{
 "cells": [
  {
   "cell_type": "markdown",
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   },
   "source": [
    "This notebook is part of the `clifford` documentation: https://clifford.readthedocs.io/."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Quick Start   (G2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This notebook gives a terse introduction to using the `clifford` module, using a two-dimensional geometric algebra  as the context."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, import clifford and instantiate a two-dimensional algebra (G2),"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import clifford as cf\n",
    "\n",
    "layout, blades = cf.Cl(2) # creates a 2-dimensional clifford algebra"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Inspect blades."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "blades "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assign blades to variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "e1 = blades['e1']\n",
    "e2 = blades['e2']\n",
    "e12 = blades['e12']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Basics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "e1*e2  # geometric product"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "e1|e2  # inner product "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "e1^e2  # outer product"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Reflection "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = e1+e2     # the vector\n",
    "n = e1        # the reflector\n",
    "-n*a*n.inv()  # reflect `a` in hyperplane normal to `n`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rotation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from math import e, pi\n",
    "\n",
    "R = e**(pi/4*e12)  # enacts rotation by pi/2 \n",
    "R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "R*e1*~R    # rotate e1 by pi/2 in the e12-plane"
   ]
  }
 ],
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