求解积分：d_t=h-int(v_y,t,0,t),v_y=sqrt(mg/k) * ( (1-e^(-2sqrt(mgk)/m t)) / (1+e^(-2sqrt(mgk)/m t)) )

首先，我们可以计算出 v_y 的不定积分：

∫v_y dt = ∫sqrt(mg/k) * ( (1-e^(-2sqrt(mgk)/m t)) / (1+e^(-2sqrt(mgk)/m t)) ) dt

令u = e^(-2sqrt(mgk)/m t)，则du/dt = (-2sqrt(mgk)/m) * e^(-2sqrt(mgk)/m t)dt，从而有：

∫v_y dt = (-m/2sqrt(kg)) * ∫(1-u)/(1+u) du

= (-m/2sqrt(kg)) * (∫du/(1+u) - ∫u du/(1+u))

= (-m/2sqrt(kg)) * (ln(1+u) - (1/2)ln(1+u)^2)

= (-m/2sqrt(kg)) * ln(1+u)/√(1+u)

= (-m/2sqrt(kg)) * ln(1+e^(-2sqrt(mgk)/m t))/√(1+e^(-2sqrt(mgk)/m t))

现在我们来计算 d_t：

d_t = h - ∫v_y dt

= h + (m/2sqrt(kg)) * ln(1+e^(-2sqrt(mgk)/m t))/√(1+e^(-2sqrt(mgk)/m t))

因此，积分的解为：

d_t = h + (m/2sqrt(kg)) * ln(1+e^(-2sqrt(mgk)/m t))/√(1+e^(-2sqrt(mgk)/m t))

相关问题

def crop_pointcloud(data_crop, x_o, y_o, x_i, y_i, R_o, R_i, z_critical): K_o = R_o ** 2 / range_z K_i = R_i ** 2 / range_z for z in range(range_z): r_o = np.sqrt(z * K_o) data_layer = data_crop[:, :, z] d_o = np.sqrt(x_o ** 2 + y_o ** 2) d_i = np.sqrt(x_i ** 2 + y_i ** 2) if z < z_critical: r_i = 0 else: r_i = np.sqrt(z * K_i) data_crop[:, :, z] = np.where((d_o > r_o) | (d_i <= r_i), 0, data_layer) return data_crop data_crop = data[:, :, :400] range_x, range_y, range_z = data_crop.shape dx = 550 dy = 530 x, y = np.meshgrid(np.arange(range_x), np.arange(range_y)) x_o = x - range_x / 2 y_o = y - range_y / 2 x_i = x - dx y_i = y - dy z_critical = 50 R_o = 550 R_i = 200 data_crop = crop_pointcloud(data_crop, x_o, y_o, x_i, y_i, R_o, R_i, z_critical) data_crop = data_crop[:, :, 10:] 转C++ Eigen::Tensor<uint8_t, 3, Eigen::RowMajor>

以下是转换后的C++代码：

#include <Eigen/Core>
#include <unsupported/Eigen/CXX11/Tensor>

Eigen::Tensor<uint8_t, 3, Eigen::RowMajor> crop_pointcloud(
    Eigen::Tensor<uint8_t, 3, Eigen::RowMajor>& data_crop,
    Eigen::Tensor<float, 2>& x_o,
    Eigen::Tensor<float, 2>& y_o,
    Eigen::Tensor<float, 2>& x_i,
    Eigen::Tensor<float, 2>& y_i,
    float R_o,
    float R_i,
    int z_critical) {
  int range_z = data_crop.dimension(2);
  float K_o = R_o * R_o / range_z;
  float K_i = R_i * R_i / range_z;
  for (int z = 0; z < range_z; ++z) {
    float r_o = std::sqrt(z * K_o);
    auto data_layer = data_crop.chip(z, 2);
    auto d_o = (x_o * x_o + y_o * y_o).sqrt();
    auto d_i = (x_i * x_i + y_i * y_i).sqrt();
    float r_i = (z < z_critical) ? 0 : std::sqrt(z * K_i);
    data_crop.chip(z, 2) = (d_o > r_o || d_i <= r_i).select(0, data_layer);
  }
  return data_crop;
}

int main() {
  Eigen::Tensor<uint8_t, 3, Eigen::RowMajor> data(100, 100, 500);
  // Initialize data
  Eigen::Tensor<uint8_t, 3, Eigen::RowMajor> data_crop = data.slice(Eigen::array<int, 3>({0, 0, 0}), Eigen::array<int, 3>({100, 100, 400}));
  int range_x = data_crop.dimension(0);
  int range_y = data_crop.dimension(1);
  int range_z = data_crop.dimension(2);
  Eigen::Tensor<float, 2> x(range_x, range_y);
  Eigen::Tensor<float, 2> y(range_x, range_y);
  for (int i = 0; i < range_x; ++i) {
    for (int j = 0; j < range_y; ++j) {
      x(i, j) = i - range_x / 2;
      y(i, j) = j - range_y / 2;
    }
  }
  Eigen::Tensor<float, 2> x_o = x;
  Eigen::Tensor<float, 2> y_o = y;
  Eigen::Tensor<float, 2> x_i = x - 550;
  Eigen::Tensor<float, 2> y_i = y - 530;
  int z_critical = 50;
  float R_o = 550;
  float R_i = 200;
  data_crop = crop_pointcloud(data_crop, x_o, y_o, x_i, y_i, R_o, R_i, z_critical);
  data_crop = data_crop.slice(Eigen::array<int, 3>({0, 0, 10}), Eigen::array<int, 3>({100, 100, 390}));
  return 0;
}

注意，为了方便实现，我使用了Eigen库来处理多维数组。在C++中，需要手动遍历数组来初始化x和y，而在Python中，可以直接使用np.meshgrid来生成网格。此外，由于C++中没有np.where这个函数，我使用了Eigen的.select()函数来实现类似的功能。最后，注意将Python代码中的data_crop[:, :, 10:]转换为C++中的.slice()函数。

x, y = np.meshgrid(np.arange(range_x), np.arange(range_y)) # np.savetxt('reshape_data.txt', x, delimiter=' ', fmt="%i") x_o = x - range_x / 2 y_o = y - range_y / 2 x_i = x - dx y_i = y - dy z_critical = 50 R_o = 550 R_i = 200 def crop_pointcloud(data_crop, x_o, y_o, x_i, y_i, R_o, R_i, z_critical): K_o = R_o ** 2 / range_z K_i = R_i ** 2 / range_z for z in range(range_z): r_o = np.sqrt(z * K_o) data_layer = data_crop[:, :, z] d_o = np.sqrt(x_o ** 2 + y_o ** 2) d_i = np.sqrt(x_i ** 2 + y_i ** 2) if z < z_critical: r_i = 0 else: r_i = np.sqrt(z * K_i) data_crop[:, :, z] = np.where((d_o > r_o) | (d_i <= r_i), 0, data_layer) return data_crop data_crop = crop_pointcloud(data_crop, x_o, y_o, x_i, y_i, R_o, R_i, z_critical) data_crop = data_crop[:, :, 10:] 转C++ data_crop的类型为Eigen::Tensor<uint8_t, 3, Eigen::RowMajor>

以下是将给定的 Python 代码转换为 C++ 代码后的结果，其中包括了对于 Eigen Tensor 类型的支持：

#include <Eigen/Core>
#include <Eigen/Tensor>

void crop_pointcloud(Eigen::Tensor<uint8_t, 3, Eigen::RowMajor>& data_crop,
                     Eigen::ArrayXXf& x_o, Eigen::ArrayXXf& y_o,
                     Eigen::ArrayXXf& x_i, Eigen::ArrayXXf& y_i,
                     const float R_o, const float R_i, const int z_critical) {
    const int range_x = x_o.rows();
    const int range_y = y_o.cols();
    const int range_z = data_crop.dimension(2);
    const float range_z_float = static_cast<float>(range_z);
    const float K_o = R_o * R_o / range_z_float;
    const float K_i = R_i * R_i / range_z_float;
    
    for (int z = 0; z < range_z; ++z) {
        const float r_o = std::sqrt(z * K_o);
        Eigen::Tensor<uint8_t, 2, Eigen::RowMajor> data_layer = data_crop.chip(z, 2);
        const Eigen::ArrayXXf d_o = (x_o * x_o + y_o * y_o).sqrt();
        const Eigen::ArrayXXf d_i = (x_i * x_i + y_i * y_i).sqrt();
        const float r_i = (z < z_critical) ? 0 : std::sqrt(z * K_i);
        data_crop.chip(z, 2) = ((d_o > r_o) || (d_i <= r_i)).select(0, data_layer);
    }
}

int main() {
    const int range_x = ...;  // 请填入具体数值
    const int range_y = ...;  // 请填入具体数值
    const int range_z = ...;  // 请填入具体数值
    const int z_critical = 50;
    const float R_o = 550.0f;
    const float R_i = 200.0f;
    Eigen::Tensor<uint8_t, 3, Eigen::RowMajor> data_crop(range_x, range_y, range_z);
    Eigen::ArrayXXf x_o, y_o, x_i, y_i;
    x_o.resize(range_x, range_y);
    y_o.resize(range_x, range_y);
    x_i.resize(range_x, range_y);
    y_i.resize(range_x, range_y);
    for (int i = 0; i < range_x; ++i) {
        for (int j = 0; j < range_y; ++j) {
            x_o(i, j) = static_cast<float>(i) - range_x / 2.0f;
            y_o(i, j) = static_cast<float>(j) - range_y / 2.0f;
            x_i(i, j) = static_cast<float>(i) - dx;
            y_i(i, j) = static_cast<float>(j) - dy;
        }
    }
    crop_pointcloud(data_crop, x_o, y_o, x_i, y_i, R_o, R_i, z_critical);
    data_crop = data_crop.slice(Eigen::array<int, 3>({0, 0, 10}), Eigen::array<int, 3>({range_x, range_y, range_z - 10}));
    return 0;
}

请根据您的具体需求对代码进行修改和适配。