solve G problem

G.ipynb

@@ -0,0 +1,183 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "16.97\n"
+     ]
+    }
+   ],
+   "source": [
+    "import math\n",
+    "\n",
+    "# 计算两点之间的欧几里得距离\n",
+    "def distance(p1, p2):\n",
+    "    return math.sqrt((p2[0] - p1[0])**2 + (p2[1] - p1[1])**2 + (p2[2] - p1[2])**2)\n",
+    "\n",
+    "# 计算最短距离\n",
+    "def calculate_shortest_distance(A, B, C, R):\n",
+    "    dAC = distance(A, C)\n",
+    "    dBC = distance(B, C)\n",
+    "    dAB = distance(A, B)\n",
+    "    \n",
+    "    # 如果A到C或B到C的距离小于等于球体半径，说明A或B在球体内\n",
+    "    if dAC <= R or dBC <= R:\n",
+    "        return -1  # 特殊情况处理（可以根据题意调整）\n",
+    "    \n",
+    "    # 如果AB的直线不穿过球体，直接返回欧几里得距离\n",
+    "    return dAB\n",
+    "\n",
+    "# 读取输入\n",
+    "A = list(map(float, input().split()))\n",
+    "B = list(map(float, input().split()))\n",
+    "C = list(map(float, input().split()))\n",
+    "R = float(input())\n",
+    "\n",
+    "# 计算并输出最短距离，保留两位小数\n",
+    "result = calculate_shortest_distance(A, B, C, R)\n",
+    "print(f\"{result:.2f}\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "19.71\n"
+     ]
+    }
+   ],
+   "source": [
+    "import math\n",
+    "\n",
+    "def read_point():\n",
+    "    return list(map(int, input().strip().split()))\n",
+    "\n",
+    "def distance(p1, p2):\n",
+    "    return math.hypot(p1[0]-p2[0], p1[1]-p2[1])\n",
+    "\n",
+    "def distance_point_to_line(A, B, C):\n",
+    "    # A, B, C are tuples (x, y)\n",
+    "    cross = abs( (B[0]-A[0])*(A[1]-C[1]) - (B[1]-A[1])*(A[0]-C[0]) )\n",
+    "    dist = cross / distance(A, B)\n",
+    "    return dist\n",
+    "\n",
+    "def tangent_points(P, C, R):\n",
+    "    # P and C are tuples (x, y)\n",
+    "    dx = P[0] - C[0]\n",
+    "    dy = P[1] - C[1]\n",
+    "    dist = math.hypot(dx, dy)\n",
+    "    if dist < R:\n",
+    "        return []  # No tangent\n",
+    "    elif dist == R:\n",
+    "        return [P]  # One tangent point (the point itself)\n",
+    "    else:\n",
+    "        angle_PC = math.atan2(dy, dx)\n",
+    "        alpha = math.acos(R / dist)\n",
+    "        t1 = angle_PC + alpha\n",
+    "        t2 = angle_PC - alpha\n",
+    "        tp1 = (C[0] + R * math.cos(t1), C[1] + R * math.sin(t1))\n",
+    "        tp2 = (C[0] + R * math.cos(t2), C[1] + R * math.sin(t2))\n",
+    "        return [tp1, tp2]\n",
+    "\n",
+    "def angle_between(C, P1, P2):\n",
+    "    # C is center, P1 and P2 are points on circle\n",
+    "    v1x = P1[0] - C[0]\n",
+    "    v1y = P1[1] - C[1]\n",
+    "    v2x = P2[0] - C[0]\n",
+    "    v2y = P2[1] - C[1]\n",
+    "    dot = v1x * v2x + v1y * v2y\n",
+    "    mag1 = math.hypot(v1x, v1y)\n",
+    "    mag2 = math.hypot(v2x, v2y)\n",
+    "    if mag1 ==0 or mag2==0:\n",
+    "        return 0\n",
+    "    cos_theta = dot / (mag1 * mag2)\n",
+    "    # Clamp due to floating point\n",
+    "    cos_theta = max(min(cos_theta,1), -1)\n",
+    "    theta = math.acos(cos_theta)\n",
+    "    return theta\n",
+    "\n",
+    "def compute_path(A, B, C, R):\n",
+    "    # A, B, C are tuples (x, y)\n",
+    "    dist_AB = distance(A, B)\n",
+    "    dist_to_line = distance_point_to_line(A, B, C)\n",
+    "    if dist_to_line >= R:\n",
+    "        return dist_AB\n",
+    "    # Compute tangent points\n",
+    "    tangents_A = tangent_points(A, C, R)\n",
+    "    tangents_B = tangent_points(B, C, R)\n",
+    "    if not tangents_A or not tangents_B:\n",
+    "        # No possible path\n",
+    "        return None\n",
+    "    min_path = float('inf')\n",
+    "    for ta in tangents_A:\n",
+    "        for tb in tangents_B:\n",
+    "            # Compute angles for arc\n",
+    "            angle = angle_between(C, ta, tb)\n",
+    "            # Two possible arcs, choose the smaller one\n",
+    "            arc = min(angle, 2*math.pi - angle) * R\n",
+    "            path = distance(A, ta) + arc + distance(B, tb)\n",
+    "            if path < min_path:\n",
+    "                min_path = path\n",
+    "    return min_path\n",
+    "\n",
+    "def main():\n",
+    "    # Read input\n",
+    "    A3D = read_point()\n",
+    "    B3D = read_point()\n",
+    "    C3D = read_point()\n",
+    "    R = int(input())\n",
+    "    # Project to 2D (x and z)\n",
+    "    A = (A3D[0], A3D[2])\n",
+    "    B = (B3D[0], B3D[2])\n",
+    "    C = (C3D[0], C3D[2])\n",
+    "    path_length = compute_path(A, B, C, R)\n",
+    "    if path_length is None:\n",
+    "        print(\"No valid path\")\n",
+    "    else:\n",
+    "        print(\"{0:.2f}\".format(path_length))\n",
+    "\n",
+    "if __name__ == \"__main__\":\n",
+    "    main()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "base",
+   "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": "3.11.5"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}