DirectCosineMatrices.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Матрицы направляющих косинусов"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Инициализация"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# если не установлены пакеты  Python - запустите эту ячейку с установочным скриптом\n",
    "# if several packages are not installed - run this cell\n",
    "!python -m pip install numpy sympy ipywidgets matplotlib --user"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%config IPCompleter.greedy=True\n",
    "# %matplotlib notebook # use in case of simple jupyter notebook\n",
    "%matplotlib widget # use in case of jupyter lab\n",
    "import sympy as sp\n",
    "from sympy import *\n",
    "from sympy.interactive import printing\n",
    "from sympy import pi\n",
    "from sympy.matrices import rot_axis1, rot_axis2, rot_axis3\n",
    "from ipywidgets import *\n",
    "import matplotlib.pyplot as plt\n",
    "from mpl_toolkits.mplot3d import axes3d   \n",
    "import numpy as np\n",
    "# init_printing(use_unicode=True)\n",
    "init_printing(use_latex=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Элементарные матрицы поворота / X, Y, Z-rotation matrices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Элементарная матрица поворота по X\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0\\\\0 & \\cos{\\left(\\alpha_x \\right)} & \\sin{\\left(\\alpha_x \\right)}\\\\0 & - \\sin{\\left(\\alpha_x \\right)} & \\cos{\\left(\\alpha_x \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1        0             0      ⎤\n",
       "⎢                              ⎥\n",
       "⎢0  cos(\\alphaₓ)   sin(\\alphaₓ)⎥\n",
       "⎢                              ⎥\n",
       "⎣0  -sin(\\alphaₓ)  cos(\\alphaₓ)⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Элементарная матрица поворота по Y\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\cos{\\left(\\alpha_y \\right)} & 0 & - \\sin{\\left(\\alpha_y \\right)}\\\\0 & 1 & 0\\\\\\sin{\\left(\\alpha_y \\right)} & 0 & \\cos{\\left(\\alpha_y \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡cos(\\alpha_y)  0  -sin(\\alpha_y)⎤\n",
       "⎢                                ⎥\n",
       "⎢      0        1        0       ⎥\n",
       "⎢                                ⎥\n",
       "⎣sin(\\alpha_y)  0  cos(\\alpha_y) ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Элементарная матрица поворота по Z\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\cos{\\left(\\alpha_z \\right)} & \\sin{\\left(\\alpha_z \\right)} & 0\\\\- \\sin{\\left(\\alpha_z \\right)} & \\cos{\\left(\\alpha_z \\right)} & 0\\\\0 & 0 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡cos(\\alpha_z)   sin(\\alpha_z)  0⎤\n",
       "⎢                                ⎥\n",
       "⎢-sin(\\alpha_z)  cos(\\alpha_z)  0⎥\n",
       "⎢                                ⎥\n",
       "⎣      0               0        1⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "alpha_x, alpha_y, alpha_z = symbols('\\\\alpha_x \\\\alpha_y \\\\alpha_z')\n",
    "print(\"Элементарная матрица поворота по X\")\n",
    "display(rot_axis1(alpha_x))\n",
    "print(\"Элементарная матрица поворота по Y\")\n",
    "display(rot_axis2(alpha_y))\n",
    "print(\"Элементарная матрица поворота по Z\")\n",
    "display(rot_axis3(alpha_z))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## НЗСК 🡆 связанная / Earth 🡆 body"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Заданы обозначения для крена, тангажа и рысканья (roll, pytch, yaw symbols introduced):\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\gamma, \\  \\vartheta, \\  \\psi\\right]$"
      ],
      "text/plain": [
       "[\\gamma, \\vartheta, \\psi]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "roll, pitch, yaw = symbols('\\\\gamma \\\\vartheta \\\\psi')\n",
    "print(\"Заданы обозначения для крена, тангажа и рысканья (roll, pytch, yaw symbols introduced):\")\n",
    "display([roll, pitch, yaw])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\cos{\\left(\\psi \\right)} \\cos{\\left(\\vartheta \\right)} & \\sin{\\left(\\vartheta \\right)} & - \\sin{\\left(\\psi \\right)} \\cos{\\left(\\vartheta \\right)}\\\\\\sin{\\left(\\gamma \\right)} \\sin{\\left(\\psi \\right)} - \\sin{\\left(\\vartheta \\right)} \\cos{\\left(\\gamma \\right)} \\cos{\\left(\\psi \\right)} & \\cos{\\left(\\gamma \\right)} \\cos{\\left(\\vartheta \\right)} & \\sin{\\left(\\gamma \\right)} \\cos{\\left(\\psi \\right)} + \\sin{\\left(\\psi \\right)} \\sin{\\left(\\vartheta \\right)} \\cos{\\left(\\gamma \\right)}\\\\\\sin{\\left(\\gamma \\right)} \\sin{\\left(\\vartheta \\right)} \\cos{\\left(\\psi \\right)} + \\sin{\\left(\\psi \\right)} \\cos{\\left(\\gamma \\right)} & - \\sin{\\left(\\gamma \\right)} \\cos{\\left(\\vartheta \\right)} & - \\sin{\\left(\\gamma \\right)} \\sin{\\left(\\psi \\right)} \\sin{\\left(\\vartheta \\right)} + \\cos{\\left(\\gamma \\right)} \\cos{\\left(\\psi \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡                  cos(\\psi)⋅cos(\\vartheta)                          sin(\\vart\n",
       "⎢                                                                             \n",
       "⎢sin(\\gamma)⋅sin(\\psi) - sin(\\vartheta)⋅cos(\\gamma)⋅cos(\\psi)  cos(\\gamma)⋅cos\n",
       "⎢                                                                             \n",
       "⎣sin(\\gamma)⋅sin(\\vartheta)⋅cos(\\psi) + sin(\\psi)⋅cos(\\gamma)  -sin(\\gamma)⋅co\n",
       "\n",
       "heta)                           -sin(\\psi)⋅cos(\\vartheta)                  ⎤\n",
       "                                                                           ⎥\n",
       "(\\vartheta)   sin(\\gamma)⋅cos(\\psi) + sin(\\psi)⋅sin(\\vartheta)⋅cos(\\gamma) ⎥\n",
       "                                                                           ⎥\n",
       "s(\\vartheta)  -sin(\\gamma)⋅sin(\\psi)⋅sin(\\vartheta) + cos(\\gamma)⋅cos(\\psi)⎦"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# результат совпадает с Дмитриевским\n",
    "MatNorm2Body=rot_axis1(roll) * rot_axis3(pitch) * rot_axis2(yaw)\n",
    "MatNorm2Body"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## НЗСК 🡆 траекторная"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Заданы обозначения для угла наклона траектории и угла поворта траектории:\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\Theta, \\  \\Psi\\right]$"
      ],
      "text/plain": [
       "[\\Theta, \\Psi]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Theta, Psi = symbols(\"\\\\Theta \\\\Psi\")\n",
    "print(\"Заданы обозначения для угла наклона траектории и угла поворта траектории:\")\n",
    "display([Theta, Psi])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\cos{\\left(\\Psi \\right)} \\cos{\\left(\\Theta \\right)} & \\sin{\\left(\\Theta \\right)} & - \\sin{\\left(\\Psi \\right)} \\cos{\\left(\\Theta \\right)}\\\\- \\sin{\\left(\\Theta \\right)} \\cos{\\left(\\Psi \\right)} & \\cos{\\left(\\Theta \\right)} & \\sin{\\left(\\Psi \\right)} \\sin{\\left(\\Theta \\right)}\\\\\\sin{\\left(\\Psi \\right)} & 0 & \\cos{\\left(\\Psi \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡cos(\\Psi)⋅cos(\\Theta)   sin(\\Theta)  -sin(\\Psi)⋅cos(\\Theta)⎤\n",
       "⎢                                                           ⎥\n",
       "⎢-sin(\\Theta)⋅cos(\\Psi)  cos(\\Theta)  sin(\\Psi)⋅sin(\\Theta) ⎥\n",
       "⎢                                                           ⎥\n",
       "⎣      sin(\\Psi)              0             cos(\\Psi)       ⎦"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# результат совпадает с Дмитриевским\n",
    "MatTraj2Norm= rot_axis3(Theta) * rot_axis2(Psi)\n",
    "# MatTraj2Norm=MatTraj2Norm.transpose()\n",
    "MatTraj2Norm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Скоростная 🡆 траекторная"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Задано обозначение для скоростного угла крена:\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\gamma_s$"
      ],
      "text/plain": [
       "\\gammaₛ"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "gamma_s = symbols(\"\\\\gamma_s\")\n",
    "print(\"Задано обозначение для скоростного угла крена:\")\n",
    "display(gamma_s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0\\\\0 & \\cos{\\left(\\gamma_s \\right)} & - \\sin{\\left(\\gamma_s \\right)}\\\\0 & \\sin{\\left(\\gamma_s \\right)} & \\cos{\\left(\\gamma_s \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1       0              0      ⎤\n",
       "⎢                              ⎥\n",
       "⎢0  cos(\\gammaₛ)  -sin(\\gammaₛ)⎥\n",
       "⎢                              ⎥\n",
       "⎣0  sin(\\gammaₛ)  cos(\\gammaₛ) ⎦"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# проверена по Дмитриевскому\n",
    "MatWind2Traj = rot_axis1(-gamma_s)\n",
    "MatWind2Traj"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Связанная 🡆 связанная с пространственным углом атаки / Body ref.system 🡆 Spatial AOA ref.system"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Заданы обозначения пространственного угла атаки и крена:\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\alpha_п, \\  \\gamma_п\\right]$"
      ],
      "text/plain": [
       "[\\alpha_п, \\gamma_п]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "alpha_p, gamma_p = symbols(\"\\\\alpha_п \\\\gamma_п\")\n",
    "print(\"Заданы обозначения пространственного угла атаки и крена:\")\n",
    "display([alpha_p, gamma_p])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\cos{\\left(\\alpha_п \\right)} & - \\sin{\\left(\\alpha_п \\right)} \\cos{\\left(\\gamma_п \\right)} & - \\sin{\\left(\\alpha_п \\right)} \\sin{\\left(\\gamma_п \\right)}\\\\\\sin{\\left(\\alpha_п \\right)} & \\cos{\\left(\\alpha_п \\right)} \\cos{\\left(\\gamma_п \\right)} & \\sin{\\left(\\gamma_п \\right)} \\cos{\\left(\\alpha_п \\right)}\\\\0 & - \\sin{\\left(\\gamma_п \\right)} & \\cos{\\left(\\gamma_п \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡cos(\\alpha_п)  -sin(\\alpha_п)⋅cos(\\gamma_п)  -sin(\\alpha_п)⋅sin(\\gamma_п)⎤\n",
       "⎢                                                                         ⎥\n",
       "⎢sin(\\alpha_п)  cos(\\alpha_п)⋅cos(\\gamma_п)   sin(\\gamma_п)⋅cos(\\alpha_п) ⎥\n",
       "⎢                                                                         ⎥\n",
       "⎣      0               -sin(\\gamma_п)                cos(\\gamma_п)        ⎦"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# НЕ ПРОВЕРЕНА - нет источников\n",
    "# проверена по отдельным матрицам\n",
    "MatBody2Spatial = rot_axis3(-alpha_p) * rot_axis1(gamma_p)\n",
    "MatBody2Spatial"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Связанная 🡆 скоростная / Body 🡆 wind ref.system"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Заданы обозначения пространственного углов атаки и спольжения (angle of attack and sideslip angle symbols introduced):\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\alpha_п, \\  \\gamma_п\\right]$"
      ],
      "text/plain": [
       "[\\alpha_п, \\gamma_п]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "alpha, beta = symbols(\"\\\\alpha \\\\beta\")\n",
    "print(\"Заданы обозначения пространственного углов атаки и спольжения (angle of attack and sideslip angle symbols introduced):\")\n",
    "display([alpha_p, gamma_p])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\cos{\\left(\\alpha \\right)} \\cos{\\left(\\beta \\right)} & - \\sin{\\left(\\alpha \\right)} \\cos{\\left(\\beta \\right)} & \\sin{\\left(\\beta \\right)}\\\\\\sin{\\left(\\alpha \\right)} & \\cos{\\left(\\alpha \\right)} & 0\\\\- \\sin{\\left(\\beta \\right)} \\cos{\\left(\\alpha \\right)} & \\sin{\\left(\\alpha \\right)} \\sin{\\left(\\beta \\right)} & \\cos{\\left(\\beta \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡cos(\\alpha)⋅cos(\\beta)   -sin(\\alpha)⋅cos(\\beta)  sin(\\beta)⎤\n",
       "⎢                                                            ⎥\n",
       "⎢      sin(\\alpha)              cos(\\alpha)            0     ⎥\n",
       "⎢                                                            ⎥\n",
       "⎣-sin(\\beta)⋅cos(\\alpha)  sin(\\alpha)⋅sin(\\beta)   cos(\\beta)⎦"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# проверена по ГОСТ\n",
    "MatBodyToWind = rot_axis2(-beta) * rot_axis3(-alpha)\n",
    "MatBodyToWind"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Геоцентрическая 🡆 нормальная СК / ECEF 🡆 Earth ref.system"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Заданы обозначения пространственного широты, долготы, азимута (latitude, longitude and azimuth introduced):\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\varphi, \\  \\lambda, \\  \\widehat{\\mathbf{Az}}\\right]$"
      ],
      "text/plain": [
       "[\\varphi, \\lambda, \\widehat{\\mathbf{Az}}]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lat, lon, azimuth = symbols('\\\\varphi \\\\lambda \\\\widehat{\\\\mathbf{Az}}')\n",
    "print(\"Заданы обозначения пространственного широты, долготы, азимута (latitude, longitude and azimuth introduced):\")\n",
    "display([lat, lon, azimuth])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\sin{\\left(\\lambda \\right)} \\sin{\\left(\\widehat{\\mathbf{Az}} \\right)} - \\sin{\\left(\\varphi \\right)} \\cos{\\left(\\lambda \\right)} \\cos{\\left(\\widehat{\\mathbf{Az}} \\right)} & \\cos{\\left(\\varphi \\right)} \\cos{\\left(\\widehat{\\mathbf{Az}} \\right)} & \\sin{\\left(\\lambda \\right)} \\sin{\\left(\\varphi \\right)} \\cos{\\left(\\widehat{\\mathbf{Az}} \\right)} - \\sin{\\left(\\widehat{\\mathbf{Az}} \\right)} \\cos{\\left(\\lambda \\right)}\\\\\\sin{\\left(\\lambda \\right)} \\cos{\\left(\\widehat{\\mathbf{Az}} \\right)} - \\sin{\\left(\\varphi \\right)} \\sin{\\left(\\widehat{\\mathbf{Az}} \\right)} \\cos{\\left(\\lambda \\right)} & \\sin{\\left(\\widehat{\\mathbf{Az}} \\right)} \\cos{\\left(\\varphi \\right)} & \\sin{\\left(\\lambda \\right)} \\sin{\\left(\\varphi \\right)} \\sin{\\left(\\widehat{\\mathbf{Az}} \\right)} + \\cos{\\left(\\lambda \\right)} \\cos{\\left(\\widehat{\\mathbf{Az}} \\right)}\\\\\\cos{\\left(\\lambda \\right)} \\cos{\\left(\\varphi \\right)} & \\sin{\\left(\\varphi \\right)} & - \\sin{\\left(\\lambda \\right)} \\cos{\\left(\\varphi \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-sin(\\lambda)⋅sin(\\widehat{\\mathbf{Az}}) - sin(\\varphi)⋅cos(\\lambda)⋅cos(\\wid\n",
       "⎢                                                                             \n",
       "⎢sin(\\lambda)⋅cos(\\widehat{\\mathbf{Az}}) - sin(\\varphi)⋅sin(\\widehat{\\mathbf{A\n",
       "⎢                                                                             \n",
       "⎣                                   cos(\\lambda)⋅cos(\\varphi)                 \n",
       "\n",
       "ehat{\\mathbf{Az}})  cos(\\varphi)⋅cos(\\widehat{\\mathbf{Az}})  sin(\\lambda)⋅sin(\n",
       "                                                                              \n",
       "z}})⋅cos(\\lambda)   sin(\\widehat{\\mathbf{Az}})⋅cos(\\varphi)  sin(\\lambda)⋅sin(\n",
       "                                                                              \n",
       "                                 sin(\\varphi)                                 \n",
       "\n",
       "\\varphi)⋅cos(\\widehat{\\mathbf{Az}}) - sin(\\widehat{\\mathbf{Az}})⋅cos(\\lambda)⎤\n",
       "                                                                             ⎥\n",
       "\\varphi)⋅sin(\\widehat{\\mathbf{Az}}) + cos(\\lambda)⋅cos(\\widehat{\\mathbf{Az}})⎥\n",
       "                                                                             ⎥\n",
       "                 -sin(\\lambda)⋅cos(\\varphi)                                  ⎦"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "MatGeocentric2Norm = rot_axis3(pi/2 - azimuth) * rot_axis1(-lat) * rot_axis2(pi/2 + lon)\n",
    "MatGeocentric2Norm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## //TODO Кинематические уравнения Эйлера"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## //TODO Углы из матриц"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\operatorname{atan_{2}}{\\left(\\sin{\\left(\\gamma \\right)} \\sin{\\left(\\vartheta \\right)} \\cos{\\left(\\psi \\right)} + \\sin{\\left(\\psi \\right)} \\cos{\\left(\\gamma \\right)},- \\sin{\\left(\\gamma \\right)} \\cos{\\left(\\vartheta \\right)} \\right)}$"
      ],
      "text/plain": [
       "atan2(sin(\\gamma)⋅sin(\\vartheta)⋅cos(\\psi) + sin(\\psi)⋅cos(\\gamma), -sin(\\gamm\n",
       "a)⋅cos(\\vartheta))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\operatorname{acos}{\\left(- \\sin{\\left(\\gamma \\right)} \\sin{\\left(\\psi \\right)} \\sin{\\left(\\vartheta \\right)} + \\cos{\\left(\\gamma \\right)} \\cos{\\left(\\psi \\right)} \\right)}$"
      ],
      "text/plain": [
       "acos(-sin(\\gamma)⋅sin(\\psi)⋅sin(\\vartheta) + cos(\\gamma)⋅cos(\\psi))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\operatorname{atan_{2}}{\\left(- \\sin{\\left(\\psi \\right)} \\cos{\\left(\\vartheta \\right)},\\sin{\\left(\\gamma \\right)} \\cos{\\left(\\psi \\right)} + \\sin{\\left(\\psi \\right)} \\sin{\\left(\\vartheta \\right)} \\cos{\\left(\\gamma \\right)} \\right)}$"
      ],
      "text/plain": [
       "-atan2(-sin(\\psi)⋅cos(\\vartheta), sin(\\gamma)⋅cos(\\psi) + sin(\\psi)⋅sin(\\varth\n",
       "eta)⋅cos(\\gamma))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# https://en.wikipedia.org/wiki/Rotation_formalisms_in_three_dimensions#Conversion_formulae_between_formalisms\n",
    "# TODO проверить численным расчётом исходных углов обратно\n",
    "display(atan2(MatNorm2Body[2,0],MatNorm2Body[2,1]))\n",
    "display(acos(MatNorm2Body[2,2]))\n",
    "display(-atan2(MatNorm2Body[0,2],MatNorm2Body[1,2]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\operatorname{atan_{2}}{\\left(\\sin{\\left(\\gamma \\right)} \\sin{\\left(\\psi \\right)} - \\sin{\\left(\\vartheta \\right)} \\cos{\\left(\\gamma \\right)} \\cos{\\left(\\psi \\right)},\\cos{\\left(\\psi \\right)} \\cos{\\left(\\vartheta \\right)} \\right)}$"
      ],
      "text/plain": [
       "atan2(sin(\\gamma)⋅sin(\\psi) - sin(\\vartheta)⋅cos(\\gamma)⋅cos(\\psi), cos(\\psi)⋅\n",
       "cos(\\vartheta))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\operatorname{atan_{2}}{\\left(- \\sin{\\left(\\gamma \\right)} \\sin{\\left(\\vartheta \\right)} \\cos{\\left(\\psi \\right)} - \\sin{\\left(\\psi \\right)} \\cos{\\left(\\gamma \\right)},\\sqrt{\\left(- \\sin{\\left(\\gamma \\right)} \\sin{\\left(\\psi \\right)} \\sin{\\left(\\vartheta \\right)} + \\cos{\\left(\\gamma \\right)} \\cos{\\left(\\psi \\right)}\\right)^{2} + \\sin^{2}{\\left(\\gamma \\right)} \\cos^{2}{\\left(\\vartheta \\right)}} \\right)}$"
      ],
      "text/plain": [
       "     ⎛                                                                  ______\n",
       "     ⎜                                                                 ╱      \n",
       "atan2⎝-sin(\\gamma)⋅sin(\\vartheta)⋅cos(\\psi) - sin(\\psi)⋅cos(\\gamma), ╲╱  (-sin\n",
       "\n",
       "______________________________________________________________________________\n",
       "                                                          2      2            \n",
       "(\\gamma)⋅sin(\\psi)⋅sin(\\vartheta) + cos(\\gamma)⋅cos(\\psi))  + sin (\\gamma)⋅cos\n",
       "\n",
       "_____________⎞\n",
       "2            ⎟\n",
       " (\\vartheta) ⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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      "text/latex": [
       "$\\displaystyle \\operatorname{atan_{2}}{\\left(- \\sin{\\left(\\gamma \\right)} \\cos{\\left(\\vartheta \\right)},- \\sin{\\left(\\gamma \\right)} \\sin{\\left(\\psi \\right)} \\sin{\\left(\\vartheta \\right)} + \\cos{\\left(\\gamma \\right)} \\cos{\\left(\\psi \\right)} \\right)}$"
      ],
      "text/plain": [
       "atan2(-sin(\\gamma)⋅cos(\\vartheta), -sin(\\gamma)⋅sin(\\psi)⋅sin(\\vartheta) + cos\n",
       "(\\gamma)⋅cos(\\psi))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# https://stackoverflow.com/a/37558238/12691808\n",
    "# TODO проверить численным расчётом исходных углов обратно\n",
    "display( atan2(MatNorm2Body[1,0], MatNorm2Body[0,0]) ) # yaw\n",
    "display( atan2(-MatNorm2Body[2,0], sqrt( MatNorm2Body[2,1]**2+MatNorm2Body[2,2]**2 ) ) ) # pitch\n",
    "display( atan2(MatNorm2Body[2,1], MatNorm2Body[2,2]) ) # roll"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}