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 #include "bits/stdc++.h"
class Point { public: double x, y, z;
public: Point() = default;
Point(double out_x, double out_y, double out_z) : x(out_x), y(out_y), z(out_z) {};
Point operator (const Point& p) const { return Point(this>x  p.x, this>y  p.y, this>z  p.z); }
Point operator +(const Point& p) const { return Point(this>x + p.x, this>y + p.y, this>z + p.z); }
Point operator *(const Point& p) const { return Point(this>y * p.z  this>z * p.y, this>z * p.x  this>x * p.z, this>x * p.y  this>y * p.x); }
Point operator /(const double& d) const { return Point(this>x / d, this>y / d, this>z / d); }
double operator ^(const Point& p) const { return (this>x * p.x + this>y * p.y + this>z * p.z); } };
class ConvexHull3D { private: struct Surface { int a, b, c; bool isBelongToConvexHull;
Surface(int out_a, int out_b, int out_c, bool outIsBelongToConvexHull) : a(out_a), b(out_b), c(out_c), isBelongToConvexHull(outIsBelongToConvexHull) {}; }; std::vector<Point> points; std::map<std::pair<double, double>, int> edgeToSurface; std::vector<Surface> surfaces;
const double EPS = 1e8; private: static double vectorLength(const Point &p) { return sqrt(pow(p.x, 2) + pow(p.y, 2) + pow(p.z, 2)); }
static double triangleArea(const Point& a, const Point& b, const Point& c) { return vectorLength((b  a) * (c  a) / 2); }
static double tetrahedronVolume(const Point& a, const Point& b, const Point& c, const Point& d) { return (b  a) * (c  a) ^ (d  a) / 6; }
bool isPointSurfaceSameDirection(const Point &point, const Surface &surface) { return tetrahedronVolume(point, points[surface.a], points[surface.b], points[surface.c]) > EPS; }
public: double Area() { double res = 0; for(const Surface& surface : surfaces) { res += triangleArea(points[surface.a], points[surface.b], points[surface.c]); } return res; }
double Volume() { double res = 0; for(const Surface& surface : surfaces) { res += tetrahedronVolume(Point(0, 0, 0), points[surface.a], points[surface.b], points[surface.c]); } return res; }
int TriangleNumber() { return static_cast<int>(surfaces.size()); }
int PolygonNumber() { auto isSame = [this](const Surface& a, const Surface& b) { return fabs(tetrahedronVolume(points[a.a], points[a.b], points[a.c], points[b.a])) < EPS && fabs(tetrahedronVolume(points[a.a], points[a.b], points[a.c], points[b.b])) < EPS && fabs(tetrahedronVolume(points[a.a], points[a.b], points[a.c], points[b.c])) < EPS; }; int res = 0; for(int i = 0; i < surfaces.size(); i++) { bool flag = true; for(int j = 0; j < i; j++) { if(isSame(surfaces[i], surfaces[j])) { flag = false; break; } } res += flag; } return res; }
private: void updateConvexHull(const int& pointIndex, const std::pair<int, int>& edge) { Surface surface = surfaces[edgeToSurface[edge]]; if(surface.isBelongToConvexHull) { if(isPointSurfaceSameDirection(points[pointIndex], surface)) { dfs(pointIndex, edgeToSurface[edge]); } else { edgeToSurface[std::pair<int, int>(pointIndex, edge.second)] = static_cast<int>(surfaces.size()); edgeToSurface[std::pair<int, int>(edge.first, pointIndex)] = static_cast<int>(surfaces.size()); edgeToSurface[std::pair<int, int>(edge.second, edge.first)] = static_cast<int>(surfaces.size());
surfaces.emplace_back(edge.second, edge.first, pointIndex, true); } } }
void dfs(const int& pointIndex, const int& surfaceIndex) { surfaces[surfaceIndex].isBelongToConvexHull = false; updateConvexHull(pointIndex, std::pair<int, int>(surfaces[surfaceIndex].b, surfaces[surfaceIndex].a)); updateConvexHull(pointIndex, std::pair<int, int>(surfaces[surfaceIndex].c, surfaces[surfaceIndex].b)); updateConvexHull(pointIndex, std::pair<int, int>(surfaces[surfaceIndex].a, surfaces[surfaceIndex].c)); }
void build() { bool isSameDirection = true; for(int i = 1; i < points.size(); i++) { if(vectorLength(points[0]  points[i]) > EPS) { std::swap(points[1], points[i]); isSameDirection = false; break; } } if(isSameDirection) { return ; }
isSameDirection = true; for(int i = 2; i < points.size(); i++) { if(vectorLength((points[i]  points[0]) * (points[i]  points[1])) > EPS) { std::swap(points[2], points[i]); isSameDirection = false; break; } } if(isSameDirection) { return ; }
isSameDirection = true; for(int i = 3; i < points.size(); i++) { if(fabs((points[1]  points[0]) * (points[2]  points[0]) ^ (points[i]  points[0])) > EPS) { std::swap(points[3], points[i]); isSameDirection = false; break; } } if(isSameDirection) { return ; }
for(int i = 0; i < 4; i++) { Surface surface((i + 1) & 3, (i + 2) & 3, (i + 3) & 3, true); if(isPointSurfaceSameDirection(points[i], surface)) { std::swap(surface.b, surface.c); } edgeToSurface[std::pair<int, int>(surface.a, surface.b)] = static_cast<int>(surfaces.size()); edgeToSurface[std::pair<int, int>(surface.b, surface.c)] = static_cast<int>(surfaces.size()); edgeToSurface[std::pair<int, int>(surface.c, surface.a)] = static_cast<int>(surfaces.size()); surfaces.push_back(surface); }
for(int i = 4; i < points.size(); i++) { for(int j = 0; j < surfaces.size(); j++) { if(surfaces[j].isBelongToConvexHull && isPointSurfaceSameDirection(points[i], surfaces[j])) { dfs(i, j); break; } } } int index = 0; for (const Surface &surface : surfaces) { if(surface.isBelongToConvexHull) { surfaces[index++] = surface; } } surfaces.erase(surfaces.begin() + index, surfaces.end()); }
public: explicit ConvexHull3D(const std::vector<Point>& out_points) { points = out_points; build(); } };
int main() { std::ios::sync_with_stdio(false);
int n; while(std::cin >> n) { std::vector<Point> points; for(int i = 0; i < n; i++) { Point point{}; std::cin >> point.x >> point.y >> point.z; points.push_back(point); } auto convexHull3D = new ConvexHull3D(points); std::cout << convexHull3D>PolygonNumber() << std::endl; delete convexHull3D; }
return 0; }
