Parametrized curves and algebraic curves | Differential Geometry 3 | NJ Wildberger

[@CU2T0D10]

I love the way you suitably introduce some historic facts in your lecture. It makes it much more interesting. All math teachers should do the same!

[@CU2T0D10]

this was really nice, I am actually a graduate student in mechanica engineering. I am researching the mobility of linkages. Nowadays differential geometry has become one of the most powerful tools in this subject. So, I am really interested in getting into differential geometry, but all books I've tried are almost unreadable for a student with an engineering background. Hopefully your videos will be more illustrative.

[@njwildberger]

That's why the parametrization is really that of a PROJECTIVE line! The point at infinity on line we are using for our parametrization contributes just as well as any other point.

[@maxwang2537]

It’s so interesting, I’m deeply attracted. I am always impressed and amused by the smart mechanisms that I found everywhere in our daily life, vehicles, home appliances, industrial machinery, etc etc. and have long been dreaming of having the ability to devise a simple mechanism that moves just any way I would want. Just started this series and love it, and would like to finish it. Would you recommend one textbook or two that’s good to read while following these lectures? Want to have one paper book handy anyway. Thanks. And I have to stretch my imagination very hard to accept the fact that there are people out there who would give this a thumbs down.

[@BillShillito]

First of all, I'm loving this series so far, just as I loved your Algebraic Topology series. I do take issue with something though. You claim that (cos θ, sin θ) is only an approximate parametrization of the unit circle. I'd say you have it backwards - the calculator's output for the cosine and sine of an angle is what's approximate. The actual cosine and sine functions, on the other hand, are defined to give the exact coordinates of each point of the unit circle. For instance, you could easily plot a line through the origin that makes a "tangle" (to use your terminology) of 1/8, a nice rational number, with the positive x-axis. Are you saying that the point (cos[1/8 τ], sin[1/8 τ]) where that line touches the circle doesn't really exist?

[@rrr00bb1]

Maybe he is saying that there is a difference between a symbol that defines a procedure to generate a value, and the value itself - in programming, this a thing with a different 'type'; so I think the definition of "exists" is what everybody is wrestling over. Maybe something isn't truly a "number" until you can forget the procedure from which it came? Incomplete vs Inconsistent choice being made? I am a programmer (not a math major), and I have always seen things from this finite perspective for practical reasons (after implementing some rational trig in lisp code and dealing with data structures required to maintain exact answers, I really suspect that NJW's ideas are informed by writing actual code that gets used in the real world - thank the gods). The most basic thing about math is substituting equals for equals. When something is a "number", you can go ahead and "forget" the procedure that generated the output. It doesn't matter how you found 6/7 if that is your answer. With a finite representation, numeric equality is as good as symbolic equality. There are places where inexact answers make results completely useless, like in cryptography. Even where intervals seem ok, it's important if you are to divide by a small number and know that you won't divide by zero, or take a computation resulting in zero as meaning something (like: "these lines are parallel" - proven *numerically*), or take a wrong sign on a small value. You can't add phi and pi and just forget that the expression originated from (phi+pi); so phi+phi isn't really a "number" in that sense. If you got a big ball of algorithm code that does emit phi+pi every time, you may not be able to prove that it's the same answer; so again - you can't forget the origin procedure that generated the number. You also can't blindly take a sqrt of a number without knowing if the originally squared number was positive or negative, or take a limit without knowing from which side you approached the number. Inverses, numeric equality checking, and finite algorithms are important if you are going to actually going to write code to do the math. So fine, you say: represent numbers to maintain this info exactly. You can do it for a lot of cases, but not for all of them. If you iteratively create an answer, then the data structure representing it could be so huge that it only exists in the same sense that a solution to inverting large RSA calculations exists. I am of the opinion that these things exists, but they are still irrelevant when you can't represent them outside of a latex paper, or just something to avoid when you can only get engineering approximations. No math problem is truly solved until it is turned into computer code that generates perfectly accurate answers for all cases, and runs within time constraints. (ie: a real-time notion of correctness, just like it doesn't matter what happens a million light-years away at 'the same time' as now...the signal from that place isn't relevant if it exists).

[@demr04]

What he is saying is that a rational parametrization, gives you rational result, which have closet form. In any curve, there is rational and irracional coordinates. If you use cosine and sine, first, you get a really bad aproximation for almost all values of the parameter because calculator use Taylor series, second, processors of any kind can't deal with base10 numbers because their operation are in binary, so there is always that problem, and third, given that a rational result is just a pithagorean triple, one could what sub-set (rational or irrational) is the dominant in the circle, I don't know but I guess that there is more rational that irrational points.

[@jmafoko]

those examples of linkages are a powerful motivation

[@lawrencesmallman]

What a brilliant lecture ... really enjoyed this. Is there a limit to the size of the gaps between the rational points on these curves?

[@ethannguyen2754]

13:54 This man learned the instant transmission technique

[@l1mmg0t]

very interesting

[@anthonycaine5962]

Around 8:20 you go on to say we will get a rational solution for P and give reason along the lines of "We already have rational root -1". However, in the formula for the coordinates of P I don't see anything stopping me from plugging in t = pi/4 which would produce an irrational point. What's going on?

[@njwildberger]

Even the existence of these so-called `irrational numbers' that you speak of is not clear once you investigate more deeply into the standard constructions of the continuum. See my MathFoundations series!

[@rameshkadambi1741]

Dr Wildberger, Is there a recommended book that goes with these lectures?

[@WahranRai]

10:44 We can see these equations of parameter t as the expression of cosinus(theta)/ sinus (theta) related to t= tangente(theta/2). You said that the parametrization with cos / sin is not precise !?

[@therealhotwatertunes]

He said it's not precise because it's a transcendental number, you rarely can express trig functions as a radical or a fraction

[@patrickdougherty5659]

Will the curve plotted out in x/y plane for x^3+y^3=1 only have transcendental solutions? It's a continuous curve but it doesn't have any rational values? What does it mean to have no rational parametrization? The curve assumes no rational values as it sweeps the x/y plane?

[@tillotman]

Maybe just that for any rational value x, it's guaranteed that any corresponding y on the curve will be irrational.

[@TheNomadic69]

Wow now I can finally connect number theory to physics.

[@Cubinator73]

Why is 〈x|y〉=〈cos θ | sin θ〉 just an approximation for the unit circle? Every vector of this function has the length (sin θ)^2+(cos θ)^2 ≡ 1, so it is a perfect unit circle. Am I missing something?

[@njwildberger]

Try to find a point on the circle using this formulation, and things become clearer. For example, try to work out the point corresponding to theta=2.

[@Cubinator73]

Well, it surely wouldn't be a rational, but the most exact result I can come up with would be 〈cos 2 | sin 2〉. I just read in another comment that only rationals were properly defined so far, in which case sticking to the rational equation is indeed a better idea.

[@mathematicaluniverse7493]

I think the point is that while (cos𝛉, sin𝛉) is exact (in itself), it only yields approximations (in practice) for general 𝛉, since you are forced to round at some point.

[@karimkhan1312]

very interesting -- thk u sir

[@karimkhan1312]

lecture is very clear and very systametic