蝙蝠俠遙控器pcb

View Graph查看圖

介紹： (Introduction:)

The circle in itself is really pretty, ain’t it? But, with some curiosity, you can go beyond a circle. It doesn’t need any calculus or any advanced concepts, just with our favourite class 10 mathematics and with some beautiful ideas, we can go beyond the circle and even create the Batman Inequation. This article is about creating inequations for whatever shapes we want. In the second part, we will see how we can improve and generalize this framework and derive commonly used functions in Machine Learning.

圓圈本身真的很漂亮，不是嗎？ 但是，出于好奇，您可以超越圈子。 它不需要任何演算或任何高級概念，只需借助我們最喜歡的10級數學和一些漂亮的想法，我們就可以超越圈子，甚至創造出蝙蝠俠不等式。 本文旨在為我們想要的任何形狀創建不等式。 在第二部分中，我們將看到如何改進和概括該框架并派生出機器學習中常用的功能。

This is a journey from the curve on the left to the one on the right這是一條從左側曲線到右側曲線的旅程

All visualisations created here are generated in openly available software like Desmos and Geogebra. I suggest the readers go ahead and play with these equations using these tools and explore the beauty by themselves.

此處創建的所有可視化效果都是通過Desmos和Geogebra等可公開獲得的軟件生成的。 我建議讀者繼續使用這些工具處理這些方程式，并自己探索美麗。

潛力一圈！ (The potential of a Circle!)

The humble circle describes multiple points which lie equidistant from a given centre point. Let’s have a look at our familiar friend in all its forms. When all points lie at a given distance, let’s consider 1 unit, we get a circle. Points with distance less than 1 lie inside the circle and ones with more than 1 lie outside the circle. In 3 dimensions a paraboloid with the equation z=x2+y2–1 forms our circle when it intersects with the XY plane.

虛圓圈描述了與給定中心點等距的多個點。 讓我們來看看我們熟悉的各種形式的朋友。 當所有點都位于給定距離上時，考慮1個單位，我們得到一個圓。 距離小于1的點位于圓內，大于1的點位于圓外。 在3維中，等式z =x2+ y2–1的拋物面與XY平面相交時形成我們的圓。

Different views of a circle圓的不同觀點

We all know that a circle normally takes the form x2+y2=1. What do you think will happen if we go beyond the power of 2? That’s where the magic lies. Let’s take the values from 2 to 10 and observe what happens.

我們都知道，圓通常采用x2+y2= 1的形式。 如果我們超越2的冪，您認為會發生什么？ 那就是魔術所在。 讓我們取2到10之間的值，觀察會發生什么。

Powers going from 3 to 10力量從3到10

For even powers, we see the circle starts looking like a square and it does become one at infinite power. But why does it do so and especially at even powers? We will first try to understand how the points lying on the curve (x^n+y^n=1) behave and then look into the nature and properties of points within the curve (x^n+y^n<=1).

對于偶數冪，我們看到圓開始看起來像一個正方形，并且在無窮大冪時確實變為一個。 但是，為什么要這樣做，尤其是在偶數權力下呢？ 我們將首先嘗試了解曲線上的點(x ^ n + y ^ n = 1)的行為，然后研究曲線內的點(x ^ n + y ^ n <= 1)的性質和特性。

Let’s understand odd powers first. If the power is odd, and say x is negative, then the result of x^n term is also negative, thus y^n term takes a value greater than 1 so that the sum remains 1. For large powers, the difference of 2 numbers becomes insignificant with an increase in power. Let’s take an example, if x? is -100,000 then the value of y? should be 100,001 to satisfy the equation. But, we are plotting x and y, so the values of x and y which justify the equation are (-10, 10.0000199..). This is extremely close to the line y=-x. This is also applicable for the case with negative y and positive x. Also note that higher the power, smaller is the deviation from y=-x. This cannot be possible with negative y and negative x, and that’s why we don’t see a part of the function in that quadrant. When x and y both are positive, we see something like a part of a square. This is because if x is significantly smaller than 1, like 0.7, then x? becomes very small very fast (0.168 here). Thus y? has to be 1-x? which is 0.832, which means y will be very close to 1 but slightly less (0.9638) here. The same logic applies the other way around. So, for x values away from 1, y takes values near 1 (like a horizontal edge) and for x values near 1, y drops to 0 fast (like a vertical edge). This makes the curve look like a part of the square. This can be seen below:

首先讓我們了解奇數冪。 如果冪是奇數，并且說x為負，則x ^ n項的結果也為負，因此y ^ n項的值大于1，因此總和仍為1。對于大冪，差2隨著功率的增加，數字變得無關緊要。 讓我們舉一個例子，如果x?是-100,000，那么y?的值應該是100,001以滿足方程。 但是，我們正在繪制x和y，因此證明該方程式正確的x和y的值為(-10，10.0000199 ..)。 這非常接近線y = -x。 這也適用于具有負y和正x的情況。 還要注意，功率越高，與y = -x的偏差就越小。 對于負y和負x不可能做到這一點，這就是為什么我們在該象限中看不到函數的一部分的原因。 當x和y均為正數時，我們看到的東西類似于正方形的一部分。 這是因為如果x顯著小于1，例如0.7，則x?很快變得非常小(此處為0.168)。 因此，y?必須為1-x?，即0.832，這意味著y將非常接近于1，但此處略小于(0.9638)。 相同的邏輯反過來適用。 因此，對于x值遠離1，y取接近1的值(如水平邊)，對于x值接近1，ySwift減小到0(如垂直邊)。 這使曲線看起來像正方形的一部分。 可以在下面看到：

Demonstrating nature with n=9.展示自然，n = 9。

Once odd powers are understood, understanding even powers becomes much easier. The case of negative values now doesn’t exist so the entire function looks like a square in all the quadrants. This can be seen below. This is what we will focus on from here onwards.

一旦理解了奇數冪，就更容易理解偶數冪。 現在不存在負值的情況，因此整個函數在所有象限中看起來像一個正方形。 可以在下面看到。 這就是我們將從現在開始關注的重點。

Graph for n=10n = 10的圖

Finally, the circle and the square are not at odds with each other and have set their powers even. With the hope that you square up with these ideas and doubts don’t encircle you, we loop back to mathematics without cutting any corners.

最后，圓圈和正方形彼此并不矛盾，甚至設定了它們的力量。 希望您能接受這些想法，而疑惑不會籠罩您，因此我們回頭繼續進行數學學習，而不用加倍努力。

Wait!!! There is a lot left in the store of mathematics. We also know how to shift coordinates. Subtraction for left and addition for right. So we can not only generate a square, but we can also position it anywhere we want. Not only that, but we can also rescale and thus stretch the square and make into a rectangle. Let’s try it:

等待！！！ 數學存儲中還有很多東西。 我們也知道如何移動坐標。 左減法，右加法。 因此，我們不僅可以生成一個正方形，而且可以將其放置在所需的任何位置。 不僅如此，我們還可以重新縮放比例，從而拉伸正方形并制成矩形。 讓我們嘗試一下：

View Graph查看圖

By now, some of you familiar with deeper mathematics would be able to see how this is related to ideas in Minkowski Distance and F-norm but we will leave them for the future.

到現在，一些熟悉更深層次的數學的人將能夠看到它與Minkowski距離和F范式中的思想之間的關系，但是我們將把它們留待將來使用。

超越圈子 (Beyond Circles)

While going beyond circles and rectangles, we have to take a slightly different perspective on these graphs. This time we look from the perspective of inequations for points within the curve which have a sum lesser than 1. If the sum of 2 positive numbers is less than 1 then both the numbers have to be less than 1. Similarly, if there are many such terms, then all of them have to be as small as possible so that the sum never exceeds 1. Even if one of the terms is greater than 1 then the inequality will not hold. Isn’t this very similar to the idea of intersection? Selected points should lie in all sets (i.e. all inequations should give values close to 0) if they are not in even one of the sets then they are not selected (even if one of the terms is greater than 1 then the inequation does not hold true). Thus, the squares made above can be seen as the region of intersection of two terms, x^(2n)<1 and y^(2n)<1 (referred to as trenches for their shape) as shown below. Higher values of n allow the terms to be as small as possible.

在超越圓形和矩形的同時，我們必須對這些圖形采取稍微不同的觀點。 這次，我們從不等式的角度來看曲線中總和小于1的點。如果2個正數的總和小于1，則兩個數都必須小于1。類似地，如果存在許多這樣的項，那么所有項都必須盡可能小，以使總和永遠不超過1。即使其中一項大于1，也不等式也不成立。 這與十字路口的想法不是很相似嗎？ 如果選擇的點不在所有集合中，則它們應該位于所有集合中(即，所有不等式的值都應接近0)，即使它們不在一組集合中，也不會被選中(即使其中一項大于1，則不等式也不成立)真正)。 因此，如下所示，上面制作的正方形可以看作是x ^(2n)<1和y ^(2n)<1的兩項的交集區域(其形狀稱為溝槽)。 n的值越高，項越小。

The squares(z = x1?+y1?) as the intersection of 2 trenches made by z=x1? and z=y1?平方(z = x +1 + y1)是由z = x +1和z = y1構成的兩個溝槽的交點

And now we have taken a humongous step. We can make very complex figures which emerge from such intersections and take our designing skills to the next level.

現在，我們邁出了巨大的一步。 我們可以制作出非常復雜的圖形，這些圖形從這樣的交叉點出現，并將我們的設計技能提高到一個新的水平。

A diamond as the intersection of diagonal trenches: z = (y-2x)1?+(y+2x)1?菱形作為對角線溝槽的交點：z =(y-2x)1+(y + 2x)1

We can get back our graph by just making z=1:

我們可以通過使z = 1來返回圖：

View Graph查看圖

Let’s try this strategy on something simple:

讓我們嘗試一些簡單的策略：

View Graph查看圖

Remember that the trenches we make have the walls at the following positions: y?—?f(x)=1 and y?—?f(x)= -1. This is because all absolute values below 1 tend to zero thus are part of the trench whereas all absolute values greater than one increase very fast thus forming the walls. So we can use the trenches shown below.

請記住，我們制作的溝槽的壁在以下位置：y — f(x)= 1和y — f(x)= -1。 這是因為所有小于1的絕對值都趨于零，因此是溝槽的一部分，而所有大于1的絕對值都非常快地增加，從而形成了壁。 因此，我們可以使用下面顯示的溝槽。

We are now completely equipped for making the batman symbol and covering more than half of the journey. The strategy is not to just intersect but to also eliminate regions from the curve to carve out the shape. This is done by taking curves one by one and refining them and their positioning to match the shape. In some places, the curves had to be inverted i.e. the region greater than 1 had to be made less than 1 and vice versa. This was done by changing the sign of the power of the curves. Note that this strategy can be applied to many shapes. All these curves have the property of being greater than 1 on one side (away from symbol) and less than 1 on the other (towards symbol). Thus every section has its own curve which is then combined using the sum of large even powers as described earlier.

現在，我們已經完全具備制作蝙蝠俠標志的能力，并且涵蓋了整個旅程的一半以上。 該策略不僅要相交，而且還要從曲線中消除區域以雕刻出形狀。 這是通過一條一條地繪制曲線并對其進行細化和使其位置與形狀匹配來完成的。 在某些地方，曲線必須反轉，即大于1的區域必須小于1，反之亦然。 這是通過更改曲線功效的符號來完成的。 注意，該策略可以應用于許多形狀。 所有這些曲線的特性是，一側(遠離符號)大于1，而另一側(朝向符號)小于1。 因此，每個部分都有自己的曲線，然后使用較大的偶數冪之和將其合并，如前所述。

The following expressions were used (selected according to the shape of the curve):

使用以下表達式(根據曲線的形狀選擇)：

• f1(x,y):(0.5(x-1.16)^(2.8))^(2) +(y+1.6): lower edge of right wing
  f1(x，y):( 0.5(x-1.16)^(2.8))^(2)+(y + 1.6)：右翼下邊緣
• f2(x,y):(0.5(x+1.16)^(2.8))^(2) +(y+1.6): lower edge of left wing
  f2(x，y):( 0.5(x + 1.16)^(2.8))^(2)+(y + 1.6)：左翼下緣
• f3(x,y):(0.5(y+1.6))^(8)+(x+3): left edge of left wing
  f3(x，y):( 0.5(y + 1.6))^(8)+(x + 3)：左翼的左邊緣
• f4(x,y):(0.5(y+1.6))^(8)+(-x+3): right edge of right wing
  f4(x，y):( 0.5(y + 1.6))^(8)+(-x + 3)：右翼的右邊緣
• f5(x,y):y+0.6: upper horizontal line
  f5(x，y)：y + 0.6：上水平線
• f6(x,y):(3(x+0.45))^(14)-y+1: left curve between head and wing
  f6(x，y):( 3(x + 0.45))^(14)-y + 1：頭和翼之間的左曲線
• f7(x,y):(3(x-0.45))^(14)-y+1: right curve between head and wing
  f7(x，y):( 3(x-??0.45))^(14)-y + 1：頭和翼之間的右曲線
• f8(x,y):e^((3(y-0.1)-258.18((1.9x+0.1)(1.9x-0.1))^(1.6))): forms the head and ears
  f8(x，y)：e ^((3(y-0.1)-258.18((1.9x + 0.1)(1.9x-0.1))^(1.6)))：形成頭和耳朵

When all these curves are combined the following figure is obtained:

將所有這些曲線組合后，可獲得下圖：

Note that there are extra bits on the sides but the original function is intact.請注意，側面還有多余的位，但原始功能未損壞。

To remove the extra bits, the function is cleaned by adding another term which gives values close to 0 near the shape we want and values greater than 1 in places we don’t want. This makes the final figure as:

為了刪除多余的位，通過添加另一個術語來清理該函數，該術語在所需形狀附近給出接近0的值，而在不需要的位置給出大于1的值。 這使得最終數字為：

View Graph; What does the first term do? It doesn’t kill our curve, so it simply makes it stronger (by eliminating unwanted parts).視圖圖 ; 第一個學期做什么？ 它不會殺死我們的曲線，因此只是使其變得更牢固(通過消除不需要的部分)。

We can identify all the parts of the curve individually as seen below:

我們可以分別識別曲線的所有部分，如下所示：

All curves of the form f(x,y)=1 shown with the inequationf(x，y)= 1形式的所有曲線均顯示為不等式

結論和下一步 (Conclusion and What’s coming next)

We have just obtained a deep understanding of circles and similar inequations with high even powers. We understood why they behave like the intersection of inequation and mastered this by creating our own batman inequation. Everything’s impossible until somebody does it. Well, the batman equation was created about a decade ago so we tried our hand at Batman Inequation. But, we still had to deal with large powers, had to trim our figure and the process still seemed complex. Part II of this blog will remove all these challenges and simplify everything. It will also explain how these ideas are relevant in machine learning in the form of Softmax(our classification friend), Softplus (well-known activation function), log-sum-exp(commonly used function and father of Softmax) and other related directions.

我們剛剛對圓和具有較高偶數冪的類似不等式有了深刻的了解。 我們了解了為什么他們表現得像不等式的交集，并通過創建自己的蝙蝠俠不等式來掌握它們。 除非有人做，否則一切都是不可能的。 好吧，蝙蝠俠方程式是大約十年前創建的，因此我們嘗試了蝙蝠俠不等式。 但是，我們仍然必須處理大國問題，必須精簡身材，而且過程似乎仍然很復雜。 本博客的第二部分將消除所有這些挑戰并簡化一切。 它還將以Softmax(我們的分類伙伴)，Softplus(眾所周知的激活函數)，log-sum-exp(Softmax的常用函數和父)的形式以及其他相關方向說明這些思想在機器學習中的相關性。 。

The math is the heaviest just before the application. And I promise you, the application is coming!

在申請之前，數學是最繁重的。 我向您保證，應用程序即將發布！

翻譯自: https://medium.com/@jasdeep.grover100/from-circle-to-ml-via-batman-part-i-51ab4cf2db66

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