the Holy Trinity of curves
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- čas přidán 21. 06. 2024
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The two line case an easy example is x^2 -y^2 = 0, factors to (x+y) (x-y) = 0; which gets 2 lines
15:56 There's also a very nice etymological harmony of the sign of this "discriminant" b^2 - 4*a*c with the corresponding curves. A parabola comes from the Greek παραβάλλειν: to compare, to be side by side, to be equal - and the discriminant is equal to zero. Likewise, an ellipse comes from έλλειψις, "deficiency", and the discriminant is < 0; finally, a hyperbola comes from υπερβάλλειν, "to be in excess", and the discriminant is > 0. These in fact are precisely the origins of these terms in the original conception of quadratic curves as sections of a cone by a plane. Namely, the angle the plane forms with the vertical is either the same as that of the cone generator (parabola), larger (hyperbola) or smaller (ellipse).
And that's also the source of the terms "parable", "ellipsis" and "hyperbole". It's really nice that these terms appear both in mathematics and in linguistics, and in both cases, they have the same basic meaning. :)
Also, a parabola is tangent to the line at infinity (the two curves are "the same" in some neighbourhood), while an ellipse has no intersection with it (it "comes short") and a hyperbola has two intersections...
@@bjornfeuerbacher5514 Exactly! In Greek, "parable" and "hyperbole" are even the exact same words as "parabola" and "hyperbola". Which leads to some rather funny jokes about Jesus dropping equations in his sermon ;-)
@@Nikolas_Davis In Germany, we also use the exact same words for both cases.
That's a very nice observation, but unfortunately you need the angle of the plane with the vertical to be _smaller_ to get a hyperbola and _larger_ to get an ellipse (e.g. 90° for a circle). Perhaps we need to consider the angle between the plane and the horizontal?
Great video, as usual. As an aside, I'm relieved to hear that my favorite on-line mathematician can't quite wrap his head around the concept of "sheaf". I've struggled, unsuccessfully, with that in connection with analytic continuation. Anyway,. this is a wonderful exposition on the fruitful general quadratic's connection to the conic sections.
If there are some intuitive videos online, I'd like to get a pointer myself. I have a love-hate relationship with Grothendieck.
This, conveniently, is directly relevant to some work I'm doing.
0:30 - Whenever I try to wrap my head around what a sheaf is, I end up with a sheaf wrapped around my head.
And then you feel sheafish.
I d like to see more about these curves in the complex plane
Oh yes.
You need an real and imaginary parted brain ;)
Yes, we would like a sequence, the trinity deserves two more videos.
(vertices of the hyperbola)
At 6:02 those two points are called the vertices of the hyperbola.
Btw, all these creatures are also called conic sections because they live inside a double cone.
More like: on a double cone.
On a double cone.
@@sophiophile Err, that's what I already wrote several hours before you, did you miss my comment somehow?
At the intersection of the surface of a double cone and a plane
Great video! Conics are one of my favorite topics. I never realized that a determinant could be also be used to determine the type of conic. Thinking about these objects in CxC blew my mind. Is there some sort of 4-D saddle happening in the last example?
There is one more degenerate case: b=r=s=t=0 and sign of a = sign of c. This yields a degenerate circle/ellipse, which is a point (the radius/axes are zero). This occurs when the cutting plane of the double cone passes through the point where the tips of the two cones touch. If you keep the plane passing through that point but tip it up until it just touches the surface of the cones you get a degenerate parabola (the single line). Keep tipping the plane inside the cones and you get a degenerate hyperbola (the two lines).
Please do a video on the equations of the rotated curves. How to find the angle of rotation from a given equation. And, how to rotate the base curve given an angle. Thanks.
I would very much like to see what x^2 - y^2 + u^2 - v^2 = 1 where xy + uv = 0 looks like in 3 dimensions. you can easily see how it produces the various 2-dimensional slices presented in the video.
The problem with many books is that they are written by people who regurgitate and don't understand that definitions require motivation. They also don't understand that the simpler more general defintions do not necessarily make sense when presented before more specific definitions. So they go yadda yadda this is a pre-sheaf we just put together some seemingly random stuff and call it a pre-sheaf. The we add more random stuff call these specific pre-sheaves, sheaves. The sensible way is to start with coordinate systems on manifolds in differential geometry, then point out that similar structures occur in unrelated areas, like models for modal predicate calculus. Then abstract the commonalities to a sheaf. Only then introduce pre-sheaves which although having less conditions are interesting precisely becauese they can be used to make sheaves as previously defined.
So this might not be the best way to look at it, but for me the concept (i.e the wrapping your head around it part) of a sheaf was simplified a lot by just considering it as the fanciest way to define the set of "well-behaved" functions on a space. Vakil's notes give a very motivating introduction with the sheaf of differentiable functions.
Now the general idea is: you're given a big, global object (a topological space, i.e a curve or smth) and want to consider functions on it. Now it often happens (continuous functions, differentiable functions, etc.) that you don't need to look at the "whole". In these cases you can basically take the little pieces the whole is made of and your function will be completely determined by their values on the little pieces.
The sheaf is just this. Take some big piece (some open set) and the sheaf spits out a fancy version of functions on your big piece such that if you split your big piece into smaller pieces (so an open cover) and consider the functions on those, you can glue the functions together in a nice way to get a function on the big piece.
Looking at it this way makes e.g the stalk at a point just a fancy way of saying yeah these are the germs of functions.
Again i highly recommend Vakil's chapter on the matter. This is (at least at the beginning) the biggest barrier towards algebraic geometry so i hope this makes sense at all (i'm sleep deprived).
The level curves of a real polynomial function in two letters of degree 1 are lines
The level curves of a real polynomial function in two letters of degree 2 are (possibly degenerate) hyperbolas, parabolas, and ellipses
Degree 3 level curves?
What is a sheaf a generalization of?
In mathematics, a sheaf (pl.: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data are well behaved in that they can be restricted to smaller open sets, and also the data assigned to an open set are equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every datum is the sum of its constituent data)
Also, sheaves are understood conceptually as general and abstract objects. Their correct definition is rather technical. They are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets.
There are also maps (or morphisms) from one sheaf to another; sheaves (of a specific type, such as sheaves of abelian groups) with their morphisms on a fixed topological space form a category. On the other hand, to each continuous map there is associated both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction. These functors, and certain variants of them, are essential parts of sheaf theory.
You copied the first paragraph off of Wikipedia
I feel the same about modern algebraic geometry. I'd much rather the classical way of ideals and varieties. I don't understand the sheaf and scheme stuff.
It’s decent that you admit your weakness in a branch of mathematics namely here algebraic geometry.
@6:00 : name of points is "hyperbola vertices"
I've seen a book which talks about algebraic curves and classifies conic sections into quadrolas and grammolas and maybe others. A line which is perpendicular to itself, i.e. its slope is √-1 (which is i in the complex numbers, but 8 in Z13) is called a null line. All circles are asymptotic to null lines, if the field has such things.
Sure! we want to see that space where parabolas are also included in the equivalence class
Good refresher on complex numbers and the interesting results you can get with them. Tank you!
🙏🙏🙏 Please, please, make a video on the claim that parabolai are in the same class as ellipseis and hyperbolai. 🙏🙏🙏
Would be nice to see that any conic section (ellipse, parabola, hyperbola, ...) is a perspective (linear fractional) transform of the unit circle.
Can't you just solve for y(x) using the quadratic formula, then just plug-in a,b,c, r, s and t to get the graph?
So i image unit circle in the u-x plane as a hole around the y axis in 3 dim u-x-y. the same is valid in the u-v-x 3d space with v give the hole around the v axis. i think the zero case covers 2 more 3d imaginations. I have a Dali picture in mind claiming a 4d instance should have 4 projections into the 3d space - correct? similar than a 3d object has 3 projections into 2d space.
fascinating! great presentation
@ 13:12 X^2 and Y^2 should be reversed.
awesome!
And that is two consecutive Fascinatings! from me
Isn't the parabola itself a degenerate case too? Of an infinitely stretched ellipse, or hyperbola.
I'd like to ask what may seem like a "dumb question" ... to those with a better handle on this stuff than I have, but: How is the "Discriminant" of a conic section related to the discriminant in the formula for the solution to the zeroes of a parabola, if at all. Is it coincidental, or is there some connection that I might be able to visualize.
z^2 + w^2 = z^2 - (iw)^2
23:08
when the big MP opens by describing a particular area of math is "such a difficult subject", smash cut to a cabinet of urns containing the ashes of several postdocs and graduates who gave it a try. or better yet, so one of them approaching you with a book they dust off from your shelf and are like, hey can I borrow this?
MP, serious suggestion: get some theater heads looking for playtime to add bits like that. For example, Sabine just does one every couple eps but it's golden. time for you to hit 1M brother
Woah ive been learning maths that the math god himself struggles with? I am making progress then! yesssss
All quadratic compositions are one and the same, a thing you can't say for cubics, for instance.
What if you use quaternions instead of vanilla complex numbers? Does the universe implode? Is everything a point? A sphere? Hahaha.
Would be fun to prove that these are all conic sections.
czcams.com/video/kvvpGW_dD1E/video.html
The points are the vertices (vertex)
Something curious to note with respect to what is said here is sqrt(|1 - x^2|) = sqrt(|x^2 - 1|). Now if we integrate either of these... :>
Well, that's bc |-x|=|x|, it has nothing to do with what's being said here.
@@ivanjorromedina4010 I don't follow. One is the explicit formula of a unit circle; the other is the explicit formula of a unit hyperbola; but these under absolute value describe both in the same formula, one or the other.
17:40 Marty, you're not thinking four-dimensionally
this is shocking, In india we are taught this in 12th standard under Coordinate Geometry and we solve a ton of questions with varying difficulties manipulating the same equation
Kind of disappointed that the first 15 minutes of the video mostly consisted of mindlessly plugging numbers into stuff, and that we then jumped at 15:40 into the main result without even a sketch of its proof. Same thing about the invariance of the 3 curve classes under affine transformation. All exploration and no theorems leaves me with mathematically blue balls