MinimalAlgebraForDirectionalScaling.ipynb

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    "## Directional Scaling"
   ]
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    " The code below is a dumb trial-and-error way to determine a mapping which implements directional scaling through spinors. This is done by \n",
    " \n",
    " 1. create a algebra and a representation (higher) algebra \n",
    " 1. guess a map  `up()` and map `down()` \n",
    " 2. Take a orthonormal frame  map it up\n",
    " 3. take each bivectors in repr space and spin the frame \n",
    " 4. map the resultant frame down and convert frame into matrices\n",
    " 5. repeat for all bivectors\n",
    " \n",
    " \n",
    " \n",
    " So far this seems to work, \n",
    " \n",
    " Given $G(n,0)$, create  $G(n,1)$ with added dimenion given by $e_0$\n",
    " \n",
    " \n",
    " $$\n",
    " \\text{up(x)} = e_0/x\\\\\n",
    " \\text{down}(X) = -e_0^2 /(X\\cdot e_0)\n",
    " $$\n",
    " \n",
    " a directional scaling of amount $r$ along $e_i$ is given by the  spinor, \n",
    " \n",
    " $$S = \\sqrt{r}e^{\\frac{\\text{arccosh}{(r)}}{2}\\quad e_i\\wedge e_0}$$ \n",
    " \n",
    " \n"
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      "generator= (1^e14) \n",
      " [[5. 0. 0.]\n",
      " [0. 1. 0.]\n",
      " [0. 0. 1.]]\n",
      "generator= (1^e24) \n",
      " [[1. 0. 0.]\n",
      " [0. 5. 0.]\n",
      " [0. 0. 1.]]\n",
      "generator= (1^e34) \n",
      " [[1. 0. 0.]\n",
      " [0. 1. 0.]\n",
      " [0. 0. 5.]]\n"
     ]
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   ],
   "source": [
    "from clifford import Cl\n",
    "from clifford.tools import mat2Frame, frame2Mat\n",
    "from pylab import * \n",
    "\n",
    "\n",
    "N = 3\n",
    "l,b  = Cl(N,1)\n",
    "locals().update(b)\n",
    "\n",
    "I_ = l.pseudoScalar   # projection space\n",
    "e_ = I_.basis()[-1]   # the projection vector\n",
    "I  = I_/e_            # the original space\n",
    "\n",
    "###### up/down projections  ################### \n",
    "up   = lambda x:  e_/x  \n",
    "down = lambda X: -(e_**2)/(X|e_)\n",
    " \n",
    "# sanity test of up/down \n",
    "x = I.layout.randomV()(I)\n",
    "assert(down(up(x)) ==x)\n",
    "\n",
    "def f2Mat(f,I):\n",
    "    '''\n",
    "    convert a geometric function to a  matrix. \n",
    "    b = f(a)  for a = ortho-normal frame of I.\n",
    "    return matrix version of b\n",
    "    '''\n",
    "    A = I.basis()\n",
    "    B = [f(x) for x in A]\n",
    "    mat, dum = frame2Mat(B=B, I=I)\n",
    "    return mat \n",
    " \n",
    "# sort bivectors by those in/out of I\n",
    "Bin  = [k for k in l.blades_of_grade(2) if k|I != 0]\n",
    "Bout = [k for k in l.blades_of_grade(2) if k|I == 0 ]\n",
    "\n",
    "for B in  Bout: # (we know what Bin do already)\n",
    "    ############ construct the spinor ###############\n",
    "    r =5\n",
    "    rho = 1/sqrt(r)\n",
    "    R = e**(arccosh(r)* B/2.)\n",
    "    S = rho*R\n",
    "    f = lambda x: down(S*up(x)*~S)\n",
    "    print('generator =',B,'\\n',f2Mat(f = f, I = I))"
   ]
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