Symbolic GA.ipynb

{
 "metadata": {
  "name": "",
  "signature": "sha256:5d684f5c68331280ed850f18ea93218968a44f5fe55d19a679ae51c4a518c86f"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "from GA.printer import  Format\n",
      "from GA.ga import Ga\n",
      "from sympy import symbols,solve\n",
      "Format(ipy=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "ga,a ,b ,c  = Ga.build('a b c')\n",
      "alpha,beta = [ga.mv(k,'scalar') for k in ['alpha','beta']]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Cramers rule"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "c = alpha*a+ beta*b\n",
      "c"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "\\begin{equation*} \\alpha  \\boldsymbol{a} + \\beta  \\boldsymbol{b} \\end{equation*}"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 4,
       "text": [
        "<GA.mv.Mv at 0xe304910>"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(a^b).inv()*(c^b)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "\\begin{equation*} \\alpha  \\end{equation*}"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": [
        "<GA.mv.Mv at 0xd0dbe50>"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(a^b).inv()*(a^c)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "\\begin{equation*} \\beta  \\end{equation*}"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "<GA.mv.Mv at 0xe32b290>"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "\n",
      " ** ?is it possible for this  to work ? ** "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# solve( alpha*a+ beta*b -c, alpha)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 20
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "\n",
      "Comparison between Coordinate and Coordinate free expressions"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "With  A Basis"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Create a ga from an 3D orthonormal basis"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(ga,e0,e1,e2) = Ga.build('e_0 e_1 e_2',g=[1,1,1])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "\n",
      "Then, create three vectors in this ga, these are expressed in terms of the previously defined basis "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a,b,c = [ga.mv(k,'vector') for k in ['a','b','c']]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If we evaluate the following expression, it will be expanded in basis. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a|(b*c)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "\\begin{equation*} \\left ( - a^{1} b^{0} c^{1} + a^{1} b^{1} c^{0} - a^{2} b^{0} c^{2} + a^{2} b^{2} c^{0}\\right ) \\boldsymbol{e_{0}} + \\left ( a^{0} b^{0} c^{1} - a^{0} b^{1} c^{0} - a^{2} b^{1} c^{2} + a^{2} b^{2} c^{1}\\right ) \\boldsymbol{e_{1}} + \\left ( a^{0} b^{0} c^{2} - a^{0} b^{2} c^{0} + a^{1} b^{1} c^{2} - a^{1} b^{2} c^{1}\\right ) \\boldsymbol{e_{2}} \\end{equation*}"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": [
        "<GA.mv.Mv at 0xe32b1d0>"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Without a Basis"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If instead we simply define the geometric algebra with three arbitrary vectors"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "ga,a ,b ,c  = Ga.build('a b c')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now, if we evaluate the same expression, we see that it is simpler "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(a|(b*c))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "\\begin{equation*} - \\left ( a\\cdot c\\right )  \\boldsymbol{b} + \\left ( a\\cdot b\\right )  \\boldsymbol{c} \\end{equation*}"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "<GA.mv.Mv at 0xe32b310>"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Projetive Geometry"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Given a ga of n-dimensions (Gn),  we can extend the algebra by one dimensions (Gn->Gn+1), and setup a projective map to relate them. "
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Projective Plane"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For the projective plane example we extend the plane by another dimension represented by the vector n, then we can define a map between the plane and 3d space through the up/down methods\n",
      "\n",
      "** ? can i implement projective/conformal  geometry using a submanifold ?**\n",
      "I would like also to map the projective split so i can re-use the same symbols. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(ga,e0,e1,n) = Ga.build('e_0 e_1 n',g=[1,1,1])\n",
      "\n",
      "\n",
      "def down(a, n=n):\n",
      "    return n* ((n^a)*(n|a).inv())\n",
      "\n",
      "def up(a,n=n):\n",
      "    return n+a"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a,b,c = [ga.mv(k,'vector') for k in ['a','b','c']]\n",
      "\n",
      "a = a.proj([e0,e1])\n",
      "a"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "\\begin{equation*} a^{0} \\boldsymbol{e_{0}} + a^{1} \\boldsymbol{e_{1}} \\end{equation*}"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 14,
       "text": [
        "<GA.mv.Mv at 0xe4bbb50>"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# is it possible to get a sub algebra "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "up(a)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "\\begin{equation*} a^{0} \\boldsymbol{e_{0}} + a^{1} \\boldsymbol{e_{1}} + \\boldsymbol{n} \\end{equation*}"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 16,
       "text": [
        "<GA.mv.Mv at 0xe3fe790>"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Projecting the vector b 'down' it can be seen that we are using homogeneous coordinates"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "down(b)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "\\begin{equation*} \\frac{b^{0}}{b^{}} \\boldsymbol{e_{0}} + \\frac{b^{1}}{b^{}} \\boldsymbol{e_{1}} \\end{equation*}"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 17,
       "text": [
        "<GA.mv.Mv at 0xe3fe290>"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "n*down(b)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "\\begin{equation*} - \\frac{b^{0}}{b^{}} \\boldsymbol{e_{0}\\wedge n} - \\frac{b^{1}}{b^{}} \\boldsymbol{e_{1}\\wedge n} \\end{equation*}"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 18,
       "text": [
        "<GA.mv.Mv at 0xe4bb190>"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": []
    }
   ],
   "metadata": {}
  }
 ]
}