the geometry of the third derivative

[@PerScientiaAdAstra]

In mechanical engineering a third derivative effect is called "jerk". The next 3 higher order effects are also called "snap", "crackle" and "pop", respectively.

[@tomholroyd7519]

Came here for Rice Krispies!

[@richardchapman1592]

​@@tomholroyd7519 horse guard boots were adequate, thanks. Meanwhile this type of derivatives analysis must annoy city of London gamblers who use receipts as similar to investments in real stock. Will they invest in me tho. When the doubts injected in my mind make me a good each way gamble. Put the odds rather long that I'll choose the latest fastest data gatherer as a worthy wife. She'll have little time for a male whilst keeping a rest of the mouse wheel of the latest techniques.

[@jneal4154]

This is usually used to describe the derivatives of position with respect to time specifically. So position, velocity, acceleration, jerk, snap, ... usually refer specifically to displacement with respect to time. When talking about functions and curves with respect to an input parameter we usually talk about value, slope, curvature and so on.

[@richardchapman1592]

@@jneal4154 oh. Wondered if high order data analysts were using derivatives methods on data from the comments pages, in particular to see how much correlation there is between the various meanings of a word like 'derivative'. This could be a case of feedback from an observation prior to it being made publically.

[@GiulioMarcello]

.. btw, in Latin, the preposition 'per' goes with accusative: so you should say "per scientiam ad astra' ...

[@ianfowler9340]

I always liked to think that "curvature" itself was "how far away a curve was from being a circle" since the circle is the curve with constant curvature. Never heard of the term aberrancy before. Very nice.

[@xinpingdonohoe3978]

May we count straight lines as circles of infinite radius under this?

[@bjornfeuerbacher5514]

"I always liked to think that "curvature" itself was "how far away a curve was from being a circle" since the circle is the curve with constant curvature" ? I don't understand. Yes, a circle has constant curvature. But that does in no way imply that curvature tells you how far away a curve is from being a circle - in contrast, it tells you simply which circle is best at approximating your curve. If you want to know how far away the curve is from being a circle, you have too look at how far away the curvature is from being _constant_, i. e. how fast the curvature is _chaning_, i. e. essentially the third derivative.

[@bjornfeuerbacher5514]

@@xinpingdonohoe3978 Yes.

[@АндрейДенькевич]

I like to think curvature be quantity of curves between line observed curve.

[@bjornfeuerbacher5514]

@@АндрейДенькевич Pardon? Sorry, I don't understand what you mean.

[@riadhalrabeh3783]

In physics the third derivative is the 'jerk'. In space curves, the third derivative is used to calculate 'torsion' and in statistics it is a measure of skewness.

[@MrRyanroberson1]

The aberrancy is zero for all even functuons, so more specifically you could say it measures oddness around a point.

[@basisTermium]

*Now that's odd*

[@AnyVideo999]

Geometric intuitions: 0th: Position of function 1st: Deviation from position locally (i.e. slope) 2nd: Deviation from line of best fit (concavity) 3rd: Deviation from parabola of best fit Higher order derivatives are harder to think about because of this. We don't have perfect intuition for what a parabola of best fit looks like compared to a line of best fit. Even worse, the higher order approximations are a bit less local since the higher powers in the Taylor expansion disappear faster on smaller scales.

[@Erudecorp]

nth: Deviation from polynomial regression of order n + 2

[@Fahumsixtysix]

Someday you will be smart enough to do math without clinging helplessly to geometric intuition

[@CalculusIsFun1]

@@Fahumsixtysixare you suggesting geometric intuition is bad? I think it’s great for understanding something when you first see how it was derived. It’s an excellent way to walk yourself through any proof or derivation. I agree it’s a bit pointless for more complex things to visualize it all the time but sometimes it’s either necessary or just beneficial. I don’t see your point.

[@NandrewNordrew]

@@Fahumsixtysixwhat hating on a 6 figure salary looks like

[@ancientmodis]

​@@FahumsixtysixHating on people trying to understand a concept is so sad, you are the reason people just memorize shit.

[@david_porthouse]

If you are laying railway track, whether model or real, then you might like to avoid sudden changes in the radius of curvature, which is effectively what the third derivative is measuring. The transitional pieces of track that are put in are called easements.

[@benbookworm]

That's an interesting use of the word easement, because I think of real property law when I think of railroad easements. When a piece of track is formally abandoned (there are states in between active and abandoned), it is common for the ownership of the underlying easement to revert to the property owner. This can make rails to trails conversions difficult, because nearby homeowners try to say it's been abandoned and is now theirs.

[@txikitofandango]

When you're leaning against a seat cushion in an accelerating vehicle, the acceleration of the vehicle is roughly proportional to your displacement of the cushion. In such a function, you lose two derivatives. Therefore, the 3rd derivative of position, how fast you're jerked forward or backward, is roughly proportional to how fast your cushion squishes or unsquishes

[@ericthegreat7805]

Thats why its called the jerk 😂

[@VoteScientist]

proof physicists are cool: position velocity acceleration jerk snap crackle pop

[@xinpingdonohoe3978]

I never thought of the cushions. I simply used the more basic definitions. I'm pressing the accelerator pedal down. My velocity is how much road I'm traversing in a set amount of time, and in which direction. My acceleration is a measure of if I'm going faster or slower than the immediate moments before. The jerk looks at the pedal. If I start to press it down slowly, but increase the rate until I slam it down near the bottom, my jerk is positive. If I start pushing it powerfully, but lighten up and let it come to a slower stop at the bottom, then my jerk is negative.

[@MarcusCactus]

This is just saying that the third is the derivative ("how fast") of the second. No additional insight there. We want to "see" it on the graph of a curve.

[@otterlyso]

The Aberrancy of Plane Curves Russell A. Gordon The Mathematical GaZette Vol. 89, No. 516 (Nov., 2005), pp. 424-436 (13 pages)

[@AR-yd2nd]

Ty

[@GameShowFanMan]

This was wonderful! Where were you when I was writing my dissertation 4 years ago? I had to learn this all by myself since I utilized the aberrancy in my thesis. I understood it but you made it into a breeze! Thank you so much for your video again, good sir!

[@Bodyknock]

Aberrancy could also reasonably have been called Lopsidedness since it’s sort of signifying how far the curve is from being symmetric about the point under consideration. But in all fairness Aberrancy is a cooler sounding word. 🙂

[@bjornfeuerbacher5514]

You probably man "symmetric about a normal axis through the point under consideration"?

[@smolboi9659]

Aberrancy at a point should be how far a curve is from being symmetrical about it's normal to that point. A quadratic at it's extremum or ellipse at it's pointy end or any even function at origin also had aberrancy 0. The argument used for the circle works here too.

[@MathIguess]

I was taught about the second derivative in the context of "concavity", where a positive second derivative (at a specific point) means that the shape is "concave up" and negative means "concave down" at that point.

[@der.Schtefan]

Pretty much every engineer will have screamed "jerk" at your video, but that is because that's what the 3rd derivative is. In a mechanical system, that's the uncomfortable part of the movement. We don't feel speed, a constant acceleration can be pleasant, it is a jerky change in acceleration that makes things feel unpleasant.

[@dean532]

Glad you kinda reminded me: Curvature and concavity are often juxtaposed-ly misconstrued terms in applied science curriculums these days; “ one which tells you “which direction the function deviates” (unitless and qualitative ) and the other how much at a particular point(quantitative) and often involves not just a double derivative as mentioned here in the video-glad that was brought up though! In higher dimensions one can expect the demand for specificity otherwise things are sleek and “smoothly understood” in the industries i.e. if one would ever mean to use one term for the other.

[@tomholroyd7519]

You can also think of it as transforming the coordinate system along the curve, a different Cartesian pair of axes at each point of the curve

[@1495978707]

18:00 I hate how people will say largest instead of highest, and smallest instead of lowest. The US debt is by no means small, but it is a negative balance. In this case, you really mean the closest or smallest interestion points. Large is far from zero

[@0alalune0]

This confused me, too.

[@JohnVKaravitis]

Wow! This video had me running around in third derivatives!

[@rainerzufall42]

16:56 y = c1 x + ε seems to be sloppy! If ε is the distance between the chord and the tangent, the chord crosses the y-axis at x=0 and y = ε / sqrt(1 + c1²). That means, that the chord has the equation y = c1 x + ε / sqrt(1 + c1²) not y = c1 x + ε.

[@skylardeslypere9909]

Surely this limit construction doesn't actually care about which quantity you take? Both go to zero

[@rainerzufall42]

@@skylardeslypere9909 That's why I called it "sloppy" and not "wrong"! We both know, that the actual value of ε doesn't matter for the limits at all!

[@sventheviking4635]

Tan θ is surely a/b not b/a if axes are in standard x y orientation

[@TheEternalVortex42]

y = x^2 also has an "aberrancy" of 0, so that means a parabola is also "like a circle"? I'm not sure the intuitive description makes as much sense. Maybe a better way to say it would be that it measures how symmetric the curvature is?

[@bjornfeuerbacher5514]

y = x² only has an aberrancy of 0 at x = 0, whereas a circle has an aberrancy of 0 everywhere, I think?

[@onradioactivewaves]

Circle, elilse, parabola, hyperbole are all conic sections.

[@3rdPartyIntervener]

@@onradioactivewaves hyperbola. hyperbole is claiming Trump will nuke the universe on Day One.

[@onradioactivewaves]

@@3rdPartyIntervener ya that was a typo, but technically was still just as true as your example of hyperbole

[@arduous222]

In statistics and quantum mechanics, Hermite polynomial is frequently used. n-th order hermite polynomial arises from taking n-th derivatives of Gaussian distribution, and it represents the n-th "moment" of the function/distribution. Specifically, 3rd moment is related to something called skewness, which is a very similar concept to aberrancy shown here -- like how variance(2nd moment) is similar concept to the curvature, and how mean(1st moment) is similar to the slope at x=0, if you think about it. Similarly, 4th moment is related to a quantity called kurtosis, and in terms of shape of the distribution, it represents how "boxy" or how "peaky" the distribution is. 4th derivative of a function is expected to represent a very similar geometric quantity.

[@edwardlulofs444]

Ugh. I had forgotten about the Hermite function. It’s been decades since I used it.

[@odysseus9672]

Fun fact: the aberrancy is tied to the skewness of the Fourier transform of the distribution, and vice versa for the aberrancy. 😄

[@pascalklein7446]

Curvature of a curve can be generalized to 3d for sufaces using darboux formulaes. Formulaes for aberrancy can they be generalised also?

[@rainerzufall42]

2:50 I assume, you've also set f'(0) = 0 as the x-axis is the tangent to f(x) at x=0.

[@gabrielfrank5142]

Can be done by replacing f(x) with: f(x)-f'(x=c)

[@rainerzufall42]

@gabrielfrank5142 Sure it can be done with f(x) - f(c) - f'(c) * (x - c), but he didn't require f'(0) or f'(c) to be 0.

[@rainerzufall42]

@@gabrielfrank5142 Sure can it be done by replacing f(x) with g(x) = f(x) - f(c) - f'(c) * (x - c), but nowhere did Michael require f'(c) = 0... or with h(x) = f(c + x) - f(c) - f'(c) * x and f'(0) = 0. But he wanted the x-axis to be the tangent at that point!

[@askcaralice]

From the beginning of the video to 2:26 when the third derivative talk starts I did not notice an explanation of the denominator of the curvature. What's wrong with the second derivative? It is equal to 0 for a straight line and it can tell the convex/concave nature of the function pretty much the same. First time seeing it divided over an unintuitive expression.

[@bjornfeuerbacher5514]

Blackpen redpen has several videos on that.

[@bjornfeuerbacher5514]

"What's wrong with the second derivative?" For a (half-)circle, i. e. the function y = sqrt(1-x²), the second derivative is not constant, although the curvature of a cirlce obviously _is_ constant. So the second derivative is not the same as the curvature. That's essentially why you need the expression in the denominator.

[@ickyelf4549]

The second derivative is identical to the curvature only when the curve is arc length-parameterized, otherwise you need this normalization factor. See DoCarmo or any othyclassical differential geometry book.

[@smolboi9659]

9:48 if u take b/a which is y/x you get the tangent of the angle ccw from the x axis. Theta is measured clockwise from y-axis so tangent theta should be x/y. 10:49 Anyway later on by substituting m = 0, we see that Sc = -1/A(c)

[@RobbieHatley]

It's easier to understand the first three derivatives of a function f if one expresses them in terms of motion. If f = position, then f' = velocity, f'' = acceleration, and f''' = whiplash.

[@bjornfeuerbacher5514]

f''' is commonly called the "jerk".

[@richardchapman1592]

Would love to see you extend this analysis to polynomials that have more than one second derivative shifted to x=zero.

[@arafathasan-ec5cj]

thank you sir...

[@tiborgrun6963]

I wonder if there are generalizations of the degenerate function that was continuous everywhere but (first) differentiable nowhere. So for example a function that's everywhere second differentiable but has nowhere a third deriviative?

[@mrsnake1737]

I think we can get there with integration. Let's take a continuous function that is first differentiable nowhere. If we integrate it, the result has, by definition, a first derivative (the original function) but no second derivative. Taking it one integration further we get a function with a second derivate but without the third and so on.

[@bscutajar]

But abberancy can be thought of as how far a curve is from being a parabola, and that would make the summary at 30:55 nicer with increasing polynomial degrees of constant, line, parabola

[@vadimpavlov6037]

Kinda reminds me how the third statistical moment measures asymmetry of the distribution

[@froyocrew]

They are related via a fourier transform

[@Kram1032]

So the curvature is the deviation from a curve of constant rise And and the aberrancy is the deviation from a curve of constant curvature So whatever the fourth derivative geometrically does presumably is related to deviation from the curve of constant aberrancy, right? What would that curve (or the family of such curves) look like?

[@paulg444]

is it the rate of change of the curvature ?

[@michalchik]

I would call the third derivative constriction and relaxation. Is the curve tightening or is it loosening

[@Fetrovsky]

16:47 strictly speaking it's not +epsilon because the normal is not parallel to the y axis.

[@Fetrovsky]

Ok, nevermind, I guess, since in 18:29 you're defininf the points with epsilon as the distance along the y axis, and not the distance proper as you had said in the beginning.

[@bjornfeuerbacher5514]

Fiddling around a bit, I just found out that one can write the aberrancy as -dk/dx / (3k² (1+y'^2)), where k is the curvature. The dk/dx makes sense: this ensures that a circle has an aberrancy of zero. But it's unclear to me how one could interpret the factor -1/(3k² (1+y'^2)) ...

[@noellelovespandas]

I thought to myself “why is this so complicated” then thought “finding the slope of a curve is also complicated”

[@richardchapman1592]

Any mileage in looking at when dx/dy=0? The assymtote of an infinite spike could be f^-1second derivative=0. Where ^-1 stands for the inverse function.

[@richardchapman1592]

Also, any mileage in finding a fundamental difference in the assymtote of dx/dy for a spiked function f(y) and parallel lines? Parallel lines at >or=to infinity looks a bit prosaic.

[@hakerfamily]

Kind of confusing because the tangent of theta is a/b not b/a. And if m is close to zero don’t you get A = -1/S? Seems like you need a minus sign in there.

[@hakerfamily]

A 3 got dropped in the end. I think what you have in the end is d/dx of the curvature. To be invariant, I think it should rather be d/ds of the curvature, where d/ds = (1+y’^2)^(-1/2) d/dx. Very interesting to see the chord interpretation!

[@smolboi9659]

17:37 what does a negative intersection mean?

[@smolboi9659]

Oh ok nvm i got it. It's just means the x coordinate of the intersection is negative.

[@michelebrun613]

If you use \epsilon as in min 17:00, \epsilon is not exactly the distance anymore.

[@odysseus9672]

I always felt the more natural definition of curvature was the magnitude of first derivative of the curve's unit tangent vector with respect to arc length. If you prefer, it's the magnitude of the second derivative of the curve's coordinates with respect to arc length along the curve. The formula is manifestly rotationally invariant, less arbitrary appearing than the (x, y) version, and generalizes obviously to higher dimensional curves. Is there a similar formula for the aberrancy?

[@joshualeopior9019]

I was fine until 21:55 hit and I did not understand how he can just do factor out the root like why and how and whats up with the polynomial equetion afterwards like I don't get it..

[@ianfowler9340]

You could also talk about the third derivative test for inflection points,

[@supratimsantra5413]

Just splendid sir.... thanks for your valuable learning video

[@nicklockard]

Decreasing radius on a curvature

[@phitsf5475]

I'm looking forward to this video

[@khanch.6807]

First derivative is like velocity, second derivatives is like acceleration, third derivative may be rate of change in acceleration. That how I visualise.

[@3rdPartyIntervener]

that's exactly what it is. just swap 'y' with 't'

[@vencik_krpo]

Why a circle, though? If you associate 1st der. with "distance from const. fn." (i.e. polynomial degree 0), the 2nd der. with "distance from linear fn." (i.e. polynomial degree 1), wouldn't it be natural to associate 3rd der. with distance from parabola (i.e. polynomial degree 2)?

[@chivoronco4853]

y’’’ = slope of curvature 0:39 0:40

[@msdmathssousdopamine8630]

30:47 missing a third

[@apimailtest]

Easy: 1st derivative = slope. 2nd derivative = slope of the slope. 3rd derivative = slope of the slope of the slope. and so on and so forth.

[@bjornfeuerbacher5514]

Right, but not intuitive at all when one wants to talk about the shape of the curve. Which was the point here.

[@АндрейДенькевич]

​​​​​​@@bjornfeuerbacher5514Right. Point has no dimension(quality) but has curvature(quantity). Except Point, no other shape(curvature&dimension) has pure curvature, only in composition with dimension For example, Euler's curvature&dimension equilibrium: space has curvature=0, it's pure dimension : -1 +3 -3 +1 =0 for triangle -1 +8 -12 + 6 -1 =0 for cube -1+20-30+12-1=0 for dodecahedron (12-hedron) ... so shape(number) is a equilibrium between curvature&dimension (quantity & quality, time & space, KER & IMG , temperature & degree of freedom).

[@bjornfeuerbacher5514]

@@АндрейДенькевич A point has curvature?!? Why on Earth do you think so? That makes no sense at all. "shape is a equilibrium between curvature&dimension" And where did you get that strange ideai from?!? That makes even less sense.

[@prodigal_southerner]

It's slopes all the way down.

[@prodigal_southerner]

@bjorn, if you think about it, it's actually a really interesting way to frame the idea. All shapes are defined by a perimeter. Any continuously differentiable perimeter will be a curve (or surface, volume, whatever - depending on how many dimensions.) if you think of a point as a circle whose radius has gone to zero, it's entirely curvature with no dimension.

[@randomlife7935]

Is there an aberranncy equivalent to the circle of curvature?

[@galoomba5559]

I think it's some kind of spiral

[@wilderuhl3450]

I’m curious about the pattern that appears in the geometries of higher order derivatives

[@jeffreyhersh908]

I know in physics we call the third derivative of displacement the jerk. :)

[@xinpingdonohoe3978]

Physicists just wanted to have fun; they're able to make many puns. Such as: Don't be a d³x/dt³ di/dt=ï

[@jeffreyhersh908]

@ yep there are Feyman diagrams called “tadpoles” and “penguins” as well. And quarks were named after the prom Finnagians Wake 😆

[@HakanaiVR]

Caveat: only when it’s the derivative with respect to time

[@aurinkona]

you really could have pared down a lot of the details here. muddled the essential idea.

[@bscutajar]

9:59 this is also incorrect as tangent of theta is a/b not b/a

[@chemicalbrother5743]

29:28 I somehow calculated a different formula, instead of (1+c1^2)c3 I calculated (c1^2-1)c3

[@artemurazmanov8457]

me too

[@krisbrandenberger544]

Yes. By multiplying the expression for A(c) both upstairs and downstairs by c1 and doing some other manipulations, I found that A(c) should be equal to c1-((c1^2-1)c3)/(2c2^2).

[@tonibat59]

When third derivative goes to zero the function for aberrancy does not vanish?

[@Utesfan100]

A statistician would call this skew. The fourth derivative measures kortosis, how heavy the tails are.

[@dakota8450]

No, those are statistical moments, not derivatives. Furthermore, statistical moments often involve higher orders of integration of say a probability density function, as opposed to differentiation.

[@MarcusCactus]

@@dakota8450is right. The mean is not a slope, the variance is not a curvature of anything. Although... moments are the derivatives at zero of the moment-generating function. So it deserves a bit more analysis.

[@arduous222]

At least for the case of Gaussian, mean can be thought as the slope at x=0. Similarly, Variance is related to the curvature at x=mu (because the variance is not raw moment but centralized; if it were raw, it would also represent curvature at x=0). These statistical moments are numbers and not functions, and to have any meaning, it is important where they're evaluated. But it is evident that they do bear very similar geometric meaning to the slope, curvature, and aberrancy in case of skewness. I'm not sure about other continuous distributions, but I believe they should also have a point where the moments coincide with (or bear meaning of) the derivative at that point.

[@erfanmohagheghian707]

You missed the absolute value on y" in the curvature formula.

[@ickyelf4549]

For planar curves we treat curvature as signed, see Do Carmo or equivalent book on classical differential geometry.

[@erfanmohagheghian707]

@@ickyelf4549 This is a lecture designed for undergrad calc students, not a lecture on diff geometry. Curvature in calc 2 or 3 (whatever you wanna call it) is defined as the norm of the derivative of the unit tangent vector wrt the arc length and norm is always >=0. Thank you anyway.

[@BlackHattie]

So fourth derivative measures how far is the plane from globe, fifth measures how far is the hyperplane from hyperglobe... Explanation. so there has to be elipse in abberancy... egg in jerk, snap in 3d abberancy and so on. I better like grad div rot. Or recursive operators. Oh operators, the recursive ones, you can make a trace from pot to crackle, from crackle to snap.. Till one scalar.

[@Juttutin]

Tangential physics: For forever, I've wondered if any human will ever know what sustained positive jerk (as in the 3rd derivative of distance) feels like.

[@Sir_Isaac_Newton_]

why is Sc not equal to the limit of a/b ? Just by the definition of tangent

[@carloshortuvia5988]

Just a slight remark, shouldn't we say how far from being a straight line, that's why you have a ratio between the slope and the slope's slope?

[@sergiogiudici6976]

Skewness ?

[@michelebrun613]

In the last formula a factor 3 is missed

[@uthoshantm]

3rd derivative means increasingly curvy or decreasingly curvy.

[@johndunn5272]

Was fascinating but i got lost

[@myself0510]

So... the circle is the only shape with abberancy 0 everywhere. Otherwise, lots of curves have a point with abberancy 0, like all Quadratics.

[@aurinkona]

fundamentally, all of this is really about 'how far a curve is from being quadratic,' as a vanishing third derivative means the curve is at most quadratic. this video went on forever with technical details, most of which were really inessential.

[@piergiorgio919]

In physics terms, it's just the speed of acceleration

[@viggosimonsen]

Can't you simply call it the 'variation of the curvature' - which is exactly 0 for a circle with constant curvature?

[@BRunoAWAY]

For me its the velocity of the aceleration😂😂

[@joga_bonito_aro]

Erectile dysfunction

[@guntramschlemminger7383]

Slope of curvature.

[@Christopher-e7o]

X,2x+5=8

[@朕是神]

Since when is the second derivative called curvature???

[@Choose.Nurture.Not.Excess]

its youtubers going insane

[@92MentalDisorders]

Hello, Michael, one of many russian subs here thanks for your videos, love your channel! p.s. i want to buy your merch so badly but, due to the stupidity of my goverment, it cannot be shipped to my country, can i do something about it or i just must wait for all political cringe to stop?((

[@BrickGriff]

... y''' equals... Wiggliness?

[@chudleyflusher7132]

I showed this to my evangelical neighbor and he told me that “Trump is going to end all this”!!!! As far as I can tell, he thinks that MAGA is going to put an end to mathematics!!!

[@xinpingdonohoe3978]

I suppose bad economics is a part of mathematics. When that's removed, Mr. MAGA will be slightly correct, due to a very tiny portion of mathematics being removed.

[@MarcusCactus]

MAGA exists on the premise that there are "alternative truths". Was it Illinois or Nebraska that legislated that π was equal to 3.2 ?

[@sorinal1234]

Drop the vocal fry.

[@POLMAZURKA]

show example first

[@franzlyonheart4362]

14:16, skip.

[@nicksklavos]

You are like drunk in this video. Home issues?

[@tommywalker1631]

I thought second derivative is instantaneous rate of change

[@ea9215]

That's just the derivative lol

[@bsmith6276]

Second derivative would be instantaneous rate of change in velocity, which is acceleration.

[@CliffSedge-nu5fv]

First derivative is rate of change. Second derivative is the rate of change of the rate of change.

[@NimArchivesYT]

That’s the first derivative, though it’s using a limit, so it’s just increadibly close to instantaneous

[@sunival]

Good math with bad hand writing and presentation.

[@cycklist]

Thank you for acknowledging that other countries exist :)

[@chrstfer2452]

No offense michael, but you seriously look like you need some more sleep. I hope you get it.

[@ferenc_l]

Cool geometric interpretation from Azerbaijan

[@puneetbajaj786]

First

[@adiaphoros6842]

🤢🤮 🫵🔪🫵🤡🪳

[@wiseguy7224]

The 2nd derivative is NOT curvature. Do your home work. Thumbs down.

[@Cpt_John_Price]

Explain

[@wiseguy7224]

@@Cpt_John_Price en.wikipedia.org/wiki/Curvature Section "In terms of a general parametrization"

[@PerScientiaAdAstra]

Shouldn't that read tan theta = aɜ/bɜ ?

[@politicsisstupidandannoyin269]

y'''=jerk (rate of change of curvature)