Wie misst man die Länge einer Kurve?

The new chapter is supposed to be about the length of a curve, that's the kind of thing that at first seems completely clear to you what it means and if you think about it more then you realize that it's actually not that clear what it is. something like like with other math terms where you first think that it's actually completely clear to me, as another example, tangent to tangent, we only have an idea of ​​what it's supposed to be, just the one that just touches a curve on a point and so on. And if you think about it a little more, you realize that it's not that obvious because there are many borderline cases where it's not that clear anymore and you finally realize that tangent is a term that you can actually just define.

You can't say. clearly what it is and you will define it. It needs derivations only then you can't say exactly what you mean by an arbitrary tangent and that will turn out, which means the length is the same. You have a very intuitive idea about what length is because it's actually not that easy except for very simple curves and I'll say a few sentences about the story. Even in ancient times there was an effort to determine the lengths of curves and there were no great successes. were achieved, I would say of course it's clear and what you can do is determine the length of the straight pieces.

That's really simple and now you have a very clear idea of ​​what it should be in today's terms. I would say that is not the case now in ancient times, but it was already the case in modern times. You can give the coordinates of the thing and then you can calculate the length of said piece using the Pythagorean theorem. So you can actually calculate that and from. Of course, that also corresponds to the idea of ​​what the length should be. It wasn't a problem in ancient times and then there was another type of curve for which length could be calculated, i.e. circles, although that was almost a bit of cheating. because the length of this path is when I want to go around in a circle, that is, the circumference.

You know in school it's two times pi times r, but that only works because you've defined pi in such a way that pi is exactly the number. More modern you have to multiply the diameter to get the circumference, that's all you have to do. What we knew for sure in ancient times was that the relationship between diameter and circumference was always the same, but the way of calculating it was different. In principle, it is only possible with such an approximation that we can obtain this mystical number pi, which is then calculated better and better. All other curves, even in supposedly simple times, are simply a parabola or something.

The mathematicians of antiquity. The times were completely lost and they didn't know how to calculate something like that and that continued until modern times. There is such an old word that is still partially used today and it means to stretch the zier bar and by that you mean that you can measure the. length of the thing, so it is a curve, you are right, official bar, if you can measure its length and the support, let's say from the 17th century, I actually thought that the only curves of the tivizen bar were sections, pieces and circles, everything else you can.

I don't understand anything because you can't measure the length. There were then two curves in which some people managed to do this with a lot of effort. Let's take a look at the so-called Indian parables, for example. a slightly different parabola than the one I drew here, but overall it was a very difficult task and is similar to the tangent I talked about at the beginning. Finally it turned out that you had to go longer, you can really only talk. sensibly if you have derivatives, which means you couldn't do that before you had differential calculus. Only with differential calculus is this possible and that's why ultimately what it boils down to mathematically is that length is something you. define, that is not the case of something that is already there and that can be measured, but rather you define what the length should be and try to get as close as possible to our intuitive idea of ​​length with this definition.

Try to explain how length can be measured and then we will make a definition that corresponds to some extent to our idea. Maybe I'll ask you another question. Let's continue with our intuitive ideas. You imagine something, something very simple, a function that comes from the open interval 0 1 and then goes so that the graph of this function is infinitely long. I put the extra mark because we don't know how long it is yet, but we know. You have an idea of ​​the functions that are in a very small piece in a finite interval. It's still defined from 0 to 1, but it's infinitely longer.

You can, for example, simply say that now point one here is the. mood, take something like it used to be and draw this curve then the clock will look like this and it will be here getting higher and higher and the closer you get to 0 the higher this curve will become i.e. it is possible to map something from an interval which has a finite length over a curve that has an infinitely longer length, so we have to be a little careful. We will see later that the solution is that these parentheses are not allowed, so with the curves we are working with We only talk about excessive length if they have brackets, that actually makes a big difference if you were to make a bracket now, then you would have to be there, there is 1 until but it is not defined and then everything would be at 0 1 with 0 inclusive, it is no longer continuous and much less differentiated, so it is just a preliminary comment.

I would like to remind you again how to be longer. I already mentioned it, measuring lengths for distances in two ways, one the way I drew it somehow, they have two points with their coordinates, you know this here is x1 and 1 this here is x2 and 2 as calculated. If you now have a computer, these two points have the length of this distance exactly with Pythagoras, this is this right triangle and you pull the components from here, this distance here has the length x2 x1. I am writing this as a precaution. With the quantity crossed out, I've now labeled it as x2 to the right of x1, but if I did the quantity, then x2 to the left of the line x1 could definitely come out as a positive length and this has the length and two oops, lon one time and now take Pythagoras, that means this year squared plus the square is now the square of the licher so I have to act the two squares and then take the root of then this length has been calculated wants this here to be the square now Can I skip the number of dashes again because through trust that becomes positive anyway that here squared and starting from the roots there is another possibility that in the end amounts to the same thing, the law as a reminder to say it again too it could be that you do this They both have points somewhere and points are always awarded for goals. case here.

You could say that the location vector for this point is the xy 1 vector but I want to say it again explicitly with vectors. So you have a vector that is time here and a vector that points to this here is the zero point. the sector is p and this route gate is q and now you want to have the length of the route, then you can form something like peru, which is sector p and take the length of this sector. This is also called rules and is usually written with two such hyphens to make it look like double hyphens if you type it specifically you will see that it comes out exactly the same.

The norm of an industry is defined as if the alarm clock were xy, for example, the rector's Scala product. with itself and from there the root and if you write that kalla point of the rector with itself you have x multiplied by x and was y and from there the root looks very suspiciously like the one up there in orange and if you now take this path door p - before and you take away enormously because pia has some coordinate let's say p1 p2 and co has some coordinate q1 q2 which you want to have tomorrow so here it says p1 - q1 p 2q 2 that is the way the goal difference is the rules and then it comes out exactly Same as above again.

Now I just need to use it here where ex and Student are, so you have p1 - q1 squared plus p22 squared of the roots, so in both cases. Naturally it's the same, these are just two different perspectives, one perspective is orange, you see them as points with coordinates, you calculate the distance directly with Pythagoras and the second perspective below in black is you see all kinds of vectors. and you calculate the distance by forming the difference vector and from this difference vector the norm, that is, the calculation of length which is actually the position in which we are, we have the possibility of calculating distances between two points, that is, the lengths of small chunks away, but of course. we want to use some curves with long excuses and the idea now is that it says "okay." Now somehow I have a curve like this.

I divide this curve into small parts, for example, this piece here and then make it here like this and then make it here in place and maybe make one more place here and at the end another one. Now I have approximated this curve with some elastic pieces in this case like this 1 2 3 4 5 and these elastic pieces I can I ever calculated the length and if I use these five adding the lengths I came up with something that is not the length of the curve black but it is somewhat similar and the idea now is that if I divide it into more parts Hopefully, I will get a better approximation of the length and if I do it like that, I will see it immediately.

That again comes down to a limit because I have to say yes to become more and more precise. I have to take more and more. more pieces and achieve perfect precision, when I mark an infinite number of pieces, so somehow there is a limit. If I define it like this, you can do it and then you get the definitions the first time, there's no talk about derivatives, but you can. then demonstrates with appropriate mathematical effort that curves can measure their length in a way that is almost nonsense to mathematicians. The difference everywhere you are means that you are not outside the program if you want to be able to measure the length of a curve, then you need a curve that is described by a parameter representation which is essentially the difference curve, that is, if we limit ourselves what we call parameterized curves, then we will have captured everything from which man can measure length, so it is not a bad thing. if the derivative will appear in our definition because with functions where you can't write a derivative, that's hope.

Anyway, it's relatively small that you can measure the length of the thing. I like the same image again, so a subdivision again and. Letter it and then we'll see how you can get familiar with it slowly, so I'll do it again here, such a curve study and now we have our idea again, which is a parameterized curve, it's just called that. and it goes from the hb interval to quite high n here Andy says again. I do all the examples in the r-squared plane but in principle it works for higher dimensions exactly the same and I will return to it again. just in blue again some points so this would be al fateh 0 this seiffert e 1 the cs lives in 2 this is alpha t3 t4 and in general I have a time interval divided into many parts and at the end I'm on alpha tn so that was like here is like sbh I called t0 b is tn and then I did a little subdivision of the interval here is t1 t2 so I measure one of these pieces, how long is this piece here?

Can I do that now with Pythagoras? I'll draw it again like we just saw before. I have a certain piece here on the x axis that I call data x1 that the distance has not been drawn and this piece on the y axis is. it is not the type solon 1 and now I can also write delta x1 it is alpha 1 c 2 - alpha 1 c 1 note that now we are talking about the component function speaking, alpha itself is a function with two component functions and to measure the distance on the x axis I just need the first component functions which told me what is happening with the iks components and consequently delta y is 1 alpha 2 of t2 alpha 2 and it is and the whole thing maybe I should also give you a name for what I want to calculate here, let's call it delta s 1, of course now I can calculate that with Pythagoras, so I could write it down now, but we need it in this one.

Don't put it on yet, that's what happens. of delta understand what happens when distances become smaller and smaller. For this we will use a differential calculus theorem. This is the so-called mean value theorem. You may have heard this before at some point, but I'll tell you the mean value theorem again if you have a more differentiated function. that maybe works like this somehow and choose a start and end point here and connect these two points with a straight line. So there must be a tangent somewhere on the curve that runs parallel to it. This is the mean differential value theorem. calculation, that is, no matter how long the curve lasts from one point to another, this must be practiced at some intermediate point, which corresponds exactly to this slope from the first point to the last.

I give my name to the thing. This could have been our t1 and this could have been our t2, this place where it's exactly this tangent here we add our names, let's call them xiii 1 if you write that now for our functions, then that means you can. Also write this here, I'm going to do that, but alsoyou can write in quantities like the derivation of alpha 1 at some point xiii one that is between t1 and t2 times the distance of the two, the only problem is that I don't do it. I know where is this number which I called taxi 1, it is somewhere between t1 and t2 which is the declaration for all component function alpha1 and I can also do the same declaration for the other component functions alpha 2 somewhere in between there is a place which Now it is not allowed to name the 1 to which I now start from the time so that this distance corresponds exactly to the one I show that maybe.

Again in the image there is a fix that it is not always clear what I have up there in the first equation. looking at the values ​​of my function then if this is my function alpha1 then I have alpha1 of t2 up here and here I only have one of t1 and what this image says the most is the distance between the two, that's what I want here I can also represent it like this here, the width t1 up to twice this green gradient which is the crescent triangle of God and that mean value theorem of differential calculus says that I also get the slope as a tangent somewhere, that is, this expression, this width x slope, I can write it down somewhere in between without knowing exactly where there must be such a number that once existed for this shape to apply here and accordingly.

Does it have to be somewhere in between? Now I would have to paint a second painting. , so this clothing applies here. It's a different number, so it's also between t1 and t2, but it's not necessarily the same. It can be used for this. Now I write alpha1, once you put it in squares multiplied by c2c 1 squared. This was the first part + alpha 2 once you added to the square on them, which I can also conveniently pull out because it's a square, so everything looks a little prettier than it is. I've got this down to the root here and now I can save the amounts. because the whole year is a square, so I have a line here at the point xiii 1 + alpha 2 line at the point where they were once squared, I still have to add a Sword, how does this help me now except that Now do you have an ugly one in the front with a root there?

Now comes what I had already announced. We're thinking about what will happen if these pieces in our image get smaller and smaller, so I want this piece to get closer and closer. smaller, that means that now I let t1 and t2 converge with each other so that the distance between t1 and t2 becomes increasingly narrower, which means that if the distance between t1 and t2 becomes increasingly narrower when you look at this down here, this is always the case for 1&1 There is less space, so to speak, they are squeezed in and at the limit these two values ​​that are between t1 and t2 inevitably go towards t1 because I imagine it like this: press t2 against t1, i.e. , when the two go against one, then in parentheses the distances become smaller and smaller.

In this expression alpha1 goes one against one because at some point this value between the two is no longer between them but becomes one with the two. here too, which means tea 1 is also written here, but I have to be up front with that thing. How do t2 - t1 get smaller and smaller when I push level 1 and level 2 together, is that what this calculation calls it? a small width and they usually write it as dt, so it is a distance on the x-axis. The Infiniti is small and what we have calculated now was to remember where we really started.

We wanted to calculate the gears and things like that here. This is just one of these pieces. We just have this here from here to here and now we have to add these stretched pieces everywhere, that is, now we have, because we have compressed infinity, enough, in fact, infinitely. Many of those pieces stretch and we have to form the sum about all of them and how they fit together. You have to integrate the sum over all of them. That's where it already is, that's what it all comes down to. That was just an intuitive idea. that we calculate the integral from A to B.

This is the range between which the values ​​of t can vary from this fragment of dt times that passes through the root. Now take a closer look at what's underneath the root. first component of the alpha dash, which is back here the second component of the alpha dash, that is, what is written here is nothing other than alpha dash from tee and from there the standard is exactly what I just wrote above, like this which is alpha script, the derivation of alpha as a vector of this sector, the length is there, that is, what is below the root with the root a, so here and that's what we want to have, that's the formula according to the definitions for the length of the parameterization curve, so see the sequence in two ways, you can say that we define.

That's just the length of alpha is what it says, period or you can see, like I did for you, that you can derive the geometry, so to speak, and say it makes sense to do it that way because then something comes out that corresponds to the idea of ​​making smaller and smaller pieces but like I said now you can buy it like this and say, well, that's the length. The nice thing would be, of course, that we get the same length for treatments where we already know what its length is, which we also have to say in everyday life, that is, now we are going to try this formula for simple curves and wait a lot . so that what we would expect actually comes out and then we can use it to create curves that we couldn't calculate before we could see how long they are, maybe I should say two things before doing the calculations.

One is that we had assumed. that the parameterization we were working with was observed, so this alpha is now smooth, for example, that means that all the derivatives exist and the derivatives also exist, all continuous, which means that the script alpha is continuous and I think It's obvious that your norms are also continuous, that's what all this means to me, what's called integra is continuous, that means, hopefully, for those of you who have wondered if it's possible. This integral always exists, the main theorem of differential and integral calculus, if it integrates a continuous function then the integral exists, so we do not have to worry as long as we have a curve parameterized according to our function and a closed one, the length is always from a to b define this integral there is always the question is if we can calculate it we will see that this is not always possible but at least the length is always defined first and second point maybe you have the alternative idea why you choose this definition for the length that comes, take our curve of colts again and Imagine that you drive for a long time.

I already had the idea last time and now you take your time as often as possible in different places, that means you always say "ok", here I was for a few seconds, for example, and look. the speedometer and say how fast it was going, for example 40 kilometers per hour and if you do it again and again in different places then you can always calculate in the middle that the last measurement was ten seconds ago and it was probably about the average for the last ten seconds 40 kilometers per hour for this you can estimate from calculations how long the last bit lasted and what you would do, then you would multiply the speed here, that is, you would multiply the speed that you read on the speedometer by the time interval since the last measurement, nothing else says You and integral just means that they don't do it frequently, but they do it infinitely, which means that they look at their watch continuously all the time and the clocks get arbitrarily small, but it's actually the same thing. idea while we calculate it and take a look.

Some examples and the first two games we take are those of which we already know the length, distance and a circle and now we will use this formula to see if the result is what we expect, so example one is al fateh which if we saw last time, then we take one - tpt multiplied by the points, that's the distance from p to and now we have to calculate the derivative here. Now I would make that component, but. the same thing comes out in the first and second component So here in front the derivative of 1 - tea is - 1 which is a constant factor so it says here - p and the derivative of you science so it says here in the plus cow or maybe that was a little bit clearer, that's great - p that's what Speed ​​Rector Length means than coupe length and I look back again that's not surprising because that's not exactly what we're supposed to have done here if now we want to use our formula, we want to calculate the length according to the new formula, so we would have to say now integral of oh something like that now too and b because Aryanization you have to do that, what interval do we use exactly zero to one? it means that now we have to integrate from zero to one over this here but that doesn't appear at all it's a constant, this is the simplest integral there is, there's a function in front of it.

Finding the function simply means this. fixed constant The circle if you leave leaves to vary the circle outside cosine t&t casinos was in the interval from 0 to 2 p what do we have to do it again? We have to derive components wisely it is not difficult the derivation of cosine is - derived sine from sine cosine from which we have to get the lengths of this sector which means we don't have to go into great details first - silos tea squared plus cosine tea squared and from there the root to the day what is that - it doesn't matter sin square plus cosine square, you know, and that's what we had last time.

Let's pretend this is the root of 1, which means the derivative always comes out as one. We already knew that the velocity vector has length 1. the unit price, so if we take beta the first sufficient and longest length, then we will have something like this again. Quite ridiculous integral in this case from 0 to 2 p of 1 dt, so just the start function dt of the past is like just tea again in the limits from 0 to 2 p, that means we write in detail and then it is used here for p2p. It is used for t0 20, so 2 p means that we get a total of 2 p as the longest circular arc and that should come out because the circumference of the circle is two times thymus radii and now let's take an example where it may not be so obvious.

The result is not difficult to calculate, but the question is if you expected that, I will take, say, gamma of t, the curve cosine t cosine tea, that is for the tea of ​​the interval 0 to 2 p Maybe you can think about it for a moment and get Get in touch if you have an idea of ​​what curve is being drawn. then the circle was cosine t but now I take the cosine teko, you had to do it so that the xy coordinates are always the same and last from 0 to 2 4 execute what comes for that curve, that means that this thought has to look like this, that all the points have to be on this diagonal because that is the set of points where the xy coordinates are always the same and if you put 0 1 here then you get 11 courses less from 0 to 1 if if you insert them then it says - 1 - 1 here and if you insert two and it says 11 again, that means I don't need a computer for that.

We can see that the thing starts here then it runs here and then it runs again.back that's the path once forward but back but they all only light up twice between points 11 up here and - 1 - 1 round trip now we can calculate the length using our formula, so here is the derivative - its rules Is it a contract of use? The environment doesn't matter anymore again justice in the square, so I had to go to the square twice and from there I can take the root of the two, so here I have the root of the double sine, I don't do it.

In fact, I have to consider that now we have to be on the safe side. We calculated it here, it says from 0 to 2 p, so from 0 to 2 p, the factor became two, we can pull it forward immediately and here it is. now it says the quantity of the sine tdt, which is always total Hessen, you don't like that kind of thing and the quantity is integral That's why it's best to divide the integral into two parts, the sine runs like this and then like this means on the top , so in the first half I can omit the amount in the second that I have to write; before we have the root of the integral twice from zero to p is this area here of the sine tdt and then I have here the integral over kibiz 2p of - ttt the root function is - pittis cosine 0 that means we have - free of pi - - cosine of 0 6 of key is - 1 - - one is one and here we have the client that from zero is one minus -1 is +1, so here comes two, which means that this area up here is 2 and it is so this area down here is - 2 and then we have here what comes out is 2+2, so what comes out four times the root of 2 is the length of the strange red path here four times became two exactly, that's true, but only because, like you said, we drove there and back, that's if you only do this route.

Driving from here to here would leave after the conference like two roots two. I have named the example here to make it clear thatThe length of a curve, of course, depends on the parameterization. If you just look at the track on this thing, you'll see Yeah, no you drove there and back, the track. of the thing just looks like this and if you look at it then you would say that the length of this curve must have two roots for us, but the root of company 2 comes out because we can only calculate the length of a curve if we have a parameterization. for this and this parameterization that I have chosen here has actually been chosen so that the route is traversed twice and that is why the incorrect length in ticks appears, so always keep that in mind.

This formula that we have seen here, which is framed here, is the function that does what it is supposed to do, but it does it in relation to the parameterization that they give me and if the parameterization, for example, runs through certain areas of the curve several times, then these are them. The parts are also measured several times. This is actually measured in the same way as a car driving around a curve, not how long the road is, but how long the car has traveled and if the car is still going back and forth on the road, then.

Of course, it was a longer distance back, as if it only happened once. I once used the Golden Gate Bridge as an application example. But we will not calculate it by hand, we will do it on computers and, as such. I said, I was also assuming that they would somehow calculate things like that. I always have to use the computer, so I took a look at the Golden Gate Bridge and there are these two pillars, not drawn to scale, and then these Hengl ropes. hang them up and now we idealizedly imagine that this hanging column in the middle is exactly the same height as the street.

We understood it pretty well, not exactly, but we were able to do it pretty well and the distance between these two towers. of the Golden Gate Bridge is 1280 meters, or approximately 1.3 kilometers, and the height of these towers above the street is 152 meters, so at the point where these routes are suspended and the crucial point is for physical reasons that I can not explain. Now, since I'm not a physicist, it is a parabola, which means that this type of bridge can be built better and more stable if the suspension is shaped like a parabola. Can you calculate it very easily?

I don't want to explain it. We don't need to let the computer do it right away, but just as a reminder, I hope you've seen something like this before. It's called problem interpolation. You have three points that you know if you can. now put a coordinate system here for example and say this is the point with ex coordinates 0 and y coordinates 152, this is the point with half the ex coordinates here, so 640 y coordinates and from this point the I have displayed 1280 and coordinates and 152 and I have a parabola drawn through these three points. It is clearly determined and the computer can calculate it for you.

This is my parameter representation and then I can use it to calculate the length of this wire, which. It's what I really want. Maybe I didn't say it at all, but I hope you thought I wanted to calculate the length of the cable up there and that's it. I'll show you this in two ways. I'll show you this in Python with its computer algebra system and then I'll show you. I'll tell you again if you can do it at Geo Gäbler and we'll get a free drawing. First I will simply load the system. , that's it and we need a variable of, we have always mentioned it, but the first thing is that we tell the system We would like to have a polynorm where the nomination is that passes through the points that I just said, the top point of the left pillar is 0 152, the middle where the cable touches the road is 640 0 and on the far right the pillar is 1280 meters high 152 in the variable t and we show it and then we get a problem like this. , so I thought now 19 x te squared times a pretty big man - 19 x te times 40 plus 152 and now we just have to do the following, maybe explained.

This is this p again that I got there, now it's a curve. that seems a bit exaggerated, wrong scale, i.e. here is my x axis and here is my x axis p, this is not yet a parameterized curve like ours. You know, to make a parameterized curve from To do this, I would have to write that alpha of t has the following coordinates, the y coordinates are bvt and the iks coordinate is t. We already did it last hour. The question was: can a graph of a function always be used as a parameter display? curve, the answer then was no, so we simply run components in the first and run classes in the second through functions.

Now we have to do again what we have already done several times. From there we derive it only once and the round. comes the derivation of tar and that justifies the rules so I have the root of once squared, so from 1 + p dash from tee to the square and then the integral that I want to integrate and that is, the root of 1 plus the derived from b to c and vices squared and I want to integrate from t equals zero on the far left to 1280 and that's what we get. Maybe it's not so good now, but I can ask you to give me an approximation of. and then I get that the cable length is about 1300 26 meters, a biogenic one, everything is almost a little bit prettier.

You can give the points 1 here 0 152 and then give them the same name, so the first point was called 32 280 352 the ntc the only problem is that the resolution does not match what we want to see. Now I have to move away a little, then to b and c. Now we can do something similar to what we did. First of all, we can say. to the team that we would like to approve a political issue. Put these three points, that means I have to enter polynorm here through the three points, it's actually quite easy a b c and you can see, first of all, he gave me that of all things and second of all. , he did it immediately and the thing can also be given the name f and now we can calculate the integral like this, root integral of does a conclusion sign root from it 1 + f dash nothing squared from 0 to 1280? and that also comes out to 1326 and we could have made everything even easier because if he knows how to steer the curves he calculates the formula that we have now laboriously derived.

Here you can simply enter the length of the curve, from zero to 1280. . You don't need to write an integral and so on, and before you do that, Mr. pressed, the solution is down there, of course, the same number comes out. Well, now you know how long these steel cables have been in the Golden. Gate Bridge. So you learned something else today and we'll continue next week.