See if you can run Desmos on your personal hardware, get a chunkier graphics card and make the limiting parameters bigger
@NoerLuin
Жыл бұрын
Fun Fact: in this case it is not about precision, the reason why it looks wrong is aliasing (the signal processing kind). The simple version is, that in each pixel on the screen there are multiple red lines, which cannot be shown correctly to you.
@someoneonyoutube8622
Жыл бұрын
@@NoerLuin what if someone invented a computer that could rotate pixels acording to the direction needed to display the best resolution for a given image
@pkmnhx43_27
Жыл бұрын
Finally, I can rotate the line y = x I've always wanted to model the values of y where it is twice and much as x, but never knew how to rotate it, I can finally live in peace
@incredulity
Жыл бұрын
Lol
@user-pr6ed3ri2k
Жыл бұрын
tan(a)x be like
@capsey_
Жыл бұрын
@@user-pr6ed3ri2k nah dude, i use noodle technique, it's taking a raw noodle on a paper, spinning it around and drawing what it looks like on canvas with oil paints
@dukeofhollow5541
Жыл бұрын
Also y = 2x and y = 0.5x graphs be like
@reeb3687
Жыл бұрын
@@incredulity is your username dakota? it uses letters ive seen in dakota
@IlTechnoDashlI
Жыл бұрын
For those people who don't want to watch the whole 16 minutes: 1) Replace all the X's in your function with "x cos(Θ) - y sin(Θ)" 2) Replace all the Y's in your function with "x sin(Θ) + y cos(Θ)" 3) Set the "Θ" parameter to whatever angle you want your graph to be rotated by And that's it!
@cemmy410
Жыл бұрын
Thank you! The video topic is very interesting and I would have watched the whole thing but I had to tap out about 4 minutes in because the pitched-up voice is a sensory nightmare 😩
@Dark-jn2pg
Жыл бұрын
Thanks so much
@kelly4187
Жыл бұрын
And if anyone has ever encountered a rotation matrix... You already know how to do it lol. It seems a weird choice to go through this and not at least mention it at the end, instead choosing to go through a million examples when one or two would have sufficed.
@starfeast
Жыл бұрын
That's literally a matrix. Thank you for saving me 14 minutes.
@manioqqqq
Жыл бұрын
Add the parametrics
@stealthgamer4620
Жыл бұрын
I really like that he basically taught the polar coordinate plane and system without actually using or saying that it is. Props to this person.
@fuschia-draws
11 ай бұрын
all while answering an age-old question math enthusiasts always ask!
@jacobbaer785
Жыл бұрын
One thing to keep in mind is that whenever you rotate a graph, it likely is no longer a function (if it was one to begin with). Some exceptions I can think of are straight lines, and sin and cos (rotated no more than 45 degrees). Otherwise,the curve will "bend over itself" and the same 'x' value can result in 2 different 'y' values. In other words, functions when rotated will, with some exceptions, always become implicit equations.
@notamouse5630
Жыл бұрын
And the proper solution to that is no longer thinking of it as a function in the cartesian plane, but instead the polar one. y=f(x) -> r=f(theta) then rotate and it can be a function. Or a parametric equation.
@astralnekomimi
Жыл бұрын
This is one way to define that a function is one-to-one: a function is one-to-one if and only if it can never be rotated about the origin in such a way that it is no longer a function.
@AliAhmed-ez2zy
Жыл бұрын
@@notamouse5630 Agreed, that's how I approached the original problem; parameterize your equation to make a vector in ℝ² as a function of t, and then apply the general rotation matrix in ℝ²: Rot_θ = {{cosθ, -sinθ}, {sinθ, cosθ}} So for a generic vector valued function v(t) in ℝ², the rotation would just be (Rot_θ) v(t). It's a generic linear algebra approach to the problem that yields the same results.
@kelly4187
Жыл бұрын
@Astralnekomimi not quite. Y=X can be rotated up to 45 degrees and still be a bijection.
@rashid.harvey
Жыл бұрын
This was missing, you can't really rotate all functions like stated at the beginning of the video
@withjoe1880
Жыл бұрын
Desmos can use degrees if you open the menu (wrench in upper right corner) and change from radians to degrees. You can also changes axis limits, ticks, polar, and more.
@Sahl0
Жыл бұрын
needs likes cos important
@AwesomeEv
Жыл бұрын
radians usually works better because you don't have to change the axis scale for a sin function specifically
@user-iz5pd7tj6q
Жыл бұрын
@@Sahl0 I think its sin important
@Sahl0
Жыл бұрын
@@user-iz5pd7tj6q inverse tan important
@jwjustjwgd
Жыл бұрын
You can also just put a degree symbol after a number while in radian mode and it will calculate that number specifically as degrees
@justinelliott4127
Жыл бұрын
If my math teacher had shown the movement and number changes like you did in the first 30 seconds here, I could have avoided so much pain. Why they expected everyone to be able to just look at the numbers and automatically understand I'll never know.
@no-bk4zx
Жыл бұрын
The best way I found to intuitively understand graphs is to just plot it on a graph paper. Sit down, draw the axes, start taking some easily calculatable values of x and just plot it. Don't use calculators for as many values as possible and when you run out of easy values, then use calculator. Usually just gives me a good enough understanding of exactly why the graph is what it is.
@justinelliott4127
Жыл бұрын
@@no-bk4zx Makes sense. I just know that I understand so much better with a corresponding automatic changing visual. There used to be this sim called Orbiter. Space flight but with hard numbers. I was grasping complex orbital mechanics through mathematical inputs while seeing the spacecraft change and also orbital trajectories change in real time.
@AlanCanon2222
Жыл бұрын
I feel the same way. I was educated before apps like this were commonplace (though computer graphics certainly existed, and such a program would be easy to write). But it can be shown just with chalk and a blackboard, using a few examples, even without animation.
@WhyneedanAlias
Жыл бұрын
I actually found out how to do this quite recently. I was playing around and noticed if I change the x in y=x² to x+y and y to x-y I would get a rotated parabola. Then by changing the ratios to like 5x+3y I'd get different rotations but they'd also always get scaled by some factor. So I also added complicated scaling factors until I tried using trig functions to scale the axes and it became so much easier. And after having taken a linear algera class it also makes much more sense because it is basically just applying a rotation matrix to [x,y]
@kelly4187
Жыл бұрын
Still, nice work! Maths is supposed to be something we play with, not simply memorise to pass tests. No matter how hard it gets, you're doing it right!
@trippstreehouse
Жыл бұрын
Thanks for the demonstration, wish you didn’t pitch shift vocals.
@shmuelalexis9836
Жыл бұрын
This questions, of rotating the graph, have been in my interest for long time. I always though a general procedure exist - glad I found your channel. Great work.
@int16_t
Жыл бұрын
So, basically you apply a 2D rotation matrix on the curve of the function. I find it interesting the inverse of x^2 (which is an even function) is sqrt(x) which is rotated 90 degree clockwise, and the inverse of x^3 (which is an odd function) is cuberoot(x), which is rotated 90 degrees (either sides), and flipped horizontally. While the inverse of 1/x is 1/x itself. Cool!!
@judecarter6095
Жыл бұрын
In fact every inverse function is equivalent to a pi/2 rotation and a reflection in the x axis, because that's functionally the same as reflecting in y=x.
@angeldude101
Жыл бұрын
Alternatively, you multiply by a complex number. Using a matrix let's you represent arbitrary linear transformations, but complex numbers restrict you to just rotations (and scaling if you let them be unnormalized) which is perfectly fine and more efficient if that's all you need.
@aliensoup2420
Жыл бұрын
Yeah, he's kind of lying when he says he doesn't use matrices. He is writing out the matrix operations long-hand as a new function, but he is still applying the standard rotational matrix transformation. It seems that a sophisticated graphing application could perform the proper substitutions without the need to write it out yourself. In a sense, he is defeating the purpose of layered abstraction, which is the general basis of higher mathematics.
@jaythegreat9211
Жыл бұрын
@@aliensoup2420 You do have to remember the target audience of this video is people in lower levels of math
@kelly4187
Жыл бұрын
Then perhaps he should have not said anything about matrices in the description, and actually mentioned them in the video at the end. Tease them with the method and examples in the video, but then say "you know how I said we rotate the x-y plane not the curve... Well there's a more general way to transform the x-y plane..." And at least simply name drop
@DoctressCalibrator
Жыл бұрын
I couldn't help myself but laugh when you added this modified voice that said "Shut up and tell us already." You did a great job!
@sander_bouwhuis
7 ай бұрын
You blew my mind with this video. The visual presentation makes it extremely clear that it indeed seems to work for all sorts of equations.
@muffinconsumer4431
Жыл бұрын
No pitch shift = 7x better video
@sportsloverbaseball
Жыл бұрын
What does it matter? Maybe they just don’t feel comfortable with their voice being heard
@muffinconsumer4431
Жыл бұрын
@@sportsloverbaseball And I don’t feel comfortable not hearing it
@EHMM
Жыл бұрын
@@muffinconsumer4431 Literally only you.
@muffinconsumer4431
Жыл бұрын
@@EHMM despite tens of other comments to the contrary. Riiiiiiight.
@EHMM
Жыл бұрын
@@muffinconsumer4431 Literally only abnormal people
@ergenarkenk1458
2 ай бұрын
I never realised this video was 16 minutes until the end! Fascinating presentation, really loved it!
@magnusalferes1143
Жыл бұрын
I need to thank you so much for this, I've been working on a video game for a while now and decided that I would spawn things as I go rather than hand build in the editor. You have bestowed the power of rotating graphs upon me and now I can build using arrays and rotate after, simply amazing!!!!
@igxniisan6996
Жыл бұрын
YOU DON'T KNOW HOW MUCH I WANTED THIS PARTICULAR VIDEO FOR DECADES BUT NO ONE MADE IT I WAS SO DISAPPOINTED... FINALLY I CAN NOW DIE IN PEACE ☮️❤️ This is what we learnt in Electromagnetic Field Theory course in details, it's called "Tensor", Tensors let you do this! This guy just derived it in a simple way, if u add one more axis it will become the tensor rotation formula.
@kelly4187
Жыл бұрын
... This is also just a simple rotation matrix from pre-college linear algebra.
@pseudonym8762
Жыл бұрын
never have i ever thought i would want to know how to rotate graphs like this. 10/10 gonna send it to my friends now
@jaafars.mahdawi6911
Жыл бұрын
Just how much energy can be felt in a simple, yet neat video like this one? Keep it up, man!
@jungtaemin1639
Жыл бұрын
You would have changed my life 10 years ago And you just made maths 10 times cooler for me, a mechanical engineering student
@kelly4187
Жыл бұрын
Learn linear algebra and you can do all of this and more in a simple formulation.
@jungtaemin1639
Жыл бұрын
@@kelly4187 i already did, but this video tells me i could have figured it out during middle school using more basic maths. I really wanted an answer to this problem and never got to actually solve it nicely back then
@RichConnerGMN
Жыл бұрын
cool video. why the pitch shift
@mrmaaf1443
Жыл бұрын
Really cool content but that voice changer is really annoying, like borderline unbearable
@stinknamazing
Жыл бұрын
I really enjoyed this! Great job! Such fun to watch.
@rusgon
Жыл бұрын
What a profound and clear explanation! Thank you!
@Aditya_196
9 ай бұрын
🙌 you have my praise from all the math holic kids and myself for creating this video
@may21136
Жыл бұрын
I know that derivation of sin theta cos theta for rotation is very confusing, and I avoid doing it that way because of this. I advocate for the usage of *complex numbers* for this purpose. Complex numbers make the concept of rotational transform much easier to grasp, but you need to learn complex numbers before doing a rotational transform with it. Math is fun. If you learn something as intricate as complex numbers, you will find other harder things getting easier for you (such as finding rotated coordinates in this case). Rotation is just a special case of complex number multiplication.
@Wishbone1977
Жыл бұрын
Honestly, when he began the explanation of how to rotate a single point, I was expecting the explanation to wind up in complex number territory and was mildly surprised when it didn't.
@kelly4187
Жыл бұрын
I think it's because of the level of the target audience. But to say "without using matrices!" In the description, and not at least even namedrop a rotation matrix, which is EXACTLY what he did here? Travesty.
@1234567zeek
Жыл бұрын
I've been fighting with this for about 30 years ... thank you!
@jakubw.2779
Жыл бұрын
Oh my god, this is amazing, i'm almost a decade after my education, but this made me want to study maths again and i'm not even joking.
@kelly4187
Жыл бұрын
Did you not learn linear algebra? This is just a rotation matrix. I did that in high school.
@andrewsemenenko8826
Жыл бұрын
I figured this thing a long time ago, partially by myself when I was in school (currently in University, making games) Was really curious how it works back then. Thank you for the explanation to others who are as curious as I was! Good luck with the channel
@supramayro434
Жыл бұрын
Українця тутка навіть не думав знайти
@guillermogil3391
Жыл бұрын
This is amazing, beautiful and such a perfect explanation!!
@dewanthenmalai4232
10 ай бұрын
You can also derive the formula using the complex domain. Define a complex function of a Real variable as Z(x) = x + i*f(x), then since rotation in the complex plane is just multiplication you multiply Z(x) by e^(it), where t is your angle. When you do that, the Real and Imaginary parts of your new rotated function are identical to your transformation rules.
@lycheejuicelichigaming2263
10 ай бұрын
First thing in my mind
@spicca4601
Жыл бұрын
20 years ago I was a high school student. And I ask exactly same question to my math teacher, but he even didn't understand the question. And after 20 years now finally I get an answer to this. Thanks a lot!
@ibrahimElKhalil55
Жыл бұрын
If we use a slide to move the "diagonal sine" diagonally, wouldn't that make it look like a moving escalator?
@CoacoBudder
Жыл бұрын
Loved the video, I went through the same process back in grade 9 when we were doing trigonometry. Here’s a calculus and trig question which I was asked in my last weeks (after the exam so we weren’t wholly wasting time) in Math C as a grade 12 student as a practical application of the calculus we’d been taught: ‘Given a continuous and differentiable function f(x), describe a general method to find all points whose distance to the nearest point(s) on f(x) is equal to a variable k’. I cant remember what the answer to this was but it was certainly a journey to get there, spent a week of my spare time on desmos :p
@phlaxyr
Жыл бұрын
I was really interested by your question so I wanted to try it out. On desmos: calculator / bxxawx6ifg. If you take the max of the upper bound and the min of the lower bound, then it appears to be a solution. But the resulting function is not continually differentiable, and it looks like you need a lot of piecewise functions to describe it -- I can't imagine that there's an explicit formula. In addition, I used parameterizations - is there some way to write in terms of only x and y?
@kelly4187
Жыл бұрын
That's an interesting problem. Now I'm going to lose hours of my life as well 😁
@eishwarpawar4171
Жыл бұрын
This is a nice intuiative demo, great work
@tuxat_
Жыл бұрын
this has the same vibe as being lost at walmart and not being able to find your parents.
@BITniki
Жыл бұрын
I love how well you explain how the rotation matrix works!
@letronix6243
Жыл бұрын
This video was what I was searching for weeks ago before I came up with my own way.
@allegrobas
Жыл бұрын
Wow!! Thanks for a great video. Love your voice !!!
@elimartin9268
4 ай бұрын
I found an easy way to put it into Desmos, have different equations: defining the original function before rotation (“f(x)=…”), defining a variable (“b=…”) and the function after rotation. And preferably, put it on “degree” mode. For the parabola at a 45º angle, it would be: f(x)=x^2 b=45 x*sin(b)+y*cos(b) = f(x*cos(b) - y*sin(b)) Change variable “b” to however many degrees rotated you want and change the “f(x)=…” equation to whatever you need to change it to and the rotated equation will update with it. Setting it up like this will also make it so u only have to update less things whenever you want to make a change
@syre7608
Жыл бұрын
i couldnt listen to this voice for 1 minute
@V11MonstersMSM
4 ай бұрын
Probably he got any grapes
@udomabasiekeme
4 ай бұрын
I do can, it's beautiful ❤
@JonnyBoi957
4 ай бұрын
Cap @@udomabasiekeme
@brayanxd4547
3 ай бұрын
@@V11MonstersMSM"waddle waddle"
@nachorodriguez6380
10 сағат бұрын
Me neither, I could actually listen to it for the whole 16 minutes.
@hub3530
4 ай бұрын
As an Algebra 2 student I am flabbergasted by the ludicrous graphs that were previewed.
@mraoz8706
Жыл бұрын
finally, I can spin x^2 + y^2 = 1 after I'm struggling for years
@atrus3823
2 ай бұрын
Desmos has another feature which is very handy for working with parametric functions. You can define P(t) = (f(t), g(t)) and then do P(t) in another cell and Desmos will draw points meeting that definition over the supplied range of t.
@Ilmari_Hirvonen
Жыл бұрын
Based on the voice I thought that this video was from 12 years ago
@mxsteri0
5 ай бұрын
you were 132 months off :]
@WaltherStuzka
22 күн бұрын
A simple example - rotation by +/-45°: y' = 2x' with x' = 1/√2 · (x - y) and y' = 1/√2 · (x + y) gives y = x/3. For example, the point (3, 1) lies on the line y = x/3. After a +45° counterclockwise rotation, this point will lie on the line y' = 2x'. Similarly, the line y = x/3 is just the same as the line y' = 2x' rotated by -45°.
@Killerkraft975
4 ай бұрын
This reminded me, that in computer graphics, coordinates are represented using matrices and you can compute rotations very easily. If rotation is in angles however, this method is required.
@KekusMagnus
Жыл бұрын
baby's first rotation matrix
@scoutgaming737
Жыл бұрын
It's beautiful I've looked at it for 5 hours now
@hrishikeshaggrawal
Жыл бұрын
it's like i ask myself a question and somehow a couple weeks later someone delivers. this has happened four times in a row now
@o.p-flyup2019
4 ай бұрын
This was freaking beautiful.
@dschamp5
Жыл бұрын
Crazy crazy crazy how I was JUST looking at my old writeup of this proof seconds before finding this video. Didn't say anything or look up anything even remotely related to this. Just looked at a physical paper in my notebook with this proof on it, then this video is reccomended to me. That's absolutely nuts.
@ValkyRiver
Жыл бұрын
You’re voice sounds so cool!
@jonas_the_lost
Жыл бұрын
Thanks, I didn't know this was possible. I've tried before and failed miserably so thank you for giving me the answer that I thought did not exist.
@Nagibator6000LoL
Жыл бұрын
Nice, now I can rotate x²+y²=R² graphic
@trevorallen3212
Жыл бұрын
Tip: If your using desmos if you wanted it based on degree angles set x degree = x*pi/180 when in radian mode on the trig functions.
@supersaiyan2
Жыл бұрын
I've always wondered how you could rotate a graph, this video answered that question!
@TheXientist
Жыл бұрын
"without matrices!" what you described is exactly the 2d rotation matrix but just formatted differently
@reubenmanzo2054
Жыл бұрын
The rotating cubic looks hypnotic.
@mrsqueaksrules
Жыл бұрын
In the words of the great Vincent Vinesauce, "SPEEN"
@localidiot4078
Жыл бұрын
I loved discovering this in desmos when i was in high school. It was such a ureka moment for me. Now my favorite method is turning everything into a vector, then you don't even need the trig. it really helps simplify the equations, and it helps me intuit dot = cos and cross = sin
@theblinkingbrownie4654
Жыл бұрын
Can you elaborate on the vector method? I also did this in hs but have never thought of that
@Moogie237
Жыл бұрын
Oh my god. Years ago I pondered with a classmate of mine in algebra 2 whether there was a way to rotate a graph like a parabola and they were like “probably not.” I finally have the method.
@autaj
Жыл бұрын
This is too good, your videos are not boring at all
@shaunrichardson3333
Жыл бұрын
Thank you! I have been asking myself how to do this since Year 8!! Thank you for a great explanation of it and with cool looking functions too (what is MOB??) Looking forward to amazing my students and your part 2 video! Looks very interesting and fun
@Thomfamily5
11 ай бұрын
couldnt find anything about MOB
@razschiffman5432
Жыл бұрын
This is definitely the best math video i have seen in a while
@Dhruvbala
11 ай бұрын
I think what you derived is the inverse of what we want, transforming P’ to P.. since we get points that follow x’=(y’)^2 _after_ (rather than before) rotation. We should instead have solved for x,y in terms of x’,y’, substituting these expressions in the original equation y=x^2 Makes sense as the graphs are later shown rotating in the negative direction. If we want to rotate by +theta, we could just negate all the thetas in the expression Nice video, in any case
@Sebbethy
Жыл бұрын
I love this video, I really do! So it hurts me to say there's probably a mistake as from 8:50: In the lower left corner shouldn't it say "y ---> X sin(θ) + ycos(θ)"?
@quantumgaming9180
Жыл бұрын
You are right
@rehandrone7146
Жыл бұрын
So cool Love the video
@Georgln
Жыл бұрын
I’ll try spinning, that’s a good trick
@pauselab5569
10 ай бұрын
my calculus book gave a nice answer to this, you can write the function in polar coordinates which makes it easier to rotate then switch back.
@InAMinMaths
Жыл бұрын
Rotating the plane 90 degrees clockwise almost replaces the x axis with the y axis and the x axis with the y axis, but the x values on the new y axis would be running positively down. So it needs to be inverted. Hence, inverse functions and why you change out x and y algebraically. Thanks for the fun video. 2:58
@Dhruvbala
11 ай бұрын
Ah, so I’m not the only one who saw that. And later in the video the graphs are shown rotating in the negative direction. Thanks for the sanity check
@howlu9086
Жыл бұрын
To do this I write the function in parametric form then I multiply it by a complex number for example rotating y = x^2 by 45 degrees (t, t^2), (t + it^2) * (cos 45 + i sin 45)
@omargoodman2999
Жыл бұрын
Desmos can't do _i_ in the graphing calculator. So it needs a... less complicated approach.
@howlu9086
Жыл бұрын
@@omargoodman2999 yes you are right so you can just do it manually because there are always only 4 multiplications in total.
@MrNess2911
Жыл бұрын
Great job Dexter! You've a new subscriber!
@williamfeng9808
4 ай бұрын
The method I have always learned is convert the graph to vectors and apply a linear transformation, which basically does the same thing.
@henterpriser5779
Жыл бұрын
I love math how this guy explains it
@LheannMichelleFlorento-xc7ux
5 ай бұрын
The "ShuT uP AnD TeLl thE wOrD it IS" was funny 😂
@emmettdja
Жыл бұрын
Very simple. Just add cos of the angle to x and sin of the angle to y...
@sincostan999
7 ай бұрын
lol this reminds me of when i was in year 7 randomly asking my maths teacher if there was a formula for graphing an ellipse. i actually DID learn it a few years later which was pretty cool!
@polyhistorphilomath
Жыл бұрын
You can also just use some functional equations and inequalities. Suppose we want to define cosine in terms of sine, or vice versa. Set c^2 = 1-s^2 Now if we have a good value of s, we don't need c to appear anywhere in our rotations. what are some valid values? well s should vary between -1 and 1. We can see that the coordinate transform is just giving us a linear combination of x and y. The scalar coefficients might as well be c and s. Let u = (1-s^2)x - (s^2)y . v= (s^2)x+(1-s^2)y. As it is, this will only cover one quadrant. But if you flip the sign of the first term in both u and v, you get the corresponding quadrant across the y-axis. Flipping the sign on the second term in each will put us below the x-axis. By the two combinations of two possible sign changes we get 4 possible quadrants. Great. Now just keep s between 1 and -1. Substitute u for x and v for y. if it's easier, just rewrite your original expression as a level curve. let y=f(x)=x^2. Now let g(x,y)=x^2-y. you can graph this by entering 0=g(x,y) in most software. If so then 0=g(u,v) will rotate the figure smoothly as s varies. again, just flip the signs to cover the other quadrants. [edit] I should add that the constraint on s isn't arbitrary. if |s| > 1 then the rotation is no longer rigid...that is, the shape is not preserved without deformation. As you can probably guess from the graph of y=sin(x), this is going to cover all possible rotations before you run out of s values. If you only let s vary from 0 to 1 then stitching together all the various sign permutations should cover the entire range of angles. [clarification] To get a mathematically positive rotation, let s vary from 1 to 0, 0 to 1, 1 to 0, then finally 0 to 1, bringing you back to y=f(x). Just switch to the next quadrant when s reaches either 0 or 1.
@brendandelear1145
Жыл бұрын
STOP its too beautiful
@jayedahmed3437
11 ай бұрын
I feel alive again*";\ After going to university and learning how hard it is to understand an university teacher
@momeet6965
Жыл бұрын
I had this same thought experiment back in high school, I think I googled if it was possible or not and then forgot about it. Now I'm kicking myself for not trying to figure it out because it's so simple. Thank you for this video, my high school self is ecstatic right now (and current self too).
@letter_o_hyphen_letter_o
Жыл бұрын
instructions unclear: my parametric equation is now spinning on 3 axes like a 90's commercial logo
@letter_o_hyphen_letter_o
Жыл бұрын
I TURNED IT INTO A CYLINDER WHAT
@ItsaMe444
Жыл бұрын
You can also multiply a rotation matrice with the vectors of the graph (x,f(x))
@eriklokensgard7487
Жыл бұрын
Awesome! This helped me answer one of my student's questions. Thanks!
@rhydderc127
Жыл бұрын
Melting your FPU for science 😂Thanks, that was cool stuff!
@bavariandave5627
Жыл бұрын
This channel must be Matt Parker doing forbidden maths stuff undercover.
@BrennaButcher
9 ай бұрын
I used a circle that ran through both points p and p’. Then found the arc length from x to x’ using the formula “L=pi*theta*r/180” Then set that length L to the integral for the function of the circle, from the bounds of x’ to x. Then solved for x’. Plugged x’ into the function for the circle, to find y’. My final results were: x’=sqrt(x^2+y^2)sin(arctan(x/y)-pi*theta/180) y’= sqrt(x^2+y^2)cos(arctan(x/y)-pi*theta/180)
@Abhishek-ji3id
Жыл бұрын
A child is teaching me high school maths🙌🏻😅Mom i'm done in this world.
@Jacob-vl6ts
Жыл бұрын
Yes! I finally have a vague concept for how this works
@amukh1_dev274
Жыл бұрын
I haven't watched the video yet, but I was going to assume a matrix transformation in the xy plane, but now looking at the description It appears I'm wrong. I'm excited to see how you did it!
@csicee
Жыл бұрын
You were gonna right
@csicee
Жыл бұрын
Basically
@ss_avsmt
Жыл бұрын
It is a matrix only. Just written out in scalar form after multiplication
@kavinbala8885
6 ай бұрын
I want to now generalize this to higher dimensions
@Catkid
Жыл бұрын
What is the MOB function I googled it and didn't get anything
@Yuritolok
Жыл бұрын
Absolutely remarkable. Matt Parker should be worried.
@725etw7w
Жыл бұрын
I was thinking about and in autumn I gave up, and finally... Thank you!!
@kratos861
8 ай бұрын
Another way to do it in your head is to figure out the equation of the new axis line, so for example if you want to move it 45 degrees you just substitute the new x axis which is not x=0 its x-y=0 and y axis is x+y=0 substitute x and y in any equation to the origin with the equation of the line around which you need to rotate and bingo, it helps if you can remember the cos and sin values in 15 degrees interval for example the x axis at 15 degrees become x= ((root 3 -1)/(root 3+1) )y and y is the inverse of it, substitute in the original equation and you get the graph rotated to around that axis.
@jixpuzzle
Жыл бұрын
Absolutely Legendary video man! Now I'll go crazy! Fucking love mathematics. Incredible video! Incredible channel!
@PlayNowWorkLater
Жыл бұрын
Interesting video. What I really liked was in the beginning, with then chalkboard background how you had the x and y coordinates in the lower part of then board changing as the parabolas etc on the actual graph changed. But when you moved on to the Trigonometric functions they just stayed the same on the bottom as the drawing rotated would be great if that could be made visual too
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