// AABB tree built upon external data set, referencing the external data by integer indices.
// The AABB tree balancing and traversal (ray casting, closest triangle of an indexed triangle mesh)
// were adapted from libigl AABB.{cpp,hpp} Copyright (C) 2015 Alec Jacobson <alecjacobson@gmail.com>
// while the implicit balanced tree representation and memory optimizations are Vojtech's.

#ifndef slic3r_AABBTreeIndirect_hpp_
#define slic3r_AABBTreeIndirect_hpp_

#include <algorithm>
#include <limits>
#include <type_traits>
#include <vector>

#include <Eigen/Geometry>

#include "Utils.hpp" // for next_highest_power_of_2()

// Definition of the ray intersection hit structure.
#include <igl/Hit.h>

namespace Slic3r {
namespace AABBTreeIndirect {

// Static balanced AABB tree for raycasting and closest triangle search.
// The balanced tree is built over a single large std::vector of nodes, where the children of nodes
// are addressed implicitely using a power of two indexing rule.
// Memory for a full balanced tree is allocated, but not all nodes at the last level are used.
// This may seem like a waste of memory, but one saves memory for the node links and there is zero
// overhead of a memory allocator management (usually the memory allocator adds at least one pointer 
// before the memory returned). However, allocating memory in a single vector is very fast even
// in multi-threaded environment and it is cache friendly.
//
// A balanced tree is built upon a vector of bounding boxes and their centroids, storing the reference
// to the source entity (a 3D triangle, a 2D segment etc, a 3D or 2D point etc).
// The source bounding boxes may have an epsilon applied to fight numeric rounding errors when 
// traversing the AABB tree.
template<int ANumDimensions, typename ACoordType>
class Tree
{
public:
    static constexpr int    NumDimensions = ANumDimensions;
	using					CoordType     = ACoordType;
    using 					VectorType 	  = Eigen::Matrix<CoordType, NumDimensions, 1, Eigen::DontAlign>;
    using  					BoundingBox   = Eigen::AlignedBox<CoordType, NumDimensions>;
    // Following could be static constexpr size_t, but that would not link in C++11
    enum : size_t {
    	// Node is not used.
        npos  = size_t(-1),
        // Inner node (not leaf).
        inner = size_t(-2)
    };

    // Single node of the implicit balanced AABB tree. There are no links to the children nodes,
    // as these links are calculated implicitely using a power of two rule.
    struct Node {
    	// Index of the external source entity, for which this AABB tree was built, npos for internal nodes.
        size_t 			idx = npos;
    	// Bounding box around this entity, possibly with epsilons applied to fight numeric rounding errors
    	// when traversing the AABB tree.
        BoundingBox 	bbox;

        bool 	is_valid() const { return this->idx != npos; }
        bool 	is_inner() const { return this->idx == inner; }
        bool 	is_leaf()  const { return ! this->is_inner(); }

    	template<typename SourceNode>
    	void set(const SourceNode &rhs) {
            this->idx  = rhs.idx();
            this->bbox = rhs.bbox();
		}
    };

	void clear() { m_nodes.clear(); }

	// SourceNode shall implement
	// size_t SourceNode::idx() const
	// 		- Index to the outside entity (triangle, edge, point etc).
	// const VectorType& SourceNode::centroid() const
	// 		- Centroid of this node. The centroid is used for balancing the tree.
	// const BoundingBox& SourceNode::bbox() const
	// 		- Bounding box of this node, likely expanded with epsilon to account for numeric rounding during tree traversal.
	//        Union of bounding boxes at a single level of the AABB tree is used for deciding the longest axis aligned dimension
	//        to split around.
	template<typename SourceNode>
	void build(std::vector<SourceNode> &&input)
	{
        if (input.empty())
			clear();
		else {
			// Allocate enough memory for a full binary tree.
            m_nodes.assign(next_highest_power_of_2(input.size()) * 2 - 1, Node());
            build_recursive(input, 0, 0, input.size() - 1);
		}
        input.clear();
	}

	const std::vector<Node>& 	nodes() const { return m_nodes; }
	const Node& 				node(size_t idx) const { return m_nodes[idx]; }
	bool 						empty() const { return m_nodes.empty(); }

	// Addressing the child nodes using the power of two rule.
    static size_t				left_child_idx(size_t idx) { return idx * 2 + 1; }
    static size_t				right_child_idx(size_t idx) { return left_child_idx(idx) + 1; }
	const Node&					left_child(size_t idx) const { return m_nodes[left_child_idx(idx)]; }
	const Node&					right_child(size_t idx) const { return m_nodes[right_child_idx(idx)]; }

	template<typename SourceNode>
    void build(const std::vector<SourceNode> &input)
	{
        std::vector<SourceNode> copy(input);
        this->build(std::move(copy));
	}

private:
	// Build a balanced tree by splitting the input sequence by an axis aligned plane at a dimension.
	template<typename SourceNode>
	void build_recursive(std::vector<SourceNode> &input, size_t node, const size_t left, const size_t right)
	{
        assert(node < m_nodes.size());
        assert(left <= right);

		if (left == right) {
			// Insert a node into the balanced tree.
			m_nodes[node].set(input[left]);
			return;
		}

		// Calculate bounding box of the input.
        BoundingBox bbox(input[left].bbox());
        for (size_t i = left + 1; i <= right; ++ i)
            bbox.extend(input[i].bbox());
        int dimension = -1;
        bbox.diagonal().maxCoeff(&dimension);

		// Partition the input to left / right pieces of the same length to produce a balanced tree.
		size_t center = (left + right) / 2;
		partition_input(input, size_t(dimension), left, right, center);
		// Insert an inner node into the tree. Inner node does not reference any input entity (triangle, line segment etc).
		m_nodes[node].idx  = inner;
		m_nodes[node].bbox = bbox;
        build_recursive(input, node * 2 + 1, left, center);
		build_recursive(input, node * 2 + 2, center + 1, right);
	}

	// Partition the input m_nodes <left, right> at "k" and "dimension" using the QuickSelect method:
	// https://en.wikipedia.org/wiki/Quickselect
	// Items left of the k'th item are lower than the k'th item in the "dimension", 
	// items right of the k'th item are higher than the k'th item in the "dimension", 
	template<typename SourceNode>
	void partition_input(std::vector<SourceNode> &input, const size_t dimension, size_t left, size_t right, const size_t k) const
	{
		while (left < right) {
			size_t center = (left + right) / 2;
			CoordType pivot;
			{
				// Bubble sort the input[left], input[center], input[right], so that a median of the three values
				// will end up in input[center].
				CoordType left_value   = input[left  ].centroid()(dimension);
				CoordType center_value = input[center].centroid()(dimension);
				CoordType right_value  = input[right ].centroid()(dimension);
				if (left_value > center_value) {
					std::swap(input[left], input[center]);
					std::swap(left_value,  center_value);
				}
				if (left_value > right_value) {
					std::swap(input[left], input[right]);
					right_value = left_value;
				}
				if (center_value > right_value) {
					std::swap(input[center], input[right]);
					center_value = right_value;
				}
				pivot = center_value;
			}
			if (right <= left + 2)
				// The <left, right> interval is already sorted.
				break;
			size_t i = left;
			size_t j = right - 1;
			std::swap(input[center], input[j]);
			// Partition the set based on the pivot.
			for (;;) {
				// Skip left points that are already at correct positions.
				// Search will certainly stop at position (right - 1), which stores the pivot.
				while (input[++ i].centroid()(dimension) < pivot) ;
				// Skip right points that are already at correct positions.
				while (input[-- j].centroid()(dimension) > pivot && i < j) ;
				if (i >= j)
					break;
				std::swap(input[i], input[j]);
			}
			// Restore pivot to the center of the sequence.
			std::swap(input[i], input[right - 1]);
			// Which side the kth element is in?
			if (k < i)
				right = i - 1;
			else if (k == i)
				// Sequence is partitioned, kth element is at its place.
				break;
			else
				left = i + 1;
		}
	}

	// The balanced tree storage.
	std::vector<Node> m_nodes;
};

using Tree2f = Tree<2, float>;
using Tree3f = Tree<3, float>;
using Tree2d = Tree<2, double>;
using Tree3d = Tree<3, double>;

namespace detail {
	template<typename AVertexType, typename AIndexedFaceType, typename ATreeType, typename AVectorType>
	struct RayIntersector {
		using VertexType 		= AVertexType;
		using IndexedFaceType 	= AIndexedFaceType;
		using TreeType			= ATreeType;
		using VectorType 		= AVectorType;

		const std::vector<VertexType> 		&vertices;
		const std::vector<IndexedFaceType> 	&faces;
		const TreeType 						&tree;

		const VectorType					 origin;
		const VectorType 					 dir;
		const VectorType					 invdir;

		// epsilon for ray-triangle intersection, see intersect_triangle1()
		const double  						 eps;
	};

    template<typename VertexType, typename IndexedFaceType, typename TreeType, typename VectorType>
    struct RayIntersectorHits : RayIntersector<VertexType, IndexedFaceType, TreeType, VectorType> {
		std::vector<igl::Hit>				 hits;
	};

	//FIXME implement SSE for float AABB trees with float ray queries.
	// SSE/SSE2 is supported by any Intel/AMD x64 processor.
	// SSE support requires 16 byte alignment of the AABB nodes, representing the bounding boxes with 4+4 floats,
	// storing the node index as the 4th element of the bounding box min value etc.
	// https://www.flipcode.com/archives/SSE_RayBox_Intersection_Test.shtml
	template <typename Derivedsource, typename Deriveddir, typename Scalar>
	inline bool ray_box_intersect_invdir(
  		const Eigen::MatrixBase<Derivedsource> 	&origin,
  		const Eigen::MatrixBase<Deriveddir> 	&inv_dir,
  		Eigen::AlignedBox<Scalar,3> 			 box,
  		const Scalar 							&t0,
  		const Scalar 							&t1) {
		// http://people.csail.mit.edu/amy/papers/box-jgt.pdf
		// "An Efficient and Robust Ray–Box Intersection Algorithm"
		if (inv_dir.x() < 0)
			std::swap(box.min().x(), box.max().x());
		if (inv_dir.y() < 0)
			std::swap(box.min().y(), box.max().y());
        Scalar tmin = (box.min().x() - origin.x()) * inv_dir.x();
		Scalar tymax = (box.max().y() - origin.y()) * inv_dir.y();
		if (tmin > tymax)
			return false;
        Scalar tmax = (box.max().x() - origin.x()) * inv_dir.x();
		Scalar tymin = (box.min().y()  - origin.y()) * inv_dir.y();
		if (tymin > tmax)
			return false;
		if (tymin > tmin)
			tmin = tymin;
		if (tymax < tmax)
			tmax = tymax;
		if (inv_dir.z() < 0)
			std::swap(box.min().z(), box.max().z());
		Scalar tzmin = (box.min().z()  - origin.z()) * inv_dir.z();
		if (tzmin > tmax)
			return false;
		Scalar tzmax = (box.max().z() - origin.z()) * inv_dir.z();
		if (tmin > tzmax)
			return false;
		if (tzmin > tmin)
			tmin = tzmin;
		if (tzmax < tmax)
			tmax = tzmax;
        return tmin < t1 && tmax > t0;
	}

	// The following intersect_triangle() is derived from raytri.c routine intersect_triangle1()
	// Ray-Triangle Intersection Test Routines
	// Different optimizations of my and Ben Trumbore's
	// code from journals of graphics tools (JGT)
	// http://www.acm.org/jgt/
	// by Tomas Moller, May 2000
	template<typename V, typename W>
	std::enable_if_t<std::is_same<typename V::Scalar, double>::value&& std::is_same<typename W::Scalar, double>::value, bool>
	intersect_triangle(const V &orig, const V &dir, const W &vert0, const W &vert1, const W &vert2, double &t, double &u, double &v, double eps)
	{
	   // find vectors for two edges sharing vert0
	   const V      edge1 = vert1 - vert0;
	   const V      edge2 = vert2 - vert0;
	   // begin calculating determinant - also used to calculate U parameter
	   const V      pvec  = dir.cross(edge2);
	   // if determinant is near zero, ray lies in plane of triangle
	   const double det   = edge1.dot(pvec);
	   V      	 	qvec;

	   if (det > eps) {
	      	// calculate distance from vert0 to ray origin
	      	V tvec = orig - vert0;
	      	// calculate U parameter and test bounds
	      	u = tvec.dot(pvec);
			if (u < 0.0 || u > det)
				return false;
	      	// prepare to test V parameter
	      	qvec = tvec.cross(edge1);
	      	// calculate V parameter and test bounds
	      	v = dir.dot(qvec);
	      	if (v < 0.0 || u + v > det)
		 		return false;
	   } else if (det < -eps) {
	      	// calculate distance from vert0 to ray origin
	      	V tvec = orig - vert0;
	      	// calculate U parameter and test bounds
	      	u = tvec.dot(pvec);
	      	if (u > 0.0 || u < det)
		 		return false;
	      	// prepare to test V parameter
	      	qvec = tvec.cross(edge1);
	      	// calculate V parameter and test bounds
	      	v = dir.dot(qvec);
	      	if (v > 0.0 || u + v < det)
		 		return false;
	   } else 
	     	// ray is parallel to the plane of the triangle
		   	return false;

	   double inv_det = 1.0 / det;
	   // calculate t, ray intersects triangle
	   t = edge2.dot(qvec) * inv_det;
	   u *= inv_det;
	   v *= inv_det;
	   return true;
	}

	template<typename V, typename W>
    std::enable_if_t<std::is_same<typename V::Scalar, double>::value && !std::is_same<typename W::Scalar, double>::value, bool>
	intersect_triangle(const V &origin, const V &dir, const W &v0, const W &v1, const W &v2, double &t, double &u, double &v, double eps) {
        return intersect_triangle(origin, dir, v0.template cast<double>(), v1.template cast<double>(), v2.template cast<double>(), t, u, v, eps);
	}

	template<typename V, typename W>
    std::enable_if_t<! std::is_same<typename V::Scalar, double>::value && std::is_same<typename W::Scalar, double>::value, bool>
	intersect_triangle(const V &origin, const V &dir, const W &v0, const W &v1, const W &v2, double &t, double &u, double &v, double eps) {
        return intersect_triangle(origin.template cast<double>(), dir.template cast<double>(), v0, v1, v2, t, u, v, eps);
	}

	template<typename V, typename W>
    std::enable_if_t<! std::is_same<typename V::Scalar, double>::value && ! std::is_same<typename W::Scalar, double>::value, bool>
	intersect_triangle(const V &origin, const V &dir, const W &v0, const W &v1, const W &v2, double &t, double &u, double &v, double eps) {
	    return intersect_triangle(origin.template cast<double>(), dir.template cast<double>(), v0.template cast<double>(), v1.template cast<double>(), v2.template cast<double>(), t, u, v, eps);
	}

	template<typename Tree>
	double intersect_triangle_epsilon(const Tree &tree) {
		double eps = 0.000001;
		if (! tree.empty()) {
			const typename Tree::BoundingBox &bbox = tree.nodes().front().bbox;
			double l = (bbox.max() - bbox.min()).cwiseMax();
			if (l > 0)
				eps /= (l * l);
		}
		return eps;
	}

    template<typename RayIntersectorType, typename Scalar>
	static inline bool intersect_ray_recursive_first_hit(
        RayIntersectorType 	   &ray_intersector,
        size_t 				    node_idx,
        Scalar                  min_t,
        igl::Hit 			   &hit)
	{
        const auto &node = ray_intersector.tree.node(node_idx);
        assert(node.is_valid());
		
        if (! ray_box_intersect_invdir(ray_intersector.origin, ray_intersector.invdir, node.bbox.template cast<Scalar>(), Scalar(0), min_t))
			return false;

	  	if (node.is_leaf()) {
		    // shoot ray, record hit
            auto   face = ray_intersector.faces[node.idx];
		    double t, u, v;
		    if (intersect_triangle(
		    		ray_intersector.origin, ray_intersector.dir, 
		    		ray_intersector.vertices[face(0)], ray_intersector.vertices[face(1)], ray_intersector.vertices[face(2)], 
                    t, u, v, ray_intersector.eps)
		    	&& t > 0.) {
                hit = igl::Hit { int(node.idx), -1, float(u), float(v), float(t) };
				return true;
		    } else
		    	return false;
	  	} else {
			// Left / right child node index.
			size_t left  = node_idx * 2 + 1;
			size_t right = left + 1;
			igl::Hit left_hit;
			igl::Hit right_hit;
	        bool left_ret = intersect_ray_recursive_first_hit(ray_intersector, left,  min_t, left_hit);
			if (left_ret && left_hit.t < min_t) {
	    		min_t = left_hit.t;
	    		hit   = left_hit;
	  		} else
			    left_ret = false;
	        bool right_ret = intersect_ray_recursive_first_hit(ray_intersector, right, min_t, right_hit);
			if (right_ret && right_hit.t < min_t)
				hit = right_hit;
			else
				right_ret = false;
			return left_ret || right_ret;
		}
	}

    template<typename RayIntersectorType>
	static inline void intersect_ray_recursive_all_hits(RayIntersectorType &ray_intersector, size_t node_idx)
	{
        using Scalar = typename RayIntersectorType::VectorType::Scalar;

		const auto &node = ray_intersector.tree.node(node_idx);
		assert(node.is_valid());

        if (! ray_box_intersect_invdir(ray_intersector.origin, ray_intersector.invdir, node.bbox.template cast<Scalar>(),
    			Scalar(0), std::numeric_limits<Scalar>::infinity()))
			return;

	  	if (node.is_leaf()) {
            auto   face = ray_intersector.faces[node.idx];
		    double t, u, v;
		    if (intersect_triangle(
		    		ray_intersector.origin, ray_intersector.dir, 
		    		ray_intersector.vertices[face(0)], ray_intersector.vertices[face(1)], ray_intersector.vertices[face(2)], 
                    t, u, v, ray_intersector.eps)
		    	&& t > 0.) {
                ray_intersector.hits.emplace_back(igl::Hit{ int(node.idx), -1, float(u), float(v), float(t) });
			}
	  	} else {
			// Left / right child node index.
			size_t left  = node_idx * 2 + 1;
			size_t right = left + 1;
		  	intersect_ray_recursive_all_hits(ray_intersector, left);
		  	intersect_ray_recursive_all_hits(ray_intersector, right);
		}
	}

    // Real-time collision detection, Ericson, Chapter 5
    template<typename Vector>
    static inline Vector closest_point_to_triangle(const Vector &p, const Vector &a, const Vector &b, const Vector &c)
    {
        using Scalar = typename Vector::Scalar;
        // Check if P in vertex region outside A
        Vector ab = b - a;
        Vector ac = c - a;
        Vector ap = p - a;
        Scalar d1 = ab.dot(ap);
        Scalar d2 = ac.dot(ap);
        if (d1 <= 0 && d2 <= 0)
          return a;
        // Check if P in vertex region outside B
        Vector bp = p - b;
        Scalar d3 = ab.dot(bp);
        Scalar d4 = ac.dot(bp);
        if (d3 >= 0 && d4 <= d3)
          return b;
        // Check if P in edge region of AB, if so return projection of P onto AB
        Scalar vc = d1*d4 - d3*d2;
        if (a != b && vc <= 0 && d1 >= 0 && d3 <= 0) {
            Scalar v = d1 / (d1 - d3);
            return a + v * ab;
        }
        // Check if P in vertex region outside C
        Vector cp = p - c;
        Scalar d5 = ab.dot(cp);
        Scalar d6 = ac.dot(cp);
        if (d6 >= 0 && d5 <= d6)
          return c;
        // Check if P in edge region of AC, if so return projection of P onto AC
        Scalar vb = d5*d2 - d1*d6;
        if (vb <= 0 && d2 >= 0 && d6 <= 0) {
          Scalar w = d2 / (d2 - d6);
          return a + w * ac;
        }
        // Check if P in edge region of BC, if so return projection of P onto BC
        Scalar va = d3*d6 - d5*d4;
        if (va <= 0 && (d4 - d3) >= 0 && (d5 - d6) >= 0) {
          Scalar w = (d4 - d3) / ((d4 - d3) + (d5 - d6));
          return b + w * (c - b);
        }
        // P inside face region. Compute Q through its barycentric coordinates (u,v,w)
        Scalar denom = Scalar(1.0) / (va + vb + vc);
        Scalar v = vb * denom;
        Scalar w = vc * denom;
        return a + ab * v + ac * w; // = u*a + v*b + w*c, u = va * denom = 1.0-v-w
    };


	// Nothing to do with COVID-19 social distancing.
	template<typename AVertexType, typename AIndexedFaceType, typename ATreeType, typename AVectorType>
	struct IndexedTriangleSetDistancer {
		using VertexType 		= AVertexType;
		using IndexedFaceType 	= AIndexedFaceType;
		using TreeType			= ATreeType;
		using VectorType 		= AVectorType;
		using ScalarType 		= typename VectorType::Scalar;

		const std::vector<VertexType> 		&vertices;
		const std::vector<IndexedFaceType> 	&faces;
		const TreeType 						&tree;

		const VectorType					 origin;

		inline VectorType closest_point_to_origin(size_t primitive_index,
		        ScalarType& squared_distance){
		    const auto &triangle = this->faces[primitive_index];
		    VectorType closest_point = closest_point_to_triangle<VectorType>(origin,
		            this->vertices[triangle(0)].template cast<ScalarType>(),
		            this->vertices[triangle(1)].template cast<ScalarType>(),
		            this->vertices[triangle(2)].template cast<ScalarType>());
		    squared_distance = (origin - closest_point).squaredNorm();
		    return closest_point;
		}
	};

	template<typename IndexedPrimitivesDistancerType, typename Scalar>
    static inline Scalar squared_distance_to_indexed_primitives_recursive(
        IndexedPrimitivesDistancerType	&distancer,
		size_t 							 node_idx,
		Scalar 							 low_sqr_d,
  		Scalar 							 up_sqr_d,
		size_t 							&i,
  		Eigen::PlainObjectBase<typename IndexedPrimitivesDistancerType::VectorType> &c)
	{
		using Vector = typename IndexedPrimitivesDistancerType::VectorType;

  		if (low_sqr_d > up_sqr_d)
			return low_sqr_d;
	  
	  	// Save the best achieved hit.
        auto set_min = [&i, &c, &up_sqr_d](const Scalar sqr_d_candidate, const size_t i_candidate, const Vector &c_candidate) {
			if (sqr_d_candidate < up_sqr_d) {
				i     	 = i_candidate;
				c     	 = c_candidate;
				up_sqr_d = sqr_d_candidate;
			}
        };

		const auto &node = distancer.tree.node(node_idx);
		assert(node.is_valid());
  		if (node.is_leaf()) 
  		{
            Scalar sqr_dist;
            Vector c_candidate = distancer.closest_point_to_origin(node.idx, sqr_dist);
            set_min(sqr_dist, node.idx, c_candidate);
  		} 
  		else
  		{
			size_t left_node_idx  = node_idx * 2 + 1;
            size_t right_node_idx = left_node_idx + 1;
			const auto &node_left  = distancer.tree.node(left_node_idx);
			const auto &node_right = distancer.tree.node(right_node_idx);
			assert(node_left.is_valid());
			assert(node_right.is_valid());

			bool   looked_left    = false;
			bool   looked_right   = false;
			const auto &look_left = [&]()
			{
                size_t	i_left;
                Vector 	c_left = c;
                Scalar	sqr_d_left = squared_distance_to_indexed_primitives_recursive(distancer, left_node_idx, low_sqr_d, up_sqr_d, i_left, c_left);
				set_min(sqr_d_left, i_left, c_left);
				looked_left = true;
			};
			const auto &look_right = [&]()
			{
                size_t	i_right;
                Vector	c_right = c;
                Scalar	sqr_d_right = squared_distance_to_indexed_primitives_recursive(distancer, right_node_idx, low_sqr_d, up_sqr_d, i_right, c_right);
				set_min(sqr_d_right, i_right, c_right);
				looked_right = true;
			};

			// must look left or right if in box
            using BBoxScalar = typename IndexedPrimitivesDistancerType::TreeType::BoundingBox::Scalar;
            if (node_left.bbox.contains(distancer.origin.template cast<BBoxScalar>()))
			  	look_left();
            if (node_right.bbox.contains(distancer.origin.template cast<BBoxScalar>()))
			  	look_right();
			// if haven't looked left and could be less than current min, then look
			Scalar left_up_sqr_d  = node_left.bbox.squaredExteriorDistance(distancer.origin);
			Scalar right_up_sqr_d = node_right.bbox.squaredExteriorDistance(distancer.origin);
			if (left_up_sqr_d < right_up_sqr_d) {
			  	if (! looked_left && left_up_sqr_d < up_sqr_d)
			    	look_left();
			  	if (! looked_right && right_up_sqr_d < up_sqr_d)
			    	look_right();
			} else {
			  	if (! looked_right && right_up_sqr_d < up_sqr_d)
			    	look_right();
			  	if (! looked_left && left_up_sqr_d < up_sqr_d)
			    	look_left();
			}
		}
		return up_sqr_d;
	}

} // namespace detail

// Build a balanced AABB Tree over an indexed triangles set, balancing the tree
// on centroids of the triangles.
// Epsilon is applied to the bounding boxes of the AABB Tree to cope with numeric inaccuracies
// during tree traversal.
template<typename VertexType, typename IndexedFaceType>
inline Tree<3, typename VertexType::Scalar> build_aabb_tree_over_indexed_triangle_set(
	// Indexed triangle set - 3D vertices.
	const std::vector<VertexType> 		&vertices, 
	// Indexed triangle set - triangular faces, references to vertices.
    const std::vector<IndexedFaceType> 	&faces,
	//FIXME do we want to apply an epsilon?
    const typename VertexType::Scalar 	 eps = 0)
{
    using 				 TreeType 		= Tree<3, typename VertexType::Scalar>;
//    using				 CoordType      = typename TreeType::CoordType;
    using 				 VectorType	    = typename TreeType::VectorType;
    using 				 BoundingBox 	= typename TreeType::BoundingBox;

	struct InputType {
        size_t 				idx()       const { return m_idx; }
        const BoundingBox& 	bbox()      const { return m_bbox; }
        const VectorType& 	centroid()  const { return m_centroid; }

		size_t 		m_idx;
		BoundingBox m_bbox;
        VectorType 	m_centroid;
	};

	std::vector<InputType> input;
	input.reserve(faces.size());
    const VectorType veps(eps, eps, eps);
	for (size_t i = 0; i < faces.size(); ++ i) {
        const IndexedFaceType &face = faces[i];
		const VertexType &v1 = vertices[face(0)];
		const VertexType &v2 = vertices[face(1)];
		const VertexType &v3 = vertices[face(2)];
		InputType n;
        n.m_idx      = i;
        n.m_centroid = (1./3.) * (v1 + v2 + v3);
        n.m_bbox = BoundingBox(v1, v1);
        n.m_bbox.extend(v2);
        n.m_bbox.extend(v3);
        n.m_bbox.min() -= veps;
        n.m_bbox.max() += veps;
        input.emplace_back(n);
	}

	TreeType out;
	out.build(std::move(input));
	return out;
}

// Find a first intersection of a ray with indexed triangle set.
// Intersection test is calculated with the accuracy of VectorType::Scalar
// even if the triangle mesh and the AABB Tree are built with floats.
template<typename VertexType, typename IndexedFaceType, typename TreeType, typename VectorType>
inline bool intersect_ray_first_hit(
	// Indexed triangle set - 3D vertices.
	const std::vector<VertexType> 		&vertices,
	// Indexed triangle set - triangular faces, references to vertices.
	const std::vector<IndexedFaceType> 	&faces,
	// AABBTreeIndirect::Tree over vertices & faces, bounding boxes built with the accuracy of vertices.
	const TreeType 						&tree,
	// Origin of the ray.
	const VectorType					&origin,
	// Direction of the ray.
	const VectorType 					&dir,
	// First intersection of the ray with the indexed triangle set.
	igl::Hit 							&hit,
	// Epsilon for the ray-triangle intersection, it should be proportional to an average triangle edge length.
	const double 						 eps = 0.000001)
{
    using Scalar = typename VectorType::Scalar;
	auto ray_intersector = detail::RayIntersector<VertexType, IndexedFaceType, TreeType, VectorType> {
		vertices, faces, tree,
        origin, dir, VectorType(dir.cwiseInverse()),
        eps
	};
	return ! tree.empty() && detail::intersect_ray_recursive_first_hit(
        ray_intersector, size_t(0), std::numeric_limits<Scalar>::infinity(), hit);
}

// Find all intersections of a ray with indexed triangle set.
// Intersection test is calculated with the accuracy of VectorType::Scalar
// even if the triangle mesh and the AABB Tree are built with floats.
// The output hits are sorted by the ray parameter.
// If the ray intersects a shared edge of two triangles, hits for both triangles are returned.
template<typename VertexType, typename IndexedFaceType, typename TreeType, typename VectorType>
inline bool intersect_ray_all_hits(
	// Indexed triangle set - 3D vertices.
	const std::vector<VertexType> 		&vertices,
	// Indexed triangle set - triangular faces, references to vertices.
	const std::vector<IndexedFaceType> 	&faces,
	// AABBTreeIndirect::Tree over vertices & faces, bounding boxes built with the accuracy of vertices.
	const TreeType 						&tree,
	// Origin of the ray.
	const VectorType					&origin,
	// Direction of the ray.
	const VectorType 					&dir,
	// All intersections of the ray with the indexed triangle set, sorted by parameter t.
	std::vector<igl::Hit> 				&hits,
	// Epsilon for the ray-triangle intersection, it should be proportional to an average triangle edge length.
	const double 						 eps = 0.000001)
{
    auto ray_intersector = detail::RayIntersectorHits<VertexType, IndexedFaceType, TreeType, VectorType> {
        { vertices, faces, {tree},
        origin, dir, VectorType(dir.cwiseInverse()),
        eps }
	};
	if (tree.empty()) {
		hits.clear();
	} else {
		// Reusing the output memory if there is some memory already pre-allocated.
        ray_intersector.hits = std::move(hits);
        ray_intersector.hits.clear();
        ray_intersector.hits.reserve(8);
		detail::intersect_ray_recursive_all_hits(ray_intersector, 0);
		hits = std::move(ray_intersector.hits);
	    std::sort(hits.begin(), hits.end(), [](const auto &l, const auto &r) { return l.t < r.t; });
	}
	return ! hits.empty();
}

// Finding a closest triangle, its closest point and squared distance to the closest point
// on a 3D indexed triangle set using a pre-built AABBTreeIndirect::Tree.
// Closest point to triangle test will be performed with the accuracy of VectorType::Scalar
// even if the triangle mesh and the AABB Tree are built with floats.
// Returns squared distance to the closest point or -1 if the input is empty.
template<typename VertexType, typename IndexedFaceType, typename TreeType, typename VectorType>
inline typename VectorType::Scalar squared_distance_to_indexed_triangle_set(
	// Indexed triangle set - 3D vertices.
	const std::vector<VertexType> 		&vertices,
	// Indexed triangle set - triangular faces, references to vertices.
	const std::vector<IndexedFaceType> 	&faces,
	// AABBTreeIndirect::Tree over vertices & faces, bounding boxes built with the accuracy of vertices.
	const TreeType 						&tree,
	// Point to which the closest point on the indexed triangle set is searched for.
	const VectorType					&point,
	// Index of the closest triangle in faces.
	size_t 								&hit_idx_out,
	// Position of the closest point on the indexed triangle set.
	Eigen::PlainObjectBase<VectorType>	&hit_point_out)
{
    using Scalar = typename VectorType::Scalar;
    auto distancer = detail::IndexedTriangleSetDistancer<VertexType, IndexedFaceType, TreeType, VectorType>
        { vertices, faces, tree, point };
    return tree.empty() ? Scalar(-1) : 
    	detail::squared_distance_to_indexed_primitives_recursive(distancer, size_t(0), Scalar(0), std::numeric_limits<Scalar>::infinity(), hit_idx_out, hit_point_out);
}

// Decides if exists some triangle in defined radius on a 3D indexed triangle set using a pre-built AABBTreeIndirect::Tree.
// Closest point to triangle test will be performed with the accuracy of VectorType::Scalar
// even if the triangle mesh and the AABB Tree are built with floats.
// Returns true if exists some triangle in defined radius, false otherwise.
template<typename VertexType, typename IndexedFaceType, typename TreeType, typename VectorType>
inline bool is_any_triangle_in_radius(
        // Indexed triangle set - 3D vertices.
        const std::vector<VertexType> 		&vertices,
        // Indexed triangle set - triangular faces, references to vertices.
        const std::vector<IndexedFaceType> 	&faces,
        // AABBTreeIndirect::Tree over vertices & faces, bounding boxes built with the accuracy of vertices.
        const TreeType 						&tree,
        // Point to which the closest point on the indexed triangle set is searched for.
        const VectorType					&point,
        //Square of maximum distance in which triangle is searched for
        typename VectorType::Scalar &max_distance_squared)
{
    using Scalar = typename VectorType::Scalar;
    auto distancer = detail::IndexedTriangleSetDistancer<VertexType, IndexedFaceType, TreeType, VectorType>
            { vertices, faces, tree, point };

    size_t hit_idx;
    VectorType hit_point = VectorType::Ones() * (NaN<typename VectorType::Scalar>);

	if(tree.empty())
	{
		return false;
	}

	detail::squared_distance_to_indexed_primitives_recursive(distancer, size_t(0), Scalar(0), max_distance_squared, hit_idx, hit_point);

    return hit_point.allFinite();
}


// Traverse the tree and return the index of an entity whose bounding box
// contains a given point. Returns size_t(-1) when the point is outside.
template<typename TreeType, typename VectorType>
void get_candidate_idxs(const TreeType& tree, const VectorType& v, std::vector<size_t>& candidates, size_t node_idx = 0)
{
    if (tree.empty() || ! tree.node(node_idx).bbox.contains(v))
        return;

    decltype(tree.node(node_idx)) node = tree.node(node_idx);
    static_assert(std::is_reference<decltype(node)>::value,
                  "Nodes shall be addressed by reference.");
    assert(node.is_valid());
    assert(node.bbox.contains(v));

    if (! node.is_leaf()) {
        if (tree.left_child(node_idx).bbox.contains(v))
            get_candidate_idxs(tree, v, candidates, tree.left_child_idx(node_idx));
        if (tree.right_child(node_idx).bbox.contains(v))
            get_candidate_idxs(tree, v, candidates, tree.right_child_idx(node_idx));
    } else
        candidates.push_back(node.idx);

    return;
}

// Predicate: need to be specialized for intersections of different geomteries
template<class G> struct Intersecting {};

// Intersection predicate specialization for box-box intersections
template<class CoordType, int NumD>
struct Intersecting<Eigen::AlignedBox<CoordType, NumD>> {
    Eigen::AlignedBox<CoordType, NumD> box;

    Intersecting(const Eigen::AlignedBox<CoordType, NumD> &bb): box{bb} {}

    bool operator() (const typename Tree<NumD, CoordType>::Node &node) const
    {
        return box.intersects(node.bbox);
    }
};

template<class G> auto intersecting(const G &g) { return Intersecting<G>{g}; }

template<class G> struct Containing {};

// Intersection predicate specialization for box-box intersections
template<class CoordType, int NumD>
struct Containing<Eigen::AlignedBox<CoordType, NumD>> {
    Eigen::AlignedBox<CoordType, NumD> box;

    Containing(const Eigen::AlignedBox<CoordType, NumD> &bb): box{bb} {}

    bool operator() (const typename Tree<NumD, CoordType>::Node &node) const
    {
        return box.contains(node.bbox);
    }
};

template<class G> auto containing(const G &g) { return Containing<G>{g}; }

namespace detail {

template<int Dims, typename T, typename Pred, typename Fn>
void traverse_recurse(const Tree<Dims, T> &tree,
                      size_t               idx,
                      Pred &&              pred,
                      Fn &&                callback)
{
    assert(tree.node(idx).is_valid());

    if (!pred(tree.node(idx))) return;

    if (tree.node(idx).is_leaf()) {
        callback(tree.node(idx).idx);
    } else {

        // call this with left and right node idx:
        auto trv = [&](size_t idx) {
            traverse_recurse(tree, idx, std::forward<Pred>(pred),
                             std::forward<Fn>(callback));
        };

        // Left / right child node index.
        trv(Tree<Dims, T>::left_child_idx(idx));
        trv(Tree<Dims, T>::right_child_idx(idx));
    }
}

} // namespace detail

// Tree traversal with a predicate. Example usage:
// traverse(tree, intersecting(QueryBox), [](size_t face_idx) {
//      /* ... */
// });
template<int Dims, typename T, typename Predicate, typename Fn>
void traverse(const Tree<Dims, T> &tree, Predicate &&pred, Fn &&callback)
{
    if (tree.empty()) return;

    detail::traverse_recurse(tree, size_t(0), std::forward<Predicate>(pred),
                             std::forward<Fn>(callback));
}

} // namespace AABBTreeIndirect
} // namespace Slic3r

#endif /* slic3r_AABBTreeIndirect_hpp_ */
