如何找到一个旋转使得x1=a1s1+a2s2变为x2=b1s1+b2s2

假设我们有两个向量x1和x2，它们被表示为x1 = a1s1 + a2s2和x2 = b1s1 + b2s2，其中s1和s2是两个正交的单位向量。现在我们想要找到一个旋转矩阵R，使得x1在旋转后等于x2，即Rx1 = x2。

我们可以将x1和x2表示为矩阵形式：

x1 = [a1, a2]T
x2 = [b1, b2]T

R也可以表示为矩阵形式：

R = [cosθ, -sinθ]
[sinθ, cosθ]

其中θ是旋转角度。因此，我们可以将Rx1表示为：

Rx1 = [cosθ, -sinθ]
[sinθ, cosθ]
[a1, a2]T

然后我们将Rx1与x2相等，即：

[cosθ, -sinθ]
[sinθ, cosθ]
[a1, a2]T = [b1, b2]T

解这个方程组可以得到：

cosθ = (a1b1 + a2b2) / sqrt(a1^2 + a2^2)
sinθ = (a1b2 - a2b1) / sqrt(a1^2 + a2^2)

因此，我们可以通过解这个方程组来找到旋转矩阵R，使得x1在旋转后等于x2。

相关问题

将下面的python代码转成高效的C语言代码 for i in range(N): b, a = signal.butter(4, [6.0/(Cfg['Nt'][k]/2), 50.0/(Cfg['Nt'][k]/2)], 'bandpass') Hin[k][i, :] = signal.lfilter(b, a, Hin[k][i, :])

以下是将Python代码转换为C语言代码的示例：

#include <stdlib.h>
#include <stdio.h>
#include <math.h>

void butter(int N, double Wn[], char* type, double b[], double a[]) {
    double c1, c2, c3, c4, c5, c6, c7, c8, c9, c10;
    double d1, d2, d3, d4, d5, d6, d7, d8, d9, d10;
    double ar1, ar2, ai1, ai2;
    double f1, f2, f3, f4, f5, f6, f7, f8, f9, f10;
    double p1, p2, p3, p4, p5, p6, p7, p8, p9, p10;
    double q1, q2, q3, q4, q5, q6, q7, q8, q9, q10;
    double r1, r2, r3, r4, r5, r6, r7, r8, r9, r10;
    double s1, s2, s3, s4, s5, s6, s7, s8, s9, s10;
    double t1, t2, t3, t4, t5, t6, t7, t8, t9, t10;
    double u1, u2, u3, u4, u5, u6, u7, u8, u9, u10;
    double v1, v2, v3, v4, v5, v6, v7, v8, v9, v10;
    double w1, w2, w3, w4, w5, w6, w7, w8, w9, w10;
    double x1, x2, x3, x4, x5, x6, x7, x8, x9, x10;
    double y1, y2, y3, y4, y5, y6, y7, y8, y9, y10;
    double z1, z2, z3, z4, z5, z6, z7, z8, z9, z10;
    double qf1, qf2, qf3, qf4, qf5, qf6, qf7, qf8, qf9, qf10;
    double qb1, qb2, qb3, qb4, qb5, qb6, qb7, qb8, qb9, qb10;
    double af1, af2, af3, af4, af5, af6, af7, af8, af9, af10;
    double ab1, ab2, ab3, ab4, ab5, ab6, ab7, ab8, ab9, ab10;
    double a0, a1, a2, a3, a4, b0, b1, b2, b3, b4;
    double s, c, alpha, beta, gamma, delta;
    double pi = 3.14159265358979323846;
    
    if (strcmp(type, "lowpass") == 0) {
        s = sin(pi*Wn[0]);
        c = cos(pi*Wn[0]);
        alpha = s/(2*N);
        beta = sqrt(1 - alpha*alpha);
        gamma = (1 - c)/2;
        delta = (1 + c)/2;
        
        a0 = delta;
        a1 = gamma + beta*I;
        a2 = alpha*2 - delta;
        a3 = -(gamma + beta*I);
        a4 = delta;
        
        b0 = alpha;
        b1 = 0;
        b2 = -alpha;
        b3 = 0;
        b4 = alpha;
    }
    else if (strcmp(type, "highpass") == 0) {
        s = sin(pi*Wn[0]);
        c = cos(pi*Wn[0]);
        alpha = s/(2*N);
        beta = sqrt(1 - alpha*alpha);
        gamma = (1 + c)/2;
        delta = (1 - c)/2;
        
        a0 = delta;
        a1 = -(gamma + beta*I);
        a2 = alpha*2 - delta;
        a3 = gamma + beta*I;
        a4 = delta;
        
        b0 = alpha;
        b1 = 0;
        b2 = -alpha;
        b3 = 0;
        b4 = alpha;
    }
    else if (strcmp(type, "bandpass") == 0) {
        s = sin(pi*Wn[0]);
        c = cos(pi*Wn[0]);
        alpha = s*sinh(log(2)/2*N*Wn[1]/s);
        beta = sqrt(1 - alpha*alpha);
        gamma = c;
        delta = 1;
        
        a0 = delta;
        a1 = -2*gamma;
        a2 = 2*alpha*beta;
        a3 = 2*gamma;
        a4 = delta;
        
        b0 = alpha;
        b1 = 0;
        b2 = -alpha;
        b3 = 0;
        b4 = alpha;
    }
    else if (strcmp(type, "bandstop") == 0) {
        s = sin(pi*Wn[0]);
        c = cos(pi*Wn[0]);
        alpha = s/sinh(log(2)/2*N*Wn[1]/s);
        beta = sqrt(1 - alpha*alpha);
        gamma = c;
        delta = 1;
        
        a0 = delta;
        a1 = -2*gamma;
        a2 = 2*alpha*beta;
        a3 = 2*gamma;
        a4 = delta;
        
        b0 = beta;
        b1 = -2*beta*c;
        b2 = beta;
        b3 = beta;
        b4 = -2*beta*c;
    }
    else {
        printf("Invalid filter type.\n");
        exit(1);
    }
    
    ar1 = a[0];
    ar2 = a[1];
    ai1 = a[2];
    ai2 = a[3];
    f1 = b[0];
    f2 = b[1];
    f3 = b[2];
    f4 = b[3];
    f5 = b[4];
    p1 = 0;
    p2 = 0;
    p3 = 0;
    p4 = 0;
    p5 = 0;
    q1 = 0;
    q2 = 0;
    q3 = 0;
    q4 = 0;
    q5 = 0;
    r1 = 0;
    r2 = 0;
    r3 = 0;
    r4 = 0;
    r5 = 0;
    s1 = 0;
    s2 = 0;
    s3 = 0;
    s4 = 0;
    s5 = 0;
    t1 = 0;
    t2 = 0;
    t3 = 0;
    t4 = 0;
    t5 = 0;
    u1 = 0;
    u2 = 0;
    u3 = 0;
    u4 = 0;
    u5 = 0;
    v1 = 0;
    v2 = 0;
    v3 = 0;
    v4 = 0;
    v5 = 0;
    w1 = 0;
    w2 = 0;
    w3 = 0;
    w4 = 0;
    w5 = 0;
    x1 = 0;
    x2 = 0;
    x3 = 0;
    x4 = 0;
    x5 = 0;
    y1 = 0;
    y2 = 0;
    y3 = 0;
    y4 = 0;
    y5 = 0;
    z1 = 0;
    z2 = 0;
    z3 = 0;
    z4 = 0;
    z5 = 0;
    qf1 = 0;
    qf2 = 0;
    qf3 = 0;
    qf4 = 0;
    qf5 = 0;
    qb1 = 0;
    qb2 = 0;
    qb3 = 0;
    qb4 = 0;
    qb5 = 0;
    af1 = 0;
    af2 = 0;
    af3 = 0;
    af4 = 0;
    af5 = 0;
    ab1 = 0;
    ab2 = 0;
    ab3 = 0;
    ab4 = 0;
    ab5 = 0;
    
    for (int i = 0; i < N; i++) {
        qf5 = qf4;
        qf4 = qf3;
        qf3 = qf2;
        qf2 = qf1;
        qf1 = Hin[k][i];
        
        qb5 = qb4;
        qb4 = qb3;
        qb3 = qb2;
        qb2 = qb1;
        qb1 = qf5*a0 + qf4*a1 + qf3*a2 + qf2*a3 + qf1*a4 + qb5*b0 + qb4*b1 + qb3*b2 + qb2*b3 + qb1*b4;
        
        af5 = af4;
        af4 = af3;
        af3 = af2;
        af2 = af1;
        af1 = qb5*f1 + qb4*f2 + qb3*f3 + qb2*f4 + qb1*f5;
        
        ab5 = ab4;
        ab4 = ab3;
        ab3 = ab2;
        ab2 = ab1;
        ab1 = qb5*f1 + qb4*f4 + qb3*f3 + qb2*f2 + qb1*f5;
        
        Hin[k][i] = af1 - ab1;
    }
}

然后，您可以使用以下代码调用上面的函数：

int main() {
    double Wn[] = {6.0/(Cfg['Nt'][k]/2), 50.0/(Cfg['Nt'][k]/2)};
    double b[5], a[5];
    int N = ...; // 请用您的代码替换此处的 N
    
    butter(4, Wn, "bandpass", b, a);
    
    return 0;
}

使用c#分别生成 4维，5维，6维数组（分别为int类型，double类型，string类型）（每个花括号里都有随机的数值） 并且遍历数组

以下是分别生成 4维、5维、6维数组的示例代码：

// 生成 int 类型的 4 维数组
int[,,,] intArray4D = new int[2, 3, 4, 5]
{
    {
        {
            {1, 2, 3, 4, 5},
            {6, 7, 8, 9, 10},
            {11, 12, 13, 14, 15},
            {16, 17, 18, 19, 20}
        },
        {
            {21, 22, 23, 24, 25},
            {26, 27, 28, 29, 30},
            {31, 32, 33, 34, 35},
            {36, 37, 38, 39, 40}
        },
        {
            {41, 42, 43, 44, 45},
            {46, 47, 48, 49, 50},
            {51, 52, 53, 54, 55},
            {56, 57, 58, 59, 60}
        }
    },
    {
        {
            {61, 62, 63, 64, 65},
            {66, 67, 68, 69, 70},
            {71, 72, 73, 74, 75},
            {76, 77, 78, 79, 80}
        },
        {
            {81, 82, 83, 84, 85},
            {86, 87, 88, 89, 90},
            {91, 92, 93, 94, 95},
            {96, 97, 98, 99, 100}
        },
        {
            {101, 102, 103, 104, 105},
            {106, 107, 108, 109, 110},
            {111, 112, 113, 114, 115},
            {116, 117, 118, 119, 120}
        }
    }
};

// 遍历 int 类型的 4 维数组
for (int i = 0; i < intArray4D.GetLength(0); i++)
{
    for (int j = 0; j < intArray4D.GetLength(1); j++)
    {
        for (int k = 0; k < intArray4D.GetLength(2); k++)
        {
            for (int l = 0; l < intArray4D.GetLength(3); l++)
            {
                Console.WriteLine("intArray4D[{0}, {1}, {2}, {3}] = {4}", i, j, k, l, intArray4D[i, j, k, l]);
            }
        }
    }
}

// 生成 double 类型的 5 维数组
double[,,,,] doubleArray5D = new double[2, 3, 4, 5, 2]
{
    {
        {
            {
                {1.1, 1.2},
                {2.1, 2.2},
                {3.1, 3.2},
                {4.1, 4.2},
                {5.1, 5.2}
            },
            {
                {6.1, 6.2},
                {7.1, 7.2},
                {8.1, 8.2},
                {9.1, 9.2},
                {10.1, 10.2}
            },
            {
                {11.1, 11.2},
                {12.1, 12.2},
                {13.1, 13.2},
                {14.1, 14.2},
                {15.1, 15.2}
            },
            {
                {16.1, 16.2},
                {17.1, 17.2},
                {18.1, 18.2},
                {19.1, 19.2},
                {20.1, 20.2}
            }
        },
        {
            {
                {21.1, 21.2},
                {22.1, 22.2},
                {23.1, 23.2},
                {24.1, 24.2},
                {25.1, 25.2}
            },
            {
                {26.1, 26.2},
                {27.1, 27.2},
                {28.1, 28.2},
                {29.1, 29.2},
                {30.1, 30.2}
            },
            {
                {31.1, 31.2},
                {32.1, 32.2},
                {33.1, 33.2},
                {34.1, 34.2},
                {35.1, 35.2}
            },
            {
                {36.1, 36.2},
                {37.1, 37.2},
                {38.1, 38.2},
                {39.1, 39.2},
                {40.1, 40.2}
            }
        },
        {
            {
                {41.1, 41.2},
                {42.1, 42.2},
                {43.1, 43.2},
                {44.1, 44.2},
                {45.1, 45.2}
            },
            {
                {46.1, 46.2},
                {47.1, 47.2},
                {48.1, 48.2},
                {49.1, 49.2},
                {50.1, 50.2}
            },
            {
                {51.1, 51.2},
                {52.1, 52.2},
                {53.1, 53.2},
                {54.1, 54.2},
                {55.1, 55.2}
            },
            {
                {56.1, 56.2},
                {57.1, 57.2},
                {58.1, 58.2},
                {59.1, 59.2},
                {60.1, 60.2}
            }
        }
    },
    {
        {
            {
                {61.1, 61.2},
                {62.1, 62.2},
                {63.1, 63.2},
                {64.1, 64.2},
                {65.1, 65.2}
            },
            {
                {66.1, 66.2},
                {67.1, 67.2},
                {68.1, 68.2},
                {69.1, 69.2},
                {70.1, 70.2}
            },
            {
                {71.1, 71.2},
                {72.1, 72.2},
                {73.1, 73.2},
                {74.1, 74.2},
                {75.1, 75.2}
            },
            {
                {76.1, 76.2},
                {77.1, 77.2},
                {78.1, 78.2},
                {79.1, 79.2},
                {80.1, 80.2}
            }
        },
        {
            {
                {81.1, 81.2},
                {82.1, 82.2},
                {83.1, 83.2},
                {84.1, 84.2},
                {85.1, 85.2}
            },
            {
                {86.1, 86.2},
                {87.1, 87.2},
                {88.1, 88.2},
                {89.1, 89.2},
                {90.1, 90.2}
            },
            {
                {91.1, 91.2},
                {92.1, 92.2},
                {93.1, 93.2},
                {94.1, 94.2},
                {95.1, 95.2}
            },
            {
                {96.1, 96.2},
                {97.1, 97.2},
                {98.1, 98.2},
                {99.1, 99.2},
                {100.1, 100.2}
            }
        },
        {
            {
                {101.1, 101.2},
                {102.1, 102.2},
                {103.1, 103.2},
                {104.1, 104.2},
                {105.1, 105.2}
            },
            {
                {106.1, 106.2},
                {107.1, 107.2},
                {108.1, 108.2},
                {109.1, 109.2},
                {110.1, 110.2}
            },
            {
                {111.1, 111.2},
                {112.1, 112.2},
                {113.1, 113.2},
                {114.1, 114.2},
                {115.1, 115.2}
            },
            {
                {116.1, 116.2},
                {117.1, 117.2},
                {118.1, 118.2},
                {119.1, 119.2},
                {120.1, 120.2}
            }
        }
    }
};

// 遍历 double 类型的 5 维数组
for (int i = 0; i < doubleArray5D.GetLength(0); i++)
{
    for (int j = 0; j < doubleArray5D.GetLength(1); j++)
    {
        for (int k = 0; k < doubleArray5D.GetLength(2); k++)
        {
            for (int l = 0; l < doubleArray5D.GetLength(3); l++)
            {
                for (int m = 0; m < doubleArray5D.GetLength(4); m++)
                {
                    Console.WriteLine("doubleArray5D[{0}, {1}, {2}, {3}, {4}] = {5}", i, j, k, l, m, doubleArray5D[i, j, k, l, m]);
                }
            }
        }
    }
}

// 生成 string 类型的 6 维数组
string[,,,,,] stringArray6D = new string[2, 3, 4, 5, 2, 2]
{
    {
        {
            {
                {
                    {"a1", "a2"},
                    {"b1", "b2"},
                    {"c1", "c2"},
                    {"d1", "d2"},
                    {"e1", "e2"}
                },
                {
                    {"f1", "f2"},
                    {"g1", "g2"},
                    {"h1", "h2"},
                    {"i1", "i2"},
                    {"j1", "j2"}
                },
                {
                    {"k1", "k2"},
                    {"l1", "l2"},
                    {"m1", "m2"},
                    {"n1", "n2"},
                    {"o1", "o2"}
                },
                {
                    {"p1", "p2"},
                    {"q1", "q2"},
                    {"r1", "r2"},
                    {"s1", "s2"},
                    {"t1", "t2"}
                }
            },
            {
                {
                    {"u1", "u2"},
                    {"v1", "v2"},
                    {"w1", "w2"},
                    {"x1", "x2"},
                    {"y1", "y2"}
                },
                {
                    {"z1", "z2"},
                    {"A1", "A2"},
                    {"B1", "B2"},
                    {"C1", "C2"},
                    {"D1", "D2"}
                },
                {
                    {"E1", "E2"},
                    {"F1", "F2"},
                    {"G1", "G2"},
                    {"H1", "H2"},
                    {"I1", "I2"}
                },
                {
                    {"J1", "J2"},
                    {"K1", "K2"},
                    {"L1", "L2"},
                    {"M1", "M2"},
                    {"N1", "N2"}
                }
            },
            {
                {
                    {"O1", "O2"},
                    {"P1", "P2"},
                    {"Q1", "Q2"},
                    {"R1", "R2"},
                    {"S1", "S2"}
                },
                {
                    {"T1", "T2"},
                    {"U1", "U2"},
                    {"V1", "V2"},
                    {"W1", "W2"},
                    {"X1", "X2"}
                },
                {
                    {"Y1", "Y2"},
                    {"Z1", "Z2"},
                    {"1a", "1b"},
                    {"2c", "2d"},
                    {"3e", "3f"}
                },
                {
                    {"4g", "4h"},
                    {"5i", "5j"},
                    {"6k", "6l"},
                    {"7m", "7n"},
                    {"8o", "8p"}
                }
            }
        },
        {
            {
                {
                    {"9q", "9r"},
                    {"0s", "0t"},
                    {"1u", "1v"},
                    {"2w", "2x"},
                    {"3y", "3z"}
                },
                {
                    {"4A", "4B"},
                    {"5C", "5D"},
                    {"6E", "6F"},
                    {"7G", "7H"},
                    {"8I", "8J"}
                },
                {
                    {"9K", "9L"},
                    {"0M", "0N"},
                    {"1O", "1P"},
                    {"2Q", "2R"},
                    {"3S", "3T"}
                },
                {
                    {"4U", "4V"},
                    {"5W", "5X"},
                    {"6Y", "6Z"},
                    {"7a", "7b"},
                    {"8c", "8d"}
                }
            },
            {
                {
                    {"9e", "9f"},
                    {"0g", "0h"},
                    {"1i", "1j"},
                    {"2k", "2l"},
                    {"3m", "3n"}
                },
                {
                    {"4o", "4p"},
                    {"5q", "5r"},
                    {"6s", "6t"},
                    {"7u", "7v"},
                    {"8w", "8x"}
                },
                {
                    {"9y", "9z"},
                    {"0A", "0B"},
                    {"1C", "1D"},
                    {"2E", "2F"},
                    {"3G", "3H"}
                },
                {
                    {"4I", "4J"},
                    {"5K", "5L"},
                    {"6M", "6N"},
                    {"7O", "7P"},
                    {"8Q", "8R"}
                }
            },
            {
                {
                    {"9S", "9T"},
                    {"0U", "0V"},
                    {"1W", "1X"},
                    {"2Y", "2Z"},
                    {"3a", "3b"}
                },
                {
                    {"4c", "4d"},
                    {"5e", "5f"},
                    {"6g", "6h"},
                    {"7i", "7j"},
                    {"8k", "8l"}
                },
                {
                    {"9m", "9n"},
                    {"0o", "0p"},
                    {"1q", "1r"},
                    {"2s", "2t"},
                    {"3u", "3v"}
                },
                {
                    {"4w", "4x"},
                    {"5y", "5z"},
                    {"6A", "6B"},
                    {"7C", "7D"},
                    {"8E", "8F"}
                }
            }
        }
    },
    {
        {
            {
                {
                    {"9G", "9H"},
                    {"0I", "0J"},
                    {"1K", "1L"},
                    {"2M", "2N"},
                    {"3O", "3P"}
                },
                {
                    {"4Q", "4R"},
                    {"5S", "5T"},
                    {"6U", "6V"},
                    {"7W", "7X"},
                    {"8Y", "8Z"}
                },
                {
                    {"9a", "9b"},
                    {"0c", "0d"},
                    {"1e", "1f"},
                    {"2g", "2h"},
                    {"3i", "3j"}
                },
                {
                    {"4k", "4l"},
                    {"5m", "5n"},
                    {"6o", "6p"},
                    {"7q", "7r"},
                    {"8s", "8t"}
                }
            },
            {
                {
                    {"9u", "9v"},
                    {"0w", "0x"},
                    {"1y", "1z"},
                    {"2A", "2B"},
                    {"3C", "3D"}
                },
                {
                    {"4E", "4F"},
                    {"5G", "5H"},
                    {"6I", "6J"},
                    {"7K", "7L"},
                    {"8M", "8N"}
                },
                {
                    {"9O", "9P"},
                    {"0Q", "0R"},
                    {"1S", "1T"},
                    {"2U", "2V"},
                    {"3W", "3X"}
                },
                {
                    {"4Y", "4Z"},
                    {"5a", "5b"},
                    {"6c", "6d"},
                    {"7e", "7f"},
                    {"8g", "8h"}
                }
            },
            {