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  {
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   "source": [
    "# Defining a Piece of a Bezier Curve"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6e4d6cfc",
   "metadata": {},
   "source": [
    "A piece of a Bezier curve is a cubic plane curve, defined by $(x(t), y(t))$ two polynomials of degree 3 in one parameter $t$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "source": [
    "\"\"\"\n",
    "On input are two arrays x and y,\n",
    "with the coordinates of the four points\n",
    "that define a piece of a Bezier curve.\n",
    "On return are the coefficients of the\n",
    "two cubics that define the piece.\n",
    "\"\"\"\n",
    "function Bezier(x,y)\n",
    "    bx = 3*(x[2]-x[1])\n",
    "    by = 3*(y[2]-y[1])\n",
    "    cx = 3*(x[3]-x[2]) - bx\n",
    "    cy = 3*(y[3]-y[2]) - by\n",
    "    dx = x[4] - x[1] - bx - cx\n",
    "    dy = y[4] - y[1] - by - cy\n",
    "    xt = [x[1], bx, cx, dx]\n",
    "    yt = [y[1], by, cy, dy]\n",
    "    return (xt, yt)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e7a03f14",
   "metadata": {},
   "source": [
    "We will define one piece that starts at (1,1) with control (1,3), ends at (2,2), with control (3,3). The four x-values are 1, 1, 3, 2. The four y-values are 1, 3, 3, 2. The coordinates of the control points are the middle values. The function piece takes two arrays of 4 numbers on input and returns the two arrays, with the four coefficients of x(t) and y(t) of the piece of the Bezier curve as specified by the input."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "574a84af",
   "metadata": {},
   "outputs": [],
   "source": [
    "a = [1, 1, 3, 2]; b = [1, 3, 3, 2];\n",
    "cffx, cffy = Bezier(a,b)\n",
    "println(\"coefficients for x(t) \", cffx)\n",
    "println(\"coefficients for y(t) \", cffy)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e9c3bc44",
   "metadata": {},
   "outputs": [],
   "source": [
    "startpoint = (evalpoly(0, cffx), evalpoly(0,cffx))\n",
    "endpoint = (evalpoly(1,cffx), evalpoly(1,cffy))\n",
    "print(\"The curve starts at \", startpoint)\n",
    "println(\" and ends at \", endpoint, \".\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f590b38b",
   "metadata": {},
   "source": [
    "# Plotting a Plane Curve"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "36ecbe8d",
   "metadata": {},
   "source": [
    "To define the curve $(x(t), y(t))$, we copy the coefficients in ``cffx`` and ``cffy`` in the definition of the coordinate functions ``x(t)`` and ``y(t)``."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6583e3d7",
   "metadata": {},
   "outputs": [],
   "source": [
    "x(t) = 1 + 6*t^2 - 5*t^3\n",
    "y(t) = 1 + 6*t - 6*t^2 + t^3\n",
    "print(\"The curve starts at (\", x(0), \",\", y(0), \")\")\n",
    "print(\" and ends at (\", x(1), \",\", y(1), \")\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a2815c84",
   "metadata": {},
   "source": [
    "In making the plot, observe that this is different from plotting y as a function of x.  The plane curve is defined by two functions ``x(t)`` and ``y(t)``."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "50b327af",
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "634d0240",
   "metadata": {},
   "source": [
    "To plot, we have to define a range where we sample the functions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "23c2cfc2",
   "metadata": {},
   "outputs": [],
   "source": [
    "r = 0:0.01:1\n",
    "xr = [x(t) for t in r];\n",
    "yr = [y(t) for t in r];"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7487b8ae",
   "metadata": {},
   "source": [
    "We define the limits of the range for x and y from 0 to 3.  To see the same spacings in both directions, we set the aspect ratio to one.  The legend is removed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e9821346",
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "plot(xr,yr,xlims=(0,3),ylims=(0,3), aspect_ratio=1, leg=false)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8d28fae2",
   "metadata": {},
   "outputs": [],
   "source": [
    "# savefig(\"bezierplot.png\")"
   ]
  }
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