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   "source": [
    "# Circle Segment Intersection (for interpolation)\n",
    "Here is an interactive plot that demonstrates the functionality of the formula to calculate the intersection of a line segment, and a circle centered at the origin."
   ]
  },
  {
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   "id": "d31dc723-a6dc-400d-8b31-fe84ea6d5e45",
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       "VBox(children=(Label(value='A and B can be moved with the mouse. One must be inside the circle, and one must b…"
      ]
     },
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     "output_type": "display_data"
    }
   ],
   "source": [
    "from bqplot import *\n",
    "import numpy as np\n",
    "import ipywidgets as widgets\n",
    "\n",
    "\n",
    "def circle_segment_intersection(p1, p2, r):\n",
    "    x1, y1 = p1\n",
    "    x2, y2 = p2\n",
    "    dx = x2 - x1\n",
    "    dy = y2 - y1\n",
    "    dr2 = dx ** 2 + dy ** 2\n",
    "    D = x1 * y2 - x2 * y1\n",
    "    d1 = x1 ** 2 + y1 ** 2\n",
    "    d2 = x2 ** 2 + y2 ** 2\n",
    "    dd = d2 - d1\n",
    "    sqrt_term = np.sqrt(r ** 2 * dr2 - D ** 2)\n",
    "    x = (D * dy + np.copysign(1.0, dd) * dx * sqrt_term) / dr2\n",
    "    y = (-D * dx + np.copysign(1.0, dd) * dy * sqrt_term) / dr2\n",
    "    return x, y\n",
    "\n",
    "\n",
    "MAX = 5.0\n",
    "x_sc = LinearScale(min=-MAX, max=MAX)\n",
    "y_sc = LinearScale(min=-MAX, max=MAX)\n",
    "\n",
    "ax_x = Axis(label=\"x\", scale=x_sc, tick_format=\"0.0f\")\n",
    "ax_y = Axis(label=\"y\", scale=y_sc, orientation=\"vertical\", tick_format=\"0.0f\")\n",
    "\n",
    "points = Scatter(\n",
    "    names=[\"A\", \"B\"], x=[0.0, 3.0], y=[2.0, 4.0], scales={\"x\": x_sc, \"y\": y_sc}, enable_move=True\n",
    ")\n",
    "\n",
    "\n",
    "def get_circle(r):\n",
    "    t = np.linspace(0, 2 * np.pi)\n",
    "    x = r * np.cos(t)\n",
    "    y = r * np.sin(t)\n",
    "    return x, y\n",
    "\n",
    "radius_slider = widgets.FloatSlider(min=0.0, max=MAX, value=3.0, description=\"Circle radius\")\n",
    "circle_x, circle_y = get_circle(radius_slider.value)\n",
    "\n",
    "circle = Lines(x=circle_x, y=circle_y, scales={\"x\": x_sc, \"y\": y_sc}, colors=[\"green\"])\n",
    "\n",
    "x1, x2 = points.x\n",
    "y1, y2 = points.y\n",
    "xi, yi = circle_segment_intersection((x1, y1), (x2, y2), radius_slider.value)\n",
    "\n",
    "intersection = Scatter(\n",
    "    names=[\"C\"],\n",
    "    x=[xi],\n",
    "    y=[yi],\n",
    "    scales={\"x\": x_sc, \"y\": y_sc},\n",
    "    enable_move=False,\n",
    "    colors=[\"purple\"],\n",
    ")\n",
    "\n",
    "fig = Figure(axes=[ax_x, ax_y], marks=[circle, points, intersection])\n",
    "\n",
    "fig.max_aspect_ratio = 1\n",
    "fig.min_aspect_ratio = 1\n",
    "\n",
    "\n",
    "def both_inside_or_both_outside_circle(points, r):\n",
    "    x1, x2 = points.x\n",
    "    y1, y2 = points.y\n",
    "    d1 = x1 ** 2 + y1 ** 2\n",
    "    d2 = x2 ** 2 + y2 ** 2\n",
    "    if d1 < r ** 2 and d2 < r ** 2:\n",
    "        return True\n",
    "    elif d1 > r ** 2 and d2 > r ** 2:\n",
    "        return True\n",
    "    else:\n",
    "        return False\n",
    "\n",
    "\n",
    "def update_circle(message):\n",
    "    circle_x, circle_y = get_circle(radius_slider.value)\n",
    "    circle.x = circle_x\n",
    "    circle.y = circle_y\n",
    "    update_intersection(message)\n",
    "\n",
    "\n",
    "def update_intersection(message):\n",
    "    x1, x2 = points.x\n",
    "    y1, y2 = points.y\n",
    "    r = radius_slider.value\n",
    "    if both_inside_or_both_outside_circle(points, r):\n",
    "        circle.colors = [\"red\"]\n",
    "        intersection.x = []\n",
    "        intersection.y = []\n",
    "    else:\n",
    "        circle.colors = [\"green\"]\n",
    "        xi, yi = circle_segment_intersection((x1, y1), (x2, y2), r)\n",
    "        intersection.x = [xi]\n",
    "        intersection.y = [yi]\n",
    "\n",
    "\n",
    "points.observe(update_intersection, [\"x\", \"y\"])\n",
    "\n",
    "radius_slider.observe(update_circle, \"value\")\n",
    "\n",
    "widgets.VBox(\n",
    "    [\n",
    "        widgets.Label(\n",
    "            \"A and B can be moved with the mouse. One must be inside the circle, and one must be outside.\",\n",
    "            fixed=True,\n",
    "        ),\n",
    "        radius_slider,\n",
    "        fig,\n",
    "    ]\n",
    ")"
   ]
  }
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