PlusOneSpace.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## vector functions of a scalar variable\n",
    "lets say we have a $N$-d vector-valued function of a scalar variable,  $f(t)= f_t$.\n",
    "we add a dimension, $e_0$, which we interpret as values of independent variable $t$. the function is now \n",
    "$r = f_t + te_0 $. \n",
    "Given a discrete set of $t$-values,  $r$ is   just a list of vectors. we take the `diff` of $r$ to compute the vector differential, $\\dot {r}$, which is also just a list of vectors. \n",
    "\n",
    "the derivative of $f_t$ with respect to $t$ is now computed by the split of $r$ with the new dimension, \n",
    "$V= \\frac{r\\wedge e_0}{r\\cdot e_0}$. change in velocity we guess is \n",
    "\n",
    "$A_1 = V_2/V_1$ and so on for many indecies , which is like a diff but with division between sequential velocities.\n",
    "\n",
    "There is also $\\Omega = \\dot{r}\\wedge \\ddot{r}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "MVArray([(1.0^e1), (0.6732^e0) + (1.45503^e1) - (0.62349^e2),\n",
       "         (1.3464^e0) + (1.56892^e1) - (0.97493^e2),\n",
       "         (2.0196^e0) + (1.58571^e1) - (0.90097^e2),\n",
       "         (2.69279^e0) + (1.79182^e1) - (0.43388^e2),\n",
       "         (3.36599^e0) + (2.39106^e1) + (0.22252^e2),\n",
       "         (4.03919^e0) + (3.4157^e1) + (0.78183^e2),\n",
       "         (4.71239^e0) + (4.71239^e1) + (1.0^e2),\n",
       "         (5.38559^e0) + (6.00908^e1) + (0.78183^e2),\n",
       "         (6.05879^e0) + (7.03371^e1) + (0.22252^e2),\n",
       "         (6.73198^e0) + (7.63295^e1) - (0.43388^e2),\n",
       "         (7.40518^e0) + (7.83907^e1) - (0.90097^e2),\n",
       "         (8.07838^e0) + (7.85586^e1) - (0.97493^e2),\n",
       "         (8.75158^e0) + (7.96975^e1) - (0.62349^e2),\n",
       "         (9.42478^e0) + (8.42478^e1)], dtype=object)"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from clifford import Cl,MVArray\n",
    "from clifford.tools import mat2Frame, frame2Mat\n",
    "from pylab import * \n",
    "\n",
    "\n",
    "\n",
    "l,b  = Cl(sig=[-1,1,1],firstIdx=0)\n",
    "locals().update(b)\n",
    "\n",
    "I_ = l.pseudoScalar\n",
    "I  = e12#I_/e0\n",
    "split = lambda x: (x^e0)/(x|e0)\n",
    "div = lambda x: [x[k+1]/x[k] for k in range(len(x)-1)] # like diff but division \n",
    "\n",
    "\n",
    "t = linspace(0,3*pi,15)\n",
    "f = t*e1+ e**(t*e12)*e1  # function  \n",
    "r = t*e0 + f\n",
    "r"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "MVArray([0.2799635198779851, 0.293391900633451, 0.2936481833541586,\n",
       "         0.25568389418796583, 0.166027328634858, 0.40672702619797746,\n",
       "         0.48356779786884124, 0.4067270261978369, 0.16602732863455358,\n",
       "         0.2556838941880583, 0.29364818335415194, 0.29339190063362774,\n",
       "         0.2799635198779439], dtype=object)"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rd  = diff(r)\n",
    "rdd = diff(rd)\n",
    "V = MVArray(list(map(split,rd)))\n",
    "A = MVArray(div(V))\n",
    "omega = MVArray([k^l for k,l in zip(rd,rdd)])\n",
    "abs(omega)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "MVArray([(0.6732^e0) + (0.45503^e1) - (0.62349^e2),\n",
       "         (0.6732^e0) + (0.11389^e1) - (0.35144^e2),\n",
       "         (0.6732^e0) + (0.01679^e1) + (0.07396^e2),\n",
       "         (0.6732^e0) + (0.20611^e1) + (0.46709^e2),\n",
       "         (0.6732^e0) + (0.59924^e1) + (0.6564^e2),\n",
       "         (0.6732^e0) + (1.02464^e1) + (0.55931^e2),\n",
       "         (0.6732^e0) + (1.29669^e1) + (0.21817^e2),\n",
       "         (0.6732^e0) + (1.29669^e1) - (0.21817^e2),\n",
       "         (0.6732^e0) + (1.02464^e1) - (0.55931^e2),\n",
       "         (0.6732^e0) + (0.59924^e1) - (0.6564^e2),\n",
       "         (0.6732^e0) + (0.20611^e1) - (0.46709^e2),\n",
       "         (0.6732^e0) + (0.01679^e1) - (0.07396^e2),\n",
       "         (0.6732^e0) + (0.11389^e1) + (0.35144^e2),\n",
       "         (0.6732^e0) + (0.45503^e1) + (0.62349^e2)], dtype=object)"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "MVArray([-(0.34114^e1) + (0.27205^e2), -(0.09709^e1) + (0.4254^e2),\n",
       "         (0.18932^e1) + (0.39313^e2), (0.39313^e1) + (0.18932^e2),\n",
       "         (0.4254^e1) - (0.09709^e2), (0.27205^e1) - (0.34114^e2),\n",
       "         -(0.43634^e2), -(0.27205^e1) - (0.34114^e2),\n",
       "         -(0.4254^e1) - (0.09709^e2), -(0.39313^e1) + (0.18932^e2),\n",
       "         -(0.18932^e1) + (0.39313^e2), (0.09709^e1) + (0.4254^e2),\n",
       "         (0.34114^e1) + (0.27205^e2)], dtype=object)"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rdd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "MVArray([(0.67592^e01) - (0.92616^e02), (0.16917^e01) - (0.52204^e02),\n",
       "         (0.02495^e01) + (0.10986^e02), (0.30617^e01) + (0.69383^e02),\n",
       "         (0.89014^e01) + (0.97505^e02), (1.52204^e01) + (0.83083^e02),\n",
       "         (1.92616^e01) + (0.32408^e02), (1.92616^e01) - (0.32408^e02),\n",
       "         (1.52204^e01) - (0.83083^e02), (0.89014^e01) - (0.97505^e02),\n",
       "         (0.30617^e01) - (0.69383^e02), (0.02495^e01) - (0.10986^e02),\n",
       "         (0.16917^e01) + (0.52204^e02), (0.67592^e01) + (0.92616^e02)],\n",
       "        dtype=object)"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "MVArray([0.45476 + (0.14922^e12), -0.17643 - (0.10496^e12),\n",
       "         6.60758 + (1.28649^e12), 1.65013 + (0.55477^e12),\n",
       "         1.24202 + (0.42713^e12), 1.06454 + (0.36817^e12),\n",
       "         0.94494 + (0.32724^e12), 0.83902 + (0.29017^e12),\n",
       "         0.71999 + (0.24761^e12), 0.54447 + (0.18305^e12),\n",
       "         0.14581 + (0.02839^e12), -4.18629 - (2.49045^e12),\n",
       "         1.98521 + (0.65143^e12)], dtype=object)"
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "MVArray([-(0.22966^e01) + (0.18314^e02) - (0.08891^e12),\n",
       "         -(0.06536^e01) + (0.28638^e02) + (0.01433^e12),\n",
       "         (0.12745^e01) + (0.26465^e02) - (0.0074^e12),\n",
       "         (0.26465^e01) + (0.12745^e02) - (0.1446^e12),\n",
       "         (0.28638^e01) - (0.06536^e02) - (0.33742^e12),\n",
       "         (0.18314^e01) - (0.22966^e02) - (0.50171^e12),\n",
       "         -(0.29374^e02) - (0.56579^e12),\n",
       "         -(0.18314^e01) - (0.22966^e02) - (0.50171^e12),\n",
       "         -(0.28638^e01) - (0.06536^e02) - (0.33742^e12),\n",
       "         -(0.26465^e01) + (0.12745^e02) - (0.1446^e12),\n",
       "         -(0.12745^e01) + (0.26465^e02) - (0.0074^e12),\n",
       "         (0.06536^e01) + (0.28638^e02) + (0.01433^e12),\n",
       "         (0.22966^e01) + (0.18314^e02) - (0.08891^e12)], dtype=object)"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "omega"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "MVArray([(0.18871^e01) + (0.23664^e02), -(0.04754^e01) - (0.01085^e02),\n",
       "         (0.0227^e01) - (0.01093^e02), (0.21359^e01) - (0.44353^e02),\n",
       "         -(0.25561^e01) - (1.11988^e02), -(1.33536^e01) - (1.06492^e02),\n",
       "         -(1.92616^e01), -(1.33536^e01) + (1.06492^e02),\n",
       "         -(0.25561^e01) + (1.11988^e02), (0.21359^e01) + (0.44353^e02),\n",
       "         (0.0227^e01) + (0.01093^e02), -(0.04754^e01) + (0.01085^e02),\n",
       "         (0.18871^e01) - (0.23664^e02)], dtype=object)"
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "split(omega)`-lqz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.7.3"
  },
  "toc": {
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": false,
   "toc_cell": false,
   "toc_position": {},
   "toc_section_display": "block",
   "toc_window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}