Today's article is only a little bit

point

About mid -January this year, I raised my head and saw the spots on the wall for the first time. In order to determine which day it was, I had to remember what I saw at the time. Now I remember the stove in the stove. Fire, a yellow fire light shines on my pages; three chrysanthemums are inserted in the round glass tank on the fireplace. Yes, it must be winter. We just finished tea because I remember that I was smoking at the time, I looked up and saw the spots on the wall for the first time ...

"Spots on the Wall" Virginia Woolf

This is a point:

In order to echo the masterpiece of Woolf, I held up the camera and picked up the long -sealed hand -painted board. For the first time, I edited this picture in the software. In order to determine the information of this picture, I will You have to open the software's file attribute panel. Now I remember the color mode of SRGB, and ...

Usually in certain disciplines (such as mathematics and physics), there is no size. So the following is also a point:

How beautiful and beautiful the picture on the right side is, its shape and orientation are such elegance and ... (cough, can’t blow it down.)

But if you move the camera a little bit to make the picture bigger, the point of the previous (right) is not so "huge".

However, the size of such needle eyes is dot, the size of the rice grains is point, and the size of the fist is also point. So what is the point? At this time, we might as well bring the "tactical eyepiece mirror" to bring the "combat value value value value of all we see things we see. "All displayed in a table:

If we can think of a way to write all its information into a form in any object. To some extent, we can express and replace the objects with tables in a meter to some extent.

But in the real world, it is necessary to describe a lot of information that a object may require a lot of information, that is, a object corresponds to a very many forms of tables. It is very troublesome to deal with a lot of information, so a simple idea is: only pay attention to a part of the part: only pay attention to a part of a part Information.

If we only retain the position of the object (that is, the information about the top and left in the figure above, or professional, called "coordinates".), Then no matter how the other characteristics of the object are so delicious, in our eyes, it is nothing but it is just that it is nothing but it is just that it is nothing but it is just that it is nothing but it is nothing more than the eyes. A simplified coordinate -or in other words, an abstract "point".

But sometimes, we can not be so "radical", can retain more information, such as quality (such as a 5 kg point), speed (such as a point of a 5cm/s vertical fall), angular movement (imagine a one The rotating point is really difficult), etc. So we got the "colorful" point:

Point to become refined (bushi

If you want some other views:

We can think of the above projection as a vector of the information of the information representing the information in the information space to the straight new operation of the location space and other space -to simplify the calculation amount; or In observation restrictions, only a part of the limited information can be observed -in order to restore the information of the original object as much as possible from the limited information.

Click, click

The boundary of the line is the point

"Geometry Originally" Ou Ji Liden ()) Maria Xu Guangqi (translated)

If we currently only consider the location of the object, ignore other information, and then look at the picture below:

We will find that although the patterns on the above cultural relics are only the most simple points (poke), but the points are combined together, it can form a regular pattern: triangle, ring band, etc. To a certain extent, the information that we originally passed the "point" model was simplified, and in the relationship between the points, it was re -expressed. So why don't we think in turn: Can we complicate some complexity? The graphics, objects, objects, and objects use some simple basic "points" (should we be familiar with the story of this point before?)?)?

As we all know, there can be Bad Apple in a colored place ...

Ah, not. It is two points to determine a straight line (probably). Just like the big white eyes in the "Super Marine Corps":

Dabai:? (Okay, I can't draw

So as long as there are two points, we can know the location information of other points on the straight line. How convenient is this way! And now that the straight line will be drawn, will it be far away from the curve? (The ancients will be far away? Yun: "Three -year -old children can hold pens and graffiti, with straight generations, there is great beauty." Bushi)

None of the curve in the picture above is used, unless I shake my hand

Therefore, if you want to use straight lines to express the curve (such as the hair "curve in the figure above"), you only need to select the appropriate feature point on the curve, and then use the straight line to connect the feature point to a certain extent. . 也就是说, 我们通过用直线上的点来代表曲线上的点的方式来近似地表现曲线. 而倘若我们能够将特征点的数量增加到一定的程度, 然后眯上眼睛——看上去这样The straight line becomes like a curve. As the number of feature points increases, the straight line is getting closer and closer to the curve, so it can be said that within a certain small range, the straight line can be used to represent the curve

(The idea of ​​calculus is probably like this)

Maybe there will be a bumper, so how can it be? What I want the spirit of the craftsmanship is the wonderful curve of this craft. Your straight line is the straight line of industrialized machinery, and there is no soul of my craftsman's spirit! The curve of my hand is pinched, The error of each line is less than ε error.

So in the face of such a bar essence, we can only do -further increase the number of feature points, for example, when the number of feature points is N ≥ N (ε), our error will be less than ε. So we can proud of it. Evil charm smile: The straight line can approach the curve. In the language of mathematics:

In the same way, we can also apply this method to the surface, three -dimensional issues. (Since you have learned one plus one, then please ...) Remember the "point" point "" point ", we will find that this choice The characteristic points represent the more complicated information of the object and the original abstraction is similar to the same work: they all use only a part of the information to achieve the description of the state of the original object.

dot dot dot

If this is to be proved and the type of computer is as I've already explained, then it's going to be necessary that everything that happens in a finite volume of space and time would have to be exactly analyzable with a finite number of logical operations Then, then, then

Richard P. Feynman, Simulating Physics with Computers, May 7, 1981

(The following is my dregs translation. For readers' consideration, the above is the original text)

If there is a general computer that I (Ferman) before, then (using a computer to simulate physics) needs to be implemented by performing limited logical operations in limited space and time.

Richard P. Ferman, use computer simulation physics, May 7, 1981

As we mentioned above, in order to simplify the model of abstracting objects into points for simplification and processing, and in order to better express other information about the object, we can add more points (or are or so Introduce the relationship between points and points) to carry more information about objects.

But please do n’t just limit how to record the data and the pattern should be opened. Why not ask the magic conch? Try to make these data instead of the real objects in reality? The method of breeding with abstract data instead of real objects in reality.

The amount of mathematics (general) in physics is a math function that continuously changes (over time).

For a solution

The corresponding physical quantity is a amount of continuous change over time. In mathematics, no matter how accurate the value is, it is possible -because SIN is a continuous function. In (general) physical images, it seems that it should also be That's it -because the space is continuous. If these "continuous" functions are drawn on the figure, then it will look like a smooth curve. (Note: Strictly define The function is a continuous function ".)

But why do we not learn from the "points" of the previous "points", through a certain "characteristic point" to describe the change of these physical quantities? So the original continuous physical quantity was divided into discrete quantities. And such things can be used to be able to do things. It is very attractive to hand over to the computer to deal with it. So the problem of using a computer to simulate the physical world looks very attractive.

首先是时间, 假如我们考虑一个简单的匀速直线运动: x(t) = v t, 那么在时间轴上选择一些特征点进行分割, (就像是用一台相机定格住一瞬的空间), 沿着The "space" divided by these feature points may wish to see it as a "state". According to specific rules, when jumping from this point point to the next time point, our state will change. That is: from the state of x (t 状) to the state of x (t ₀ + Δt) = x (t₀) + v Δt. If you want some other views:

May wish to transform such a state as an expanded version of the automatic machine.

Do you remember the problem of the accuracy of the accuracy? Similarly, we will encounter accuracy problems when we simulate. If we encounter very large exercise of exercise, we need more feature points. But as a touch of touching Fish big households, I saw a almost straight line full, and a snake -shaped curve cleverly "avoided" ...

dot dot dot…

Then come to some examples of "reality". (Reference: UNDERSTANDING The FinITE ELEMENT METHOD (YouTube), and "Theoretical Mechanics" Liang Kunmao) to end: If we want to understand the shape of a elastic cube under a specific force, Then, then, then

We regard a cube as a collection of high -quality points. It is well known that objects with elastic (non -rigid) will change when they are external. The size, assuming that it is a cube in a three -dimensional European space (or the right -angle coordinate system in the three -dimensional space), then its displacement can be represented by weight.

Well known

Elastic objects can be compressed, and the compressed objects will have a "trend" back to the original state -that is, there is a rebound force (that is, the compressed elastic object will generate an approximate and compression volume proportional to a certain extent. The recovery power). Write in a formula called "Hu Ke's Law": f = -k Δr.

About Hu Ke's Law:

In the book of Mr. Liang's theoretical mechanics, this defines: "When the elastic rod is elongated or shortened, the interaction force F of any cross -section of the cross -section is proportional to the elongation or shortening L.

According to Wikipedia, in the earliest publication (from the production of the spring clock): Hu Ke's law is a string of disrupted Latin puzzles: "CEIIINOSSSSTTTUV" Published after the order of disruption of letters, and then announced the conclusion after confirmation. Well, it is really a method.), The real order is: "UT Tensio SIC VIS", that is, "Eligibility (Generally Change) "(In fact, this feels the same as the literature of our ancient Chinese people. It is very simple and obscure, such as" Tao Ke Tao, Very Tao "and the like, while modern science needs to be more stricter and no ambiguous explanations. In this way, we can be regarded as the development of the giant's shoulders.)

However, it is said that the phrase "Stepping on the shoulder of the giant" is a kind of taunting Hu Ke -ridicule Hu Ke's short. Ke is a very glorious scientist. It describes Hick in English Wikipedia: "Polymath" (in Chinese is "a doctrist"). Indeed, as an early famous scientist First, Hu Ke's research involves very widely and has a basic contribution to a variety of scientific branches.

(Well, to some extent, there is a little benefit to born earlier.)

And connect the different parts of the same object to each other, so we can get a series of stress and displacement relationships, and combine these relationships together, then we can know the distribution of force in the object. For example (super simple simple (super simple example of):

A simple example, a bunch of springs with different elastic coefficients are connected in series.

So we will say that the equation of these stress is:

But for each part, there is a part of the interrelated part:

Finally, the overall balance relationship:

So we can calculate the distribution of external forces.

Can it be a bit more complicated?

No problem, we just need to expand what we need to consider, for example, consider a spring that is upside down in the gravity field at this time.

A bunch of little monkeys hang on the tree and want to fish the moon ...

Compared with the previous problems, it is not difficult to find that the changes are only the equation part of the force:

So you can calculate the final result in this way. (Calculating process is slightly?)

Regarding the calculation process and some ideas:

If we write the force equation of a single spring into the form shown in the figure below:

Similarly, we can expand the stress equation of multiple spring:

So we can calculate the results. And this practice can also help us consider more complicated and general problems, such as writing the matrix of the recovery coefficient into a general form:

So you can express more complex elastic properties. What cube?

To some extent, we can think of cubes as a PRO MAX version of the above problem through a little expansion. That is to say, we can "discrete" a cube like this:

Suddenly remembering MC ...

Or or like the image below:

Triangle always reminds me of the tower of the transmission line on the mountain ...

So the problem became the "100 million point" of the above processing method. However, in the above problems, there are some more things that need to be calculated, so at this time we can apply a computer to assist us in simulating calculation calculations And we can also find that during the calculation process above, our characteristic points are not as analyzed earlier: "densely dense places, sparse in small changes." It is also a place that can be optimized and improved.

This calculation method seems to be called "finite element analysis". But the name is not very important. Our ideas are just to reduce the original complicated and continuous problems to reduce the scale of research. Come to achieve the purpose of simplifying calculations. In short, you have learned 1+1. Then try it ...?

Edit: Mu Zi

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