# RubicCube

**Repository Path**: arrivezenith/RubicCube

## Basic Information

- **Project Name**: RubicCube
- **Description**: c#三阶魔方实现及自动还原
- **Primary Language**: C#
- **License**: GPL-2.0
- **Default Branch**: master
- **Homepage**: None
- **GVP Project**: No

## Statistics

- **Stars**: 11
- **Forks**: 3
- **Created**: 2019-06-17
- **Last Updated**: 2024-08-12

## Categories & Tags

**Categories**: Uncategorized

**Tags**: None

## README

# README
魔方是一种很受欢迎的益智玩具，同时魔方运动也被称为“手指三大极限运动”之一。本篇博文将就如何使用程序实现一个三阶魔方的旋转提出我自己的思路
## 魔方的结构
一个普通的三阶魔方具有六个面、二十六个块。其中，有六个中心块，十二个棱块，八个角块。中心块的相对位置是不会变的，无论你如何旋转魔方，六个中心块的位置始终是不会变化的。棱块处于两个面交界的位置，拥有两个颜色，因此也有两种朝向。所以一个棱块的位置可能有12×2=24种情况。角块处于三个面交界的位置，有三个颜色，因此有三种朝向。一个角块的位置可能有8×3=24种情况。
## 思考一个问题
如何确定一个魔方？按照日常经验，只要魔方的六个面都确定了，那么这个魔方的情况就完全确定了。这的确是一种思路。我们可以考虑使用一个二维数组来存放魔方六个面的情况：
``` c++
#define WHITE 0
#define YELLOW 1
#define RED 2
#define ORANGE 3
#define BLUE 4
#define GREEN 5

int[6][9] = {0};
void init_cube()
{
    for(int i=0; i<6; i++)
    {
        for(int j=0; j<9; j++)
        {
            int[i][j] = i;
        }
    }
}
```
用数字0代表白色，数字1代表黄色...这样我们便有了一个6*9的数组，代表每个面的每个色块分别是什么颜色，这样的一个数组就可以表示一个魔方。但是这种方式真的合适吗？仔细考虑一下，一方面，一个魔方面与面的颜色之间是存在关联的，比如，一个棱块上的两个颜色，必然不可能是两个相对面的颜色，一个角块上的三个颜色，必然是三个相邻的面的颜色。而我们分别储存六个面的颜色信息，这些数据之间就缺少了由魔方的结构所带来的约束关系，因此我们必然会写更多的代码来弥补这种缺失的关联。另一方面，魔方是一个三维空间中的结构，我们用这种方式表示一个魔方，相当于是将魔方六个面展开，降成了二维。如此一来，在处理三维的旋转时，必然会遇到困难。因此这种方式虽然很容易想到，也比较简单，但是后续的实现会相当复杂，因此是不合适的。
## 魔方结构的数字化
综合以上分析，我们对魔方结构的表示一定要注意几个关键点：
- 要以三维的形式来表示
- 不能丢失魔方结构决定的约束
- 易于处理旋转等操作

所以，我们可以考虑以魔方的块为单位，用26个方块的位置和朝向来决定一个魔方。说到位置，我们很自然地就能想到利用控件坐标系中的一组向量来表示方块的位置。首先，建立空间坐标系。我们选择以立方体的中心为原点，向右为X轴正向，向前为Y轴正向，向上为Z轴正向：
![坐标系建立][img1]
这样一来，魔方中所有方块的位置就都可以用一组向量来表示了。比如下图中，红色标明的中心块可以表示为 $\{0,1,0\}^T$，蓝色标明的棱块位置可以表示为 $\{-1,0,1\}^T$，绿色标明的角块位置可以表示为 $\{1,-1,1\}^T$。
![坐标系示例][img2]
接下来，我们考虑一下如何表示每个方块的朝向。所谓朝向，无非就是哪个方向对应哪个颜色。因此，我们也可以使用与位置类似的方法，用一组向量来表示方块的朝向。我们用R代表红色，O代表橙色，W代表白色，Y代表红色，B代表蓝色，G代表绿色。考虑到棱块只有两个方向有颜色，中心块只有一个方向有颜色，为了统一，我们引入X，代表某个方向没有颜色。如下图所示，中心块的颜色可以表示为 $\{X,B,Z\}^T$，右边棱块的颜色可以表示为 $\{R,G,X\}^T$，右上角角块的颜色可以表示为 $\{G,Y,R\}^T$。
![颜色坐标示例][img3]
这样，我们就可以用两个向量来表示魔方中一个块的位置和颜色。比如上图中的角块可以表示为：
$$
 \begin{cases}
 位置：\{1,1,1\}^T\\
 颜色：\{G,Y,R\}^T
 \end{cases}
$$
这样，只需要26组向量，我们便可以用数据表示出一个魔方。
## 魔方的旋转
魔方的旋转可以理解为空间中的坐标变换，而我们刚才已经可以用用向量来表示魔方的结构，因此直接使用旋转矩阵，即可完成对魔方的旋转。向量旋转公式：
$$
\begin{Bmatrix}
    x'\\y'\\z'
\end{Bmatrix}
=R*
\begin{Bmatrix}
    x\\y\\z
\end{Bmatrix}
$$
绕$x,y,z$轴旋转时，$R$的值分别如下：
$$
R_x=
\begin{Bmatrix}
    1 & 0 & 0\\
    0 & cos\theta & -sin\theta\\
    0 & sin\theta & cos\theta
\end{Bmatrix}
$$
$$
R_y=
\begin{Bmatrix}
    cos\theta & 0 & sin\theta\\
    0 & 1 & 0\\
    -sin\theta & 0 & cos\theta
\end{Bmatrix}
$$
$$
R_z=
\begin{Bmatrix}
    cos\theta & -sin\theta & 0\\
    sin\theta & cos\theta & 0 \\
    0 & 0 & 1
\end{Bmatrix}
$$
其中$\theta$是顺旋转的角度。取$\theta=90\degree$，即为顺时针旋转90度，反之为逆时针旋转90度。
让我们来测试一下。假如将上图角块所在的顶层沿顺时针方向旋转90度，那么旋转前后该角块的数据：
$$
 \begin{cases}
 位置：\{1,1,1\}^T\\
 颜色：\{G,Y,R\}^T
 \end{cases}\tag前
$$
$$
 \begin{cases}
 位置：\{-1,1,1\}^T\\
 颜色：\{Y,G,R\}^T
 \end{cases}\tag后
$$
如果我们用旋转矩阵进行变换，得到以下结果：
$$
\begin{Bmatrix}
    cos\theta & -sin\theta & 0\\
    sin\theta & cos\theta & 0 \\
    0 & 0 & 1
\end{Bmatrix}
*\begin{Bmatrix}
    1\\1\\1
\end{Bmatrix}=
\begin{Bmatrix}
    -1\\1\\1
\end{Bmatrix}
$$
$$
\begin{Bmatrix}
    cos\theta & -sin\theta & 0\\
    sin\theta & cos\theta & 0 \\
    0 & 0 & 1
\end{Bmatrix}
*\begin{Bmatrix}
    G\\Y\\R
\end{Bmatrix}=
\begin{Bmatrix}
    -Y\\G\\R
\end{Bmatrix}
$$
颜色向量中的符号代表颜色面向的方向与坐标轴方向相反。舍弃掉符号，运算所得的结果与魔方实际旋转后的结果完全一致，这就证明了我们的方法是正确的。这样一来，我们就解决了魔方用数据来表示这一问题，也解决了魔方的旋转。你可以顺着这个思路，尝试自己编写代码来实现。

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RU2wLVykDsKBcDbztK4J3d3e7im1hhQvAqHqMbSG4AIym19gWnjTVOfPLZn65epzdurFt7We27J/BOVyASsY4V9qKWWxnkerlzz3PljIAWzUf2962kecJLgBbI7aPCC4AWyG2iwQXgI0T2+MEF4CNEtvHE1wANkZsn8xtQQAcs8ptO3t7e2L7FTz4onPml838aMUYsU3/+26FC8Ba5mNbpIdxW5zDBWBlR2PLk1nhAlSS/mhHsV2OFS4ASxPb5QkuAEsR29UILgAnJrarE1wATkRs1yO4ADyV2K7Pgy86Z37ZzI9tOHr1tNhuhtuCAHiiEttTpw43Q2t/wKv9v78uW8oAPNZ8bFmfnyQAx4jt5tlSBmDBmTNnnK/dAsEFqKil85Jnz549+Fex3Q7BBWA4d+6c0G6ZDXqAzs1iu7OzU/tQJk1wATo2H1sr3O0SXIBOie24PGmqc+aXzfxy1Z7dLLatPNSiBy6aAujMfGwft7JtMb7lOFs8rmXYUgboyNNiy/YILkAnxLYuwQXogNjWJ7gAEye2bXDRFEAlR793dht2d3fFthFWuAATJbZtEVyACRLb9njwRefML5v55drm7MS2Tc7hAkzILLaz0K4a9tY+zE3hg4PgAkzEfGynEKijWvsQsCzncAEmYOqxnQLBBQgnthkEFyCY2OYQXIBQYptFcAECiW0eVykDVLTKlbd7e3tiG8iDLzpnftnMrz9im8sKFyDEfGyLbX3gGuNLFXrkHC5AgKOxJY/gAjRObKfBljJAw8T2kfRtbitcgEaJ7bQILkCDxHZ6BBegMWI7TYIL0BCxnS4XTQFUcvR+V7GdNk+a6pz5ZTO/6SixPXXqcNPRXKfJChegsvnYtkL0N6+tCQN0psXYsh1WuACVnDlzxvnajqwd3Fu3bm3iOKjE/LKZXzax7cvawX3++ec3cRxU8Pdf+1fD6W9dq30YrOiPL1z3+gt27969g1+cXEvnlc+ePbv07xHczp351m/WPgRW9YHXX7Lbt28LbqgS252dnaU/ADhTDwAnNItt+bUsF031rp0dGoCmrRPbwoMvIJjXH4yjxLbcvlV+rfq6s8LtnjdsqKXFD0yunD5uFttVV7YzggvAMS1+GKhhU7EtXDQFAI+xydgWggsAR2w6toUt5d7ZNQJYsI3YFoLbPcUFmNlWbAu3BUEwr79cZre8bf/MthnbwjlcALq37kMtTkJwAejaGLEtnMMFqKi1reXeHnwxi+06T5A6KcHtXlsvdoCxzMd2DILbO70FjigrvdZW3ps2dmwL53AB6EqN2BaCC0A3asW2cB9u50wvm9dfLrMbX83YFs7hds+LHpi+2rEtbCkDMGktxLawwu2dBS4QYpVt+HPnzjUR20JwASrp4fabmmaxLQ/zaOHnXD/5ALBh87FthRVu9+p/6gPYpBZjW7gtCIJ5/cGiVmNbWOECsCD13HLLsS0Et3d5rymAY1qPbeGiKQCiJcS2sMLtniUukCsltoUVLgCRkmJbWOECEGH+Qq5ZbI/+31smuAAVpcSiJSW2p0/n5ct9uBDM64/epMa2yDxqNscbNhAiObaFi6YAiJAc20JwAWAE2R8XANiKlq4PKF8gPwWC2712XlQAR5XYltt/WvoAsCrB7V3+32FgomaxLb/u379f+3DW5hwuAM2ZxfbUqVOTWN0W7sOFYF5/uczuyeZjO6+1n9myj5S0pdy9tv4CQ29ai0htT4rtFEzvTwRApCnHtpjmnwqAKFOPbTHdPxkAEXqIbeEcbu+cPwIq6iW2xfT/hAA0qafYFla4ACwoV05v++rp3mJbCG7nnnFbEDCyEtvyzT89xbYQXABGM1vZlodGLLuKbu2e5dEffNHaDwB64vWXq8fZ9biNPM8Kt3f9veahGWOcK21F77EtBLd7fbzYgXrE9lDff3oAtkpsH/ETAGArxHaRnwIAGye2xzmH2z3ncIHNEtvHE9ze6S2wQWL7ZIILwDGr3K507ty5lR9qsa1jaokHX0Awrz9aMR9bHs8Kt3vesIH1iO3J2GQHYGVie3JWuACVpD/aUWyXY4ULwNLEdnlWuL0L/nQN1CG2q7HCBeDExHZ1ggvAiYjtemwpd8+WMvB0s9gWyRd61eTBFxDM649tOHr19Hxsa0r/+25LGYAnKrE9fdpm6Cb4KfYu+wMjsEViu1l+kt1TXOA4sd08P00AFpTQ+nq9zRPczrm4H+pq6UKg8l22hdhuh58qAF9+cTzbY4XbvXY+XQN1zGJbft2/f7/24UyW4PZOb6Fr87FluwQXoFMltuV8bfk1fy65pfPKM1N4nKQnTUEwr79ctWc3i23Syrb2z2xdVrjdy/4LDCwvMbZT4CplgI6IbT2CC9AJsa3LlnLvws+JACcjtvVZ4QJMnNi2wQoXoJKj3zu7DWLbDitcgInyUIu2WOF2zzlcmKJ1Y9vaPa8efDG0NxToiddfrm3Obhbbo0+Qoi4rXIAJmY/t1KR/eBDc3mX//QXmTDm2UyC43VNcmAKxbZ/JAIQT2wymAxBMbHOYEEAosc3iHG73nMOFRGKbR3B7p7dQ1Sq3upw7d26rsR3jkZM98uALCOb1159ZbMuTl8w/i70IgBDzsSWPLeXOPWNPGSKIbT7BBWic2B5K30K3pQzQMLGdDivc3oV/YoQpE9tpscIFaJDYTo/gAjRGbKdJcAEaIrbT5Rxu95zDhVqOPtFpFtvZ/6+m2v/7U+RJUxDM6286SmxPn7YGmjJbygCViW0fTLh3FkhQldj2w5S7p7hQSwmtr9frx9rBvXXr1iaOgwr+9v6vuze+X/swWNUrv+/1F05sl9PSNQt7e3vDxYsXl/o9awf3lVdeWfe/gkp+MPy++SW7ccP8gt2+ffvgF3lmsb1+/fpSv8/HKwA4oVVjWwguAJzAOrEt3IfbOfPLZn4wjnVjW7hKGaCSFj8wzR4p2eKx1bKJ2Ba2lAHgCTYV20JwAeAxNhnbQnAB4IhNx7YQXACYs43YFoILAF/YVmwLtwV1zvyymV8us1vetn9m58+fH775zW9uJbaFFS4A3ZvF9tVXX93a/4bgAtC1MWJbePAFQEWtbS3PHnzRi7FiWwguAAvKh4DWPghsw5ixLWwpA9CdsWNbCC4AXakR20JwAehGrdgW7sPtnPllM79cZje+mrEtrHABmLzasS0EF4BJayG2hduCAIiwyjb8hQsXmohtIbgAlfRyv2stLcW2sKUMwOS0FttCcAGYlBZjW7gtqHPml8382Ibkre5WY1tY4QIwCS3HthBcAOK1HttCcAGIlhDbQnABiJUS20JwAYiUFNvCgy8AiDB/5XRabAvBBago9fabmkpsL168GBXbwn24nTO/bOZHb1JjWziHC0CE5NgWggtAhOTYFs7hAnBMS6crzp49O+zu7kbHthBcAJq1t7d3sLK9fv167UNZmy1lAJo0pdgWggtAc6YW20JwAWjKFGNbuA+3c+aXzfxymd3jTTW2hYumACoS3kemHNvCljIA1U09toXgAlBVD7EtBBeAanqJbSG4AFTRU2wLF00BsKBcyLXti7nOnz9/8H22vcS2sMIFYFSz2KY/G3lZggvAaHqNbSG4AIyi59gWnjTVOfPLZn65eptd77EtXDQFUMkYFye1QGwP2VIGYGvE9hHBBWArxHaR4AKwcWJ7nOACsFFi+3iCC8DGiO2TuUoZgGNWuXr6woULYvsVrHABWJvYPp0HX3TO/LKZHy0Q25OxwgVgZWJ7coILwErEdjkumgKoJPnRjmK7PCtcAJYitqsRXABOTGxXJ7gAnIjYrkdwAXgqsV2fi6YAWHD0Yi6x3QwPvuic+WUzP7atxPbixYtiuwG2lAF4LLHdLMEF4Bix3TzBBWDBmTNnxHYLBBeABS+88ILYboGrlAEqaunCt729vYOV7fXr12sfyiRZ4QIgtiMQXIDOie04BBegY2I7HsEF6JTYjsuTpjpnftnML1ft2Ynt+KxwATojtnUILkBHxLYewQXohNjWJbgAHRDb+gQXYOLEtg0e7QhQydEvet+G8+fPH3x5vNjWZ4ULMFGz2PoigjYILsAEiW17PPiic+aXzfxybXN2YtsmK1yACRHbdgkuwESIbdsEF2ACxLZ9ggsQTmwzCC5AMLHNIbgAocQ2i+ACBBLbPB7tCFDRKvfjXrhwQWwDefBF58wvm/n1R2xz2VIGCCG22QQXIIDY5hNcgMaJ7TQILkDDxHY6BBegUWI7LYIL0CCxnR7BBWiM2E6T4AI0RGyny5OmACopDy6Zf3iJ2E6bJ011zvyymd90lNhevHhRbCfMljJAZWLbB8EFqEhs+yG4AJWcOXNGbDuy9jncW7dubeI4qMT8splfthdeeEFsO/LMQ1ddAMDW2VIGgBEILgCMQHABYASCCwAjEFwAGIHgAsAIBBcARiC4ADACwQWAEQguAIxAcAFgBIILACP4SwSyFyYJnEEUAAAAAElFTkSuQmCC

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