PrusaSlicer-NonPlainar/xs/src/libslic3r/TriangleMesh.cpp

#include "TriangleMesh.hpp"
#include "ClipperUtils.hpp"
#include "Geometry.hpp"
#include <cmath>
#include <deque>
#include <queue>
#include <set>
#include <vector>
#include <map>
#include <unordered_map>
#include <utility>
#include <algorithm>
#include <math.h>

#if 0
    #define DEBUG
    #define _DEBUG
    #undef NDEBUG
#endif

#include <assert.h>

#ifdef SLIC3R_DEBUG
// #define SLIC3R_TRIANGLEMESH_DEBUG
#include "SVG.hpp"
#endif

namespace Slic3r {

TriangleMesh::TriangleMesh()
    : repaired(false)
{
    stl_initialize(&this->stl);
}

TriangleMesh::TriangleMesh(const Pointf3s &points, const std::vector<Point3>& facets )
    : repaired(false)
{
    stl_initialize(&this->stl);
    stl_file &stl = this->stl;
    stl.error = 0;
    stl.stats.type = inmemory;

    // count facets and allocate memory
    stl.stats.number_of_facets = facets.size();
    stl.stats.original_num_facets = stl.stats.number_of_facets;
    stl_allocate(&stl);

    for (int i = 0; i < stl.stats.number_of_facets; i++) {
        stl_facet facet;
        facet.normal.x = 0;
        facet.normal.y = 0;
        facet.normal.z = 0;

        const Pointf3& ref_f1 = points[facets[i].x];
        facet.vertex[0].x = ref_f1.x;
        facet.vertex[0].y = ref_f1.y;
        facet.vertex[0].z = ref_f1.z;

        const Pointf3& ref_f2 = points[facets[i].y];
        facet.vertex[1].x = ref_f2.x;
        facet.vertex[1].y = ref_f2.y;
        facet.vertex[1].z = ref_f2.z;

        const Pointf3& ref_f3 = points[facets[i].z];
        facet.vertex[2].x = ref_f3.x;
        facet.vertex[2].y = ref_f3.y;
        facet.vertex[2].z = ref_f3.z;
        
        facet.extra[0] = 0;
        facet.extra[1] = 0;

        stl.facet_start[i] = facet;
    }
    stl_get_size(&stl);
}

TriangleMesh::TriangleMesh(const TriangleMesh &other)
    : stl(other.stl), repaired(other.repaired)
{
    this->stl.heads = NULL;
    this->stl.tail  = NULL;
    this->stl.error = other.stl.error;
    if (other.stl.facet_start != NULL) {
        this->stl.facet_start = (stl_facet*)calloc(other.stl.stats.number_of_facets, sizeof(stl_facet));
        std::copy(other.stl.facet_start, other.stl.facet_start + other.stl.stats.number_of_facets, this->stl.facet_start);
    }
    if (other.stl.neighbors_start != NULL) {
        this->stl.neighbors_start = (stl_neighbors*)calloc(other.stl.stats.number_of_facets, sizeof(stl_neighbors));
        std::copy(other.stl.neighbors_start, other.stl.neighbors_start + other.stl.stats.number_of_facets, this->stl.neighbors_start);
    }
    if (other.stl.v_indices != NULL) {
        this->stl.v_indices = (v_indices_struct*)calloc(other.stl.stats.number_of_facets, sizeof(v_indices_struct));
        std::copy(other.stl.v_indices, other.stl.v_indices + other.stl.stats.number_of_facets, this->stl.v_indices);
    }
    if (other.stl.v_shared != NULL) {
        this->stl.v_shared = (stl_vertex*)calloc(other.stl.stats.shared_vertices, sizeof(stl_vertex));
        std::copy(other.stl.v_shared, other.stl.v_shared + other.stl.stats.shared_vertices, this->stl.v_shared);
    }
}

TriangleMesh::TriangleMesh(TriangleMesh &&other) : 
    repaired(false)
{
    stl_initialize(&this->stl);
    this->swap(other);
}

TriangleMesh& TriangleMesh::operator= (TriangleMesh other)
{
    this->swap(other);
    return *this;
}

TriangleMesh& TriangleMesh::operator=(TriangleMesh &&other)
{
    this->swap(other);
    return *this;
}

void
TriangleMesh::swap(TriangleMesh &other)
{
    std::swap(this->stl,      other.stl);
    std::swap(this->repaired, other.repaired);
}

TriangleMesh::~TriangleMesh() {
    stl_close(&this->stl);
}

void
TriangleMesh::ReadSTLFile(const char* input_file) {
    stl_open(&stl, input_file);
}

void
TriangleMesh::write_ascii(const char* output_file)
{
    stl_write_ascii(&this->stl, output_file, "");
}

void
TriangleMesh::write_binary(const char* output_file)
{
    stl_write_binary(&this->stl, output_file, "");
}

void
TriangleMesh::repair() {
    if (this->repaired) return;
    
    // admesh fails when repairing empty meshes
    if (this->stl.stats.number_of_facets == 0) return;
    
    // checking exact
    stl_check_facets_exact(&stl);
    stl.stats.facets_w_1_bad_edge = (stl.stats.connected_facets_2_edge - stl.stats.connected_facets_3_edge);
    stl.stats.facets_w_2_bad_edge = (stl.stats.connected_facets_1_edge - stl.stats.connected_facets_2_edge);
    stl.stats.facets_w_3_bad_edge = (stl.stats.number_of_facets - stl.stats.connected_facets_1_edge);
    
    // checking nearby
    //int last_edges_fixed = 0;
    float tolerance = stl.stats.shortest_edge;
    float increment = stl.stats.bounding_diameter / 10000.0;
    int iterations = 2;
    if (stl.stats.connected_facets_3_edge < stl.stats.number_of_facets) {
        for (int i = 0; i < iterations; i++) {
            if (stl.stats.connected_facets_3_edge < stl.stats.number_of_facets) {
                //printf("Checking nearby. Tolerance= %f Iteration=%d of %d...", tolerance, i + 1, iterations);
                stl_check_facets_nearby(&stl, tolerance);
                //printf("  Fixed %d edges.\n", stl.stats.edges_fixed - last_edges_fixed);
                //last_edges_fixed = stl.stats.edges_fixed;
                tolerance += increment;
            } else {
                break;
            }
        }
    }
    
    // remove_unconnected
    if (stl.stats.connected_facets_3_edge <  stl.stats.number_of_facets) {
        stl_remove_unconnected_facets(&stl);
    }
    
    // fill_holes
    if (stl.stats.connected_facets_3_edge < stl.stats.number_of_facets) {
        stl_fill_holes(&stl);
        stl_clear_error(&stl);
    }
    
    // normal_directions
    stl_fix_normal_directions(&stl);
    
    // normal_values
    stl_fix_normal_values(&stl);
    
    // always calculate the volume and reverse all normals if volume is negative
    stl_calculate_volume(&stl);
    
    // neighbors
    stl_verify_neighbors(&stl);
    
    this->repaired = true;
}

void
TriangleMesh::reset_repair_stats() {
    this->stl.stats.degenerate_facets   = 0;
    this->stl.stats.edges_fixed         = 0;
    this->stl.stats.facets_removed      = 0;
    this->stl.stats.facets_added        = 0;
    this->stl.stats.facets_reversed     = 0;
    this->stl.stats.backwards_edges     = 0;
    this->stl.stats.normals_fixed       = 0;
}

bool
TriangleMesh::needed_repair() const
{
    return this->stl.stats.degenerate_facets    > 0
        || this->stl.stats.edges_fixed          > 0
        || this->stl.stats.facets_removed       > 0
        || this->stl.stats.facets_added         > 0
        || this->stl.stats.facets_reversed      > 0
        || this->stl.stats.backwards_edges      > 0;
}

size_t
TriangleMesh::facets_count() const
{
    return this->stl.stats.number_of_facets;
}

void
TriangleMesh::WriteOBJFile(char* output_file) {
    stl_generate_shared_vertices(&stl);
    stl_write_obj(&stl, output_file);
}

void TriangleMesh::scale(float factor)
{
    stl_scale(&(this->stl), factor);
    stl_invalidate_shared_vertices(&this->stl);
}

void TriangleMesh::scale(const Pointf3 &versor)
{
    float fversor[3];
    fversor[0] = versor.x;
    fversor[1] = versor.y;
    fversor[2] = versor.z;
    stl_scale_versor(&this->stl, fversor);
    stl_invalidate_shared_vertices(&this->stl);
}

void TriangleMesh::translate(float x, float y, float z)
{
    if (x == 0.f && y == 0.f && z == 0.f)
        return;
    stl_translate_relative(&(this->stl), x, y, z);
    stl_invalidate_shared_vertices(&this->stl);
}

void TriangleMesh::rotate(float angle, const Axis &axis)
{
    if (angle == 0.f)
        return;

    // admesh uses degrees
    angle = Slic3r::Geometry::rad2deg(angle);
    
    if (axis == X) {
        stl_rotate_x(&(this->stl), angle);
    } else if (axis == Y) {
        stl_rotate_y(&(this->stl), angle);
    } else if (axis == Z) {
        stl_rotate_z(&(this->stl), angle);
    }
    stl_invalidate_shared_vertices(&this->stl);
}

void TriangleMesh::rotate_x(float angle)
{
    this->rotate(angle, X);
}

void TriangleMesh::rotate_y(float angle)
{
    this->rotate(angle, Y);
}

void TriangleMesh::rotate_z(float angle)
{
    this->rotate(angle, Z);
}

void TriangleMesh::mirror(const Axis &axis)
{
    if (axis == X) {
        stl_mirror_yz(&this->stl);
    } else if (axis == Y) {
        stl_mirror_xz(&this->stl);
    } else if (axis == Z) {
        stl_mirror_xy(&this->stl);
    }
    stl_invalidate_shared_vertices(&this->stl);
}

void TriangleMesh::mirror_x()
{
    this->mirror(X);
}

void TriangleMesh::mirror_y()
{
    this->mirror(Y);
}

void TriangleMesh::mirror_z()
{
    this->mirror(Z);
}

void TriangleMesh::align_to_origin()
{
    this->translate(
        -(this->stl.stats.min.x),
        -(this->stl.stats.min.y),
        -(this->stl.stats.min.z)
    );
}

void TriangleMesh::rotate(double angle, Point* center)
{
    if (angle == 0.)
        return;
    this->translate(float(-center->x), float(-center->y), 0);
    stl_rotate_z(&(this->stl), (float)angle);
    this->translate(float(+center->x), float(+center->y), 0);
}

bool TriangleMesh::has_multiple_patches() const
{
    // we need neighbors
    if (!this->repaired) CONFESS("split() requires repair()");
    
    if (this->stl.stats.number_of_facets == 0)
        return false;

    std::vector<int>  facet_queue(this->stl.stats.number_of_facets, 0);
    std::vector<char> facet_visited(this->stl.stats.number_of_facets, false);
    int               facet_queue_cnt = 1;
    facet_queue[0] = 0;
    facet_visited[0] = true;
    while (facet_queue_cnt > 0) {
        int facet_idx = facet_queue[-- facet_queue_cnt];
        facet_visited[facet_idx] = true;
        for (int j = 0; j < 3; ++ j) {
            int neighbor_idx = this->stl.neighbors_start[facet_idx].neighbor[j];
            if (! facet_visited[neighbor_idx])
                facet_queue[facet_queue_cnt ++] = neighbor_idx;
        }
    }

    // If any of the face was not visited at the first time, return "multiple bodies".
    for (int facet_idx = 0; facet_idx < this->stl.stats.number_of_facets; ++ facet_idx)
        if (! facet_visited[facet_idx])
            return true;
    return false;
}

size_t TriangleMesh::number_of_patches() const
{
    // we need neighbors
    if (!this->repaired) CONFESS("split() requires repair()");
    
    if (this->stl.stats.number_of_facets == 0)
        return false;

    std::vector<int>  facet_queue(this->stl.stats.number_of_facets, 0);
    std::vector<char> facet_visited(this->stl.stats.number_of_facets, false);
    int               facet_queue_cnt = 0;
    size_t            num_bodies = 0;
    for (;;) {
        // Find a seeding triangle for a new body.
        int facet_idx = 0;
        for (; facet_idx < this->stl.stats.number_of_facets; ++ facet_idx)
            if (! facet_visited[facet_idx]) {
                // A seed triangle was found.
                facet_queue[facet_queue_cnt ++] = facet_idx;
                facet_visited[facet_idx] = true;
                break;
            }
        if (facet_idx == this->stl.stats.number_of_facets)
            // No seed found.
            break;
        ++ num_bodies;
        while (facet_queue_cnt > 0) {
            int facet_idx = facet_queue[-- facet_queue_cnt];
            facet_visited[facet_idx] = true;
            for (int j = 0; j < 3; ++ j) {
                int neighbor_idx = this->stl.neighbors_start[facet_idx].neighbor[j];
                if (! facet_visited[neighbor_idx])
                    facet_queue[facet_queue_cnt ++] = neighbor_idx;
            }
        }
    }

    return num_bodies;
}

TriangleMeshPtrs
TriangleMesh::split() const
{
    TriangleMeshPtrs meshes;
    std::set<int> seen_facets;
    
    // we need neighbors
    if (!this->repaired) CONFESS("split() requires repair()");
    
    // loop while we have remaining facets
    while (1) {
        // get the first facet
        std::queue<int> facet_queue;
        std::deque<int> facets;
        for (int facet_idx = 0; facet_idx < this->stl.stats.number_of_facets; facet_idx++) {
            if (seen_facets.find(facet_idx) == seen_facets.end()) {
                // if facet was not seen put it into queue and start searching
                facet_queue.push(facet_idx);
                break;
            }
        }
        if (facet_queue.empty()) break;
        
        while (!facet_queue.empty()) {
            int facet_idx = facet_queue.front();
            facet_queue.pop();
            if (seen_facets.find(facet_idx) != seen_facets.end()) continue;
            facets.push_back(facet_idx);
            for (int j = 0; j <= 2; j++) {
                facet_queue.push(this->stl.neighbors_start[facet_idx].neighbor[j]);
            }
            seen_facets.insert(facet_idx);
        }
        
        TriangleMesh* mesh = new TriangleMesh;
        meshes.push_back(mesh);
        mesh->stl.stats.type = inmemory;
        mesh->stl.stats.number_of_facets = facets.size();
        mesh->stl.stats.original_num_facets = mesh->stl.stats.number_of_facets;
        stl_clear_error(&mesh->stl);
        stl_allocate(&mesh->stl);
        
        int first = 1;
        for (std::deque<int>::const_iterator facet = facets.begin(); facet != facets.end(); ++facet) {
            mesh->stl.facet_start[facet - facets.begin()] = this->stl.facet_start[*facet];
            stl_facet_stats(&mesh->stl, this->stl.facet_start[*facet], first);
            first = 0;
        }
    }
    
    return meshes;
}

void
TriangleMesh::merge(const TriangleMesh &mesh)
{
    // reset stats and metadata
    int number_of_facets = this->stl.stats.number_of_facets;
    stl_invalidate_shared_vertices(&this->stl);
    this->repaired = false;
    
    // update facet count and allocate more memory
    this->stl.stats.number_of_facets = number_of_facets + mesh.stl.stats.number_of_facets;
    this->stl.stats.original_num_facets = this->stl.stats.number_of_facets;
    stl_reallocate(&this->stl);
    
    // copy facets
    for (int i = 0; i < mesh.stl.stats.number_of_facets; i++) {
        this->stl.facet_start[number_of_facets + i] = mesh.stl.facet_start[i];
    }
    
    // update size
    stl_get_size(&this->stl);
}

/* this will return scaled ExPolygons */
ExPolygons
TriangleMesh::horizontal_projection() const
{
    Polygons pp;
    pp.reserve(this->stl.stats.number_of_facets);
    for (int i = 0; i < this->stl.stats.number_of_facets; i++) {
        stl_facet* facet = &this->stl.facet_start[i];
        Polygon p;
        p.points.resize(3);
        p.points[0] = Point(facet->vertex[0].x / SCALING_FACTOR, facet->vertex[0].y / SCALING_FACTOR);
        p.points[1] = Point(facet->vertex[1].x / SCALING_FACTOR, facet->vertex[1].y / SCALING_FACTOR);
        p.points[2] = Point(facet->vertex[2].x / SCALING_FACTOR, facet->vertex[2].y / SCALING_FACTOR);
        p.make_counter_clockwise();  // do this after scaling, as winding order might change while doing that
        pp.push_back(p);
    }
    
    // the offset factor was tuned using groovemount.stl
    return union_ex(offset(pp, 0.01 / SCALING_FACTOR), true);
}

Polygon
TriangleMesh::convex_hull()
{
    this->require_shared_vertices();
    Points pp;
    pp.reserve(this->stl.stats.shared_vertices);
    for (int i = 0; i < this->stl.stats.shared_vertices; i++) {
        stl_vertex* v = &this->stl.v_shared[i];
        pp.push_back(Point(v->x / SCALING_FACTOR, v->y / SCALING_FACTOR));
    }
    return Slic3r::Geometry::convex_hull(pp);
}

BoundingBoxf3
TriangleMesh::bounding_box() const
{
    BoundingBoxf3 bb;
    bb.min.x = this->stl.stats.min.x;
    bb.min.y = this->stl.stats.min.y;
    bb.min.z = this->stl.stats.min.z;
    bb.max.x = this->stl.stats.max.x;
    bb.max.y = this->stl.stats.max.y;
    bb.max.z = this->stl.stats.max.z;
    return bb;
}

void
TriangleMesh::require_shared_vertices()
{
    if (!this->repaired) this->repair();
    if (this->stl.v_shared == NULL) stl_generate_shared_vertices(&(this->stl));
}


TriangleMeshSlicer::TriangleMeshSlicer(TriangleMesh* _mesh) : 
    mesh(_mesh)
{
    _mesh->require_shared_vertices();
    facets_edges.assign(_mesh->stl.stats.number_of_facets * 3, -1);
    v_scaled_shared.assign(_mesh->stl.v_shared, _mesh->stl.v_shared + _mesh->stl.stats.shared_vertices);
    // Scale the copied vertices.
    for (int i = 0; i < this->mesh->stl.stats.shared_vertices; ++ i) {
        this->v_scaled_shared[i].x /= float(SCALING_FACTOR);
        this->v_scaled_shared[i].y /= float(SCALING_FACTOR);
        this->v_scaled_shared[i].z /= float(SCALING_FACTOR);
    }

    // build a table to map a facet_idx to its three edge indices
    // a_id,b_id => edge_idx
    struct pairhash {
        std::size_t operator()(const std::pair<int, int> &x) const 
            { return std::hash<int>()(x.first) ^ std::hash<int>()(x.second); }
    };
    std::unordered_map<std::pair<int, int>, int, pairhash>  edges_map;
    int                                                     num_edges = 0;
    for (int facet_idx = 0; facet_idx < this->mesh->stl.stats.number_of_facets; ++ facet_idx) {
        for (int i = 0; i < 3; ++ i) {
            // Vertex indices of th ith edge of facet_idx.
            int a_id = this->mesh->stl.v_indices[facet_idx].vertex[i];
            int b_id = this->mesh->stl.v_indices[facet_idx].vertex[(i + 1) % 3];
            int edge_idx;
            auto my_edge = edges_map.find(std::make_pair(b_id, a_id));
            if (my_edge == edges_map.end()) {
                /* admesh can assign the same edge ID to more than two facets (which is 
                   still topologically correct), so we have to search for a duplicate of 
                   this edge too in case it was already seen in this orientation */
                my_edge = edges_map.find(std::make_pair(a_id, b_id));
                if (my_edge != edges_map.end()) {
                    edge_idx = my_edge->second;
                } else {
                    // edge isn't listed in table, so we insert it
                    edges_map[std::make_pair(a_id, b_id)] = edge_idx = num_edges ++;
                }
            } else
                edge_idx = my_edge->second;
            this->facets_edges[facet_idx * 3 + i] = edge_idx;

            #ifdef SLIC3R_TRIANGLEMESH_DEBUG
            printf("  [facet %d, edge %d] a_id = %d, b_id = %d   --> edge %d\n", facet_idx, i, a_id, b_id, edge_idx);
            #endif
        }
    }
}

void
TriangleMeshSlicer::slice(const std::vector<float> &z, std::vector<Polygons>* layers) const
{
    /*
       This method gets called with a list of unscaled Z coordinates and outputs
       a vector pointer having the same number of items as the original list.
       Each item is a vector of polygons created by slicing our mesh at the 
       given heights.
       
       This method should basically combine the behavior of the existing
       Perl methods defined in lib/Slic3r/TriangleMesh.pm:
       
       - analyze(): this creates the 'facets_edges' and the 'edges_facets'
            tables (we don't need the 'edges' table)
       
       - slice_facet(): this has to be done for each facet. It generates 
            intersection lines with each plane identified by the Z list.
            The get_layer_range() binary search used to identify the Z range
            of the facet is already ported to C++ (see Object.xsp)
       
       - make_loops(): this has to be done for each layer. It creates polygons
            from the lines generated by the previous step.
        
        At the end, we free the tables generated by analyze() as we don't 
        need them anymore.
        
        NOTE: this method accepts a vector of floats because the mesh coordinate
        type is float.
    */
    
    std::vector<IntersectionLines> lines(z.size());
    {
        boost::mutex lines_mutex;
        parallelize<int>(
            0,
            this->mesh->stl.stats.number_of_facets-1,
            boost::bind(&TriangleMeshSlicer::_slice_do, this, _1, &lines, &lines_mutex, z)
        );
    }
    
    // v_scaled_shared could be freed here
    
    // build loops
    layers->resize(z.size());
    parallelize<size_t>(
        0,
        lines.size()-1,
        boost::bind(&TriangleMeshSlicer::_make_loops_do, this, _1, &lines, layers)
    );
}

void TriangleMeshSlicer::_slice_do(size_t facet_idx, std::vector<IntersectionLines>* lines, boost::mutex* lines_mutex, 
    const std::vector<float> &z) const
{
    const stl_facet &facet = this->mesh->stl.facet_start[facet_idx];
    
    // find facet extents
    const float min_z = fminf(facet.vertex[0].z, fminf(facet.vertex[1].z, facet.vertex[2].z));
    const float max_z = fmaxf(facet.vertex[0].z, fmaxf(facet.vertex[1].z, facet.vertex[2].z));
    
    #ifdef SLIC3R_DEBUG
    printf("\n==> FACET %d (%f,%f,%f - %f,%f,%f - %f,%f,%f):\n", facet_idx,
        facet.vertex[0].x, facet.vertex[0].y, facet.vertex[0].z,
        facet.vertex[1].x, facet.vertex[1].y, facet.vertex[1].z,
        facet.vertex[2].x, facet.vertex[2].y, facet.vertex[2].z);
    printf("z: min = %.2f, max = %.2f\n", min_z, max_z);
    #endif
    
    // find layer extents
    std::vector<float>::const_iterator min_layer, max_layer;
    min_layer = std::lower_bound(z.begin(), z.end(), min_z); // first layer whose slice_z is >= min_z
    max_layer = std::upper_bound(z.begin() + (min_layer - z.begin()), z.end(), max_z) - 1; // last layer whose slice_z is <= max_z
    #ifdef SLIC3R_DEBUG
    printf("layers: min = %d, max = %d\n", (int)(min_layer - z.begin()), (int)(max_layer - z.begin()));
    #endif
    
    for (std::vector<float>::const_iterator it = min_layer; it != max_layer + 1; ++it) {
        std::vector<float>::size_type layer_idx = it - z.begin();
        IntersectionLine il;
        if (this->slice_facet(*it / SCALING_FACTOR, facet, facet_idx, min_z, max_z, &il)) {
            boost::lock_guard<boost::mutex> l(*lines_mutex);
            if (il.edge_type == feHorizontal) {
                // Insert all three edges of the face.
                const int *vertices = this->mesh->stl.v_indices[facet_idx].vertex;
                const bool reverse  = this->mesh->stl.facet_start[facet_idx].normal.z < 0;
                for (int j = 0; j < 3; ++ j) {
                    int               a_id     = vertices[j % 3];
                    int               b_id     = vertices[(j+1) % 3];
                    if (reverse)
                        std::swap(a_id, b_id);
                    const stl_vertex *a = &this->v_scaled_shared[a_id];
                    const stl_vertex *b = &this->v_scaled_shared[b_id];
                    il.a.x    = a->x;
                    il.a.y    = a->y;
                    il.b.x    = b->x;
                    il.b.y    = b->y;
                    il.a_id   = a_id;
                    il.b_id   = b_id;
                    (*lines)[layer_idx].push_back(il);
                }
            } else
                (*lines)[layer_idx].push_back(il);
        }
    }
}

void
TriangleMeshSlicer::slice(const std::vector<float> &z, std::vector<ExPolygons>* layers) const
{
    std::vector<Polygons> layers_p;
    this->slice(z, &layers_p);
    
    layers->resize(z.size());
    for (std::vector<Polygons>::const_iterator loops = layers_p.begin(); loops != layers_p.end(); ++loops) {
        #ifdef SLIC3R_TRIANGLEMESH_DEBUG
        size_t layer_id = loops - layers_p.begin();
        printf("Layer " PRINTF_ZU " (slice_z = %.2f):\n", layer_id, z[layer_id]);
        #endif
        
        this->make_expolygons(*loops, &(*layers)[ loops - layers_p.begin() ]);
    }
}

// Return true, if the facet has been sliced and line_out has been filled.
bool TriangleMeshSlicer::slice_facet(
    float slice_z, const stl_facet &facet, const int facet_idx,
    const float min_z, const float max_z, 
    IntersectionLine *line_out) const
{
    IntersectionPoint points[3];
    size_t            num_points = 0;
    size_t            points_on_layer[3];
    size_t            num_points_on_layer = 0;
    
    // Reorder vertices so that the first one is the one with lowest Z.
    // This is needed to get all intersection lines in a consistent order
    // (external on the right of the line)
    int i = (facet.vertex[1].z == min_z) ? 1 : ((facet.vertex[2].z == min_z) ? 2 : 0);
    for (int j = i; j - i < 3; ++ j) {  // loop through facet edges
        int               edge_id  = this->facets_edges[facet_idx * 3 + (j % 3)];
        const int        *vertices = this->mesh->stl.v_indices[facet_idx].vertex;
        int               a_id     = vertices[j % 3];
        int               b_id     = vertices[(j+1) % 3];
        const stl_vertex *a = &this->v_scaled_shared[a_id];
        const stl_vertex *b = &this->v_scaled_shared[b_id];
        
        // Is edge or face aligned with the cutting plane?
        if (a->z == slice_z && b->z == slice_z) {
            // Edge is horizontal and belongs to the current layer.
            const stl_vertex &v0 = this->v_scaled_shared[vertices[0]];
            const stl_vertex &v1 = this->v_scaled_shared[vertices[1]];
            const stl_vertex &v2 = this->v_scaled_shared[vertices[2]];
            if (min_z == max_z) {
                // All three vertices are aligned with slice_z.
                line_out->edge_type = feHorizontal;
                if (this->mesh->stl.facet_start[facet_idx].normal.z < 0) {
                    // If normal points downwards this is a bottom horizontal facet so we reverse its point order.
                    std::swap(a, b);
                    std::swap(a_id, b_id);
                }
            } else if (v0.z < slice_z || v1.z < slice_z || v2.z < slice_z) {
                // Two vertices are aligned with the cutting plane, the third vertex is below the cutting plane.
                line_out->edge_type = feTop;
                std::swap(a, b);
                std::swap(a_id, b_id);
            } else {
                // Two vertices are aligned with the cutting plane, the third vertex is above the cutting plane.
                line_out->edge_type = feBottom;
            }
            line_out->a.x    = a->x;
            line_out->a.y    = a->y;
            line_out->b.x    = b->x;
            line_out->b.y    = b->y;
            line_out->a_id   = a_id;
            line_out->b_id   = b_id;
            return true;
        }

        if (a->z == slice_z) {
            // Only point a alings with the cutting plane.
            points_on_layer[num_points_on_layer ++] = num_points;
            IntersectionPoint &point = points[num_points ++];
            point.x         = a->x;
            point.y         = a->y;
            point.point_id  = a_id;
        } else if (b->z == slice_z) {
            // Only point b alings with the cutting plane.
            points_on_layer[num_points_on_layer ++] = num_points;
            IntersectionPoint &point = points[num_points ++];
            point.x         = b->x;
            point.y         = b->y;
            point.point_id  = b_id;
        } else if ((a->z < slice_z && b->z > slice_z) || (b->z < slice_z && a->z > slice_z)) {
            // A general case. The face edge intersects the cutting plane. Calculate the intersection point.
            IntersectionPoint &point = points[num_points ++];
            point.x         = b->x + (a->x - b->x) * (slice_z - b->z) / (a->z - b->z);
            point.y         = b->y + (a->y - b->y) * (slice_z - b->z) / (a->z - b->z);
            point.edge_id   = edge_id;
        }
    }

    // We can't have only one point on layer because each vertex gets detected
    // twice (once for each edge), and we can't have three points on layer,
    // because we assume this code is not getting called for horizontal facets.
    assert(num_points_on_layer == 0 || num_points_on_layer == 2);
    if (num_points_on_layer > 0) {
        assert(points[points_on_layer[0]].point_id == points[points_on_layer[1]].point_id);
        assert(num_points == 2 || num_points == 3);
        if (num_points < 3)
            // This triangle touches the cutting plane with a single vertex. Ignore it.
            return false;
        // Erase one of the duplicate points.
        -- num_points;
        for (int i = points_on_layer[1]; i < num_points; ++ i)
            points[i] = points[i + 1];
    }
    
    // Facets must intersect each plane 0 or 2 times.
    assert(num_points == 0 || num_points == 2);
    if (num_points == 2) {
        line_out->edge_type  = feNone;
        line_out->a          = (Point)points[1];
        line_out->b          = (Point)points[0];
        line_out->a_id       = points[1].point_id;
        line_out->b_id       = points[0].point_id;
        line_out->edge_a_id  = points[1].edge_id;
        line_out->edge_b_id  = points[0].edge_id;
        return true;
    }
    return false;
}

void
TriangleMeshSlicer::_make_loops_do(size_t i, std::vector<IntersectionLines>* lines, std::vector<Polygons>* layers) const
{
    this->make_loops((*lines)[i], &(*layers)[i]);
}

void TriangleMeshSlicer::make_loops(std::vector<IntersectionLine> &lines, Polygons* loops) const
{
    // Remove tangent edges.
    //FIXME This is O(n^2) in rare cases when many faces intersect the cutting plane.
    for (IntersectionLines::iterator line = lines.begin(); line != lines.end(); ++ line)
        if (! line->skip && line->edge_type != feNone) {
            // This line is af facet edge. There may be a duplicate line with the same end vertices.
            // If the line is is an edge connecting two facets, find another facet edge
            // having the same endpoints but in reverse order.
            for (IntersectionLines::iterator line2 = line + 1; line2 != lines.end(); ++ line2)
                if (! line2->skip && line2->edge_type != feNone) {
                    // Are these facets adjacent? (sharing a common edge on this layer)
                    if (line->a_id == line2->a_id && line->b_id == line2->b_id) {
                        line2->skip = true;
                        /* if they are both oriented upwards or downwards (like a 'V')
                           then we can remove both edges from this layer since it won't 
                           affect the sliced shape */
                        /* if one of them is oriented upwards and the other is oriented
                           downwards, let's only keep one of them (it doesn't matter which
                           one since all 'top' lines were reversed at slicing) */
                        if (line->edge_type == line2->edge_type) {
                            line->skip = true;
                            break;
                        }
                    } else if (line->a_id == line2->b_id && line->b_id == line2->a_id) {
                        /* if this edge joins two horizontal facets, remove both of them */
                        if (line->edge_type == feHorizontal && line2->edge_type == feHorizontal) {
                            line->skip = true;
                            line2->skip = true;
                            break;
                        }
                    }
                }
        }
    
    // build a map of lines by edge_a_id and a_id
    //FIXME replace the vectors of vectors by vectors of indices to a continuous memory.
    std::vector<IntersectionLinePtrs> by_edge_a_id(this->mesh->stl.stats.number_of_facets * 3);
    std::vector<IntersectionLinePtrs> by_a_id(this->mesh->stl.stats.shared_vertices);
    for (IntersectionLines::iterator line = lines.begin(); line != lines.end(); ++ line) {
        if (! line->skip) {
            if (line->edge_a_id != -1)
                by_edge_a_id[line->edge_a_id].push_back(&(*line));
            if (line->a_id != -1) 
                by_a_id[line->a_id].push_back(&(*line));
        }
    }

    IntersectionLines::iterator it_line_seed = lines.begin();
    CYCLE: while (1) {
        // take first spare line and start a new loop
        IntersectionLine *first_line = nullptr;
        for (; it_line_seed != lines.end(); ++ it_line_seed)
            if (! it_line_seed->skip) {
                first_line = &(*it_line_seed ++);
                break;
            }
        if (first_line == nullptr)
            break;
        first_line->skip = true;
        Points loop_pts;
        loop_pts.push_back(first_line->a);
        IntersectionLine *last_line = first_line;
        
        /*
        printf("first_line edge_a_id = %d, edge_b_id = %d, a_id = %d, b_id = %d, a = %d,%d, b = %d,%d\n", 
            first_line->edge_a_id, first_line->edge_b_id, first_line->a_id, first_line->b_id,
            first_line->a.x, first_line->a.y, first_line->b.x, first_line->b.y);
        */
        
        for (;;) {
            // find a line starting where last one finishes
            IntersectionLine* next_line = nullptr;
            if (last_line->edge_b_id != -1) {
                IntersectionLinePtrs &candidates = by_edge_a_id[last_line->edge_b_id];
                for (IntersectionLinePtrs::iterator lineptr = candidates.begin(); lineptr != candidates.end(); ++ lineptr)
                    if (! (*lineptr)->skip) {
                        next_line = *lineptr;
                        break;
                    }
            }
            if (next_line == nullptr && last_line->b_id != -1) {
                IntersectionLinePtrs &candidates = by_a_id[last_line->b_id];
                for (IntersectionLinePtrs::iterator lineptr = candidates.begin(); lineptr != candidates.end(); ++ lineptr)
                    if (! (*lineptr)->skip) {
                        next_line = *lineptr;
                        break;
                    }
            }
            if (next_line == nullptr) {
                // check whether we closed this loop
                if ((first_line->edge_a_id != -1 && first_line->edge_a_id == last_line->edge_b_id) || 
                    (first_line->a_id      != -1 && first_line->a_id      == last_line->b_id)) {
                    // loop is complete
                    loops->emplace_back(std::move(loop_pts));
                    #ifdef SLIC3R_TRIANGLEMESH_DEBUG
                    printf("  Discovered %s polygon of %d points\n", (p.is_counter_clockwise() ? "ccw" : "cw"), (int)p.points.size());
                    #endif
                    goto CYCLE;
                }
                // we can't close this loop!
                //// push @failed_loops, [@loop];
                //#ifdef SLIC3R_TRIANGLEMESH_DEBUG
                printf("  Unable to close this loop having %d points\n", (int)loop_pts.size());
                //#endif
                goto CYCLE;
            }
            /*
            printf("next_line edge_a_id = %d, edge_b_id = %d, a_id = %d, b_id = %d, a = %d,%d, b = %d,%d\n", 
                next_line->edge_a_id, next_line->edge_b_id, next_line->a_id, next_line->b_id,
                next_line->a.x, next_line->a.y, next_line->b.x, next_line->b.y);
            */
            loop_pts.push_back(next_line->a);
            last_line = next_line;
            next_line->skip = true;
        }
    }
}

// Only used to cut the mesh into two halves.
void TriangleMeshSlicer::make_expolygons_simple(std::vector<IntersectionLine> &lines, ExPolygons* slices) const
{
    assert(slices->empty());

    Polygons loops;
    this->make_loops(lines, &loops);
    
    Polygons holes;
    for (Polygons::const_iterator loop = loops.begin(); loop != loops.end(); ++ loop) {
        if (loop->area() >= 0.) {
            ExPolygon ex;
            ex.contour = *loop;
            slices->push_back(ex);
        } else {
            holes.push_back(*loop);
        }
    }

    // If there are holes, then there should also be outer contours.
    assert(holes.empty() || ! slices->empty());
    if (slices->empty())
        return;
    
    // Assign holes to outer contours.
    for (Polygons::const_iterator hole = holes.begin(); hole != holes.end(); ++ hole) {
        // Find an outer contour to a hole.
        int     slice_idx            = -1;
        double  current_contour_area = std::numeric_limits<double>::max();
        for (ExPolygons::iterator slice = slices->begin(); slice != slices->end(); ++ slice) {
            if (slice->contour.contains(hole->points.front())) {
                double area = slice->contour.area();
                if (area < current_contour_area) {
                    slice_idx = slice - slices->begin();
                    current_contour_area = area;
                }
            }
        }
        // assert(slice_idx != -1);
        if (slice_idx == -1)
            // Ignore this hole.
            continue;
		assert(current_contour_area < std::numeric_limits<double>::max() && current_contour_area >= -hole->area());
		(*slices)[slice_idx].holes.emplace_back(std::move(*hole));
    }

#if 0
    // If the input mesh is not valid, the holes may intersect with the external contour.
    // Rather subtract them from the outer contour.
    Polygons poly;
	for (auto it_slice = slices->begin(); it_slice != slices->end(); ++ it_slice) {
        if (it_slice->holes.empty()) {
            poly.emplace_back(std::move(it_slice->contour));
        } else {
            Polygons contours;
            contours.emplace_back(std::move(it_slice->contour));
            for (auto it = it_slice->holes.begin(); it != it_slice->holes.end(); ++ it)
                it->reverse();
            polygons_append(poly, diff(contours, it_slice->holes));
        }
	}
    // If the input mesh is not valid, the input contours may intersect.
    *slices = union_ex(poly);
#endif

#if 0
    // If the input mesh is not valid, the holes may intersect with the external contour.
    // Rather subtract them from the outer contour.
    ExPolygons poly;
    for (auto it_slice = slices->begin(); it_slice != slices->end(); ++ it_slice) {
        Polygons contours;
        contours.emplace_back(std::move(it_slice->contour));
        for (auto it = it_slice->holes.begin(); it != it_slice->holes.end(); ++ it)
            it->reverse();
        expolygons_append(poly, diff_ex(contours, it_slice->holes));
    }
    // If the input mesh is not valid, the input contours may intersect.
    *slices = std::move(poly);
#endif
}

void TriangleMeshSlicer::make_expolygons(const Polygons &loops, ExPolygons* slices) const
{
    /*
        Input loops are not suitable for evenodd nor nonzero fill types, as we might get
        two consecutive concentric loops having the same winding order - and we have to 
        respect such order. In that case, evenodd would create wrong inversions, and nonzero
        would ignore holes inside two concentric contours.
        So we're ordering loops and collapse consecutive concentric loops having the same 
        winding order.
        TODO: find a faster algorithm for this, maybe with some sort of binary search.
        If we computed a "nesting tree" we could also just remove the consecutive loops
        having the same winding order, and remove the extra one(s) so that we could just
        supply everything to offset() instead of performing several union/diff calls.
    
        we sort by area assuming that the outermost loops have larger area;
        the previous sorting method, based on $b->contains($a->[0]), failed to nest
        loops correctly in some edge cases when original model had overlapping facets
    */

    std::vector<double> area;
    std::vector<size_t> sorted_area;  // vector of indices
    for (Polygons::const_iterator loop = loops.begin(); loop != loops.end(); ++ loop) {
        area.push_back(loop->area());
        sorted_area.push_back(loop - loops.begin());
    }
    
    // outer first
    std::sort(sorted_area.begin(), sorted_area.end(),
        [&area](size_t a, size_t b) { return std::abs(area[a]) > std::abs(area[b]); });

    // we don't perform a safety offset now because it might reverse cw loops
    Polygons p_slices;
    for (std::vector<size_t>::const_iterator loop_idx = sorted_area.begin(); loop_idx != sorted_area.end(); ++ loop_idx) {
        /* we rely on the already computed area to determine the winding order
           of the loops, since the Orientation() function provided by Clipper
           would do the same, thus repeating the calculation */
        Polygons::const_iterator loop = loops.begin() + *loop_idx;
        if (area[*loop_idx] > +EPSILON)
            p_slices.push_back(*loop);
        else if (area[*loop_idx] < -EPSILON)
            //FIXME This is arbitrary and possibly very slow.
            // If the hole is inside a polygon, then there is no need to diff.
            // If the hole intersects a polygon boundary, then diff it, but then
            // there is no guarantee of an ordering of the loops.
            // Maybe we can test for the intersection before running the expensive diff algorithm?
            p_slices = diff(p_slices, *loop);
    }

    // perform a safety offset to merge very close facets (TODO: find test case for this)
    double safety_offset = scale_(0.0499);
    ExPolygons ex_slices = offset2_ex(p_slices, +safety_offset, -safety_offset);
    
    #ifdef SLIC3R_TRIANGLEMESH_DEBUG
    size_t holes_count = 0;
    for (ExPolygons::const_iterator e = ex_slices.begin(); e != ex_slices.end(); ++ e)
        holes_count += e->holes.size();
    printf(PRINTF_ZU " surface(s) having " PRINTF_ZU " holes detected from " PRINTF_ZU " polylines\n",
        ex_slices.size(), holes_count, loops.size());
    #endif
    
    // append to the supplied collection
    expolygons_append(*slices, ex_slices);
}

void TriangleMeshSlicer::make_expolygons(std::vector<IntersectionLine> &lines, ExPolygons* slices) const
{
    Polygons pp;
    this->make_loops(lines, &pp);
    this->make_expolygons(pp, slices);
}

void TriangleMeshSlicer::cut(float z, TriangleMesh* upper, TriangleMesh* lower) const
{
    IntersectionLines upper_lines, lower_lines;
    
    float scaled_z = scale_(z);
    for (int facet_idx = 0; facet_idx < this->mesh->stl.stats.number_of_facets; ++ facet_idx) {
        stl_facet* facet = &this->mesh->stl.facet_start[facet_idx];
        
        // find facet extents
        float min_z = std::min(facet->vertex[0].z, std::min(facet->vertex[1].z, facet->vertex[2].z));
        float max_z = std::max(facet->vertex[0].z, std::max(facet->vertex[1].z, facet->vertex[2].z));
        
        // intersect facet with cutting plane
        IntersectionLine line;
        if (this->slice_facet(scaled_z, *facet, facet_idx, min_z, max_z, &line)) {
            // Save intersection lines for generating correct triangulations.
            if (line.edge_type == feTop) {
                lower_lines.push_back(line);
            } else if (line.edge_type == feBottom) {
                upper_lines.push_back(line);
            } else if (line.edge_type != feHorizontal) {
                lower_lines.push_back(line);
                upper_lines.push_back(line);
            }
        }
        
        if (min_z > z || (min_z == z && max_z > z)) {
            // facet is above the cut plane and does not belong to it
            if (upper != NULL) stl_add_facet(&upper->stl, facet);
        } else if (max_z < z || (max_z == z && min_z < z)) {
            // facet is below the cut plane and does not belong to it
            if (lower != NULL) stl_add_facet(&lower->stl, facet);
        } else if (min_z < z && max_z > z) {
            // Facet is cut by the slicing plane.

            // look for the vertex on whose side of the slicing plane there are no other vertices
            int isolated_vertex;
            if ( (facet->vertex[0].z > z) == (facet->vertex[1].z > z) ) {
                isolated_vertex = 2;
            } else if ( (facet->vertex[1].z > z) == (facet->vertex[2].z > z) ) {
                isolated_vertex = 0;
            } else {
                isolated_vertex = 1;
            }
            
            // get vertices starting from the isolated one
            stl_vertex* v0 = &facet->vertex[isolated_vertex];
            stl_vertex* v1 = &facet->vertex[(isolated_vertex+1) % 3];
            stl_vertex* v2 = &facet->vertex[(isolated_vertex+2) % 3];
            
            // intersect v0-v1 and v2-v0 with cutting plane and make new vertices
            stl_vertex v0v1, v2v0;
            v0v1.x = v1->x + (v0->x - v1->x) * (z - v1->z) / (v0->z - v1->z);
            v0v1.y = v1->y + (v0->y - v1->y) * (z - v1->z) / (v0->z - v1->z);
            v0v1.z = z;
            v2v0.x = v2->x + (v0->x - v2->x) * (z - v2->z) / (v0->z - v2->z);
            v2v0.y = v2->y + (v0->y - v2->y) * (z - v2->z) / (v0->z - v2->z);
            v2v0.z = z;
            
            // build the triangular facet
            stl_facet triangle;
            triangle.normal = facet->normal;
            triangle.vertex[0] = *v0;
            triangle.vertex[1] = v0v1;
            triangle.vertex[2] = v2v0;
            
            // build the facets forming a quadrilateral on the other side
            stl_facet quadrilateral[2];
            quadrilateral[0].normal = facet->normal;
            quadrilateral[0].vertex[0] = *v1;
            quadrilateral[0].vertex[1] = *v2;
            quadrilateral[0].vertex[2] = v0v1;
            quadrilateral[1].normal = facet->normal;
            quadrilateral[1].vertex[0] = *v2;
            quadrilateral[1].vertex[1] = v2v0;
            quadrilateral[1].vertex[2] = v0v1;
            
            if (v0->z > z) {
                if (upper != NULL) stl_add_facet(&upper->stl, &triangle);
                if (lower != NULL) {
                    stl_add_facet(&lower->stl, &quadrilateral[0]);
                    stl_add_facet(&lower->stl, &quadrilateral[1]);
                }
            } else {
                if (upper != NULL) {
                    stl_add_facet(&upper->stl, &quadrilateral[0]);
                    stl_add_facet(&upper->stl, &quadrilateral[1]);
                }
                if (lower != NULL) stl_add_facet(&lower->stl, &triangle);
            }
        }
    }
    
    // triangulate holes of upper mesh
    if (upper != NULL) {
        // compute shape of section
        ExPolygons section;
        this->make_expolygons_simple(upper_lines, &section);
        
        // triangulate section
        Polygons triangles;
        for (ExPolygons::const_iterator expolygon = section.begin(); expolygon != section.end(); ++expolygon)
            expolygon->triangulate_p2t(&triangles);
        
        // convert triangles to facets and append them to mesh
        for (Polygons::const_iterator polygon = triangles.begin(); polygon != triangles.end(); ++polygon) {
            Polygon p = *polygon;
            p.reverse();
            stl_facet facet;
            facet.normal.x = 0;
            facet.normal.y = 0;
            facet.normal.z = -1;
            for (size_t i = 0; i <= 2; ++i) {
                facet.vertex[i].x = unscale(p.points[i].x);
                facet.vertex[i].y = unscale(p.points[i].y);
                facet.vertex[i].z = z;
            }
            stl_add_facet(&upper->stl, &facet);
        }
    }
    
    // triangulate holes of lower mesh
    if (lower != NULL) {
        // compute shape of section
        ExPolygons section;
        this->make_expolygons_simple(lower_lines, &section);
        
        // triangulate section
        Polygons triangles;
        for (ExPolygons::const_iterator expolygon = section.begin(); expolygon != section.end(); ++expolygon)
            expolygon->triangulate_p2t(&triangles);
        
        // convert triangles to facets and append them to mesh
        for (Polygons::const_iterator polygon = triangles.begin(); polygon != triangles.end(); ++polygon) {
            stl_facet facet;
            facet.normal.x = 0;
            facet.normal.y = 0;
            facet.normal.z = 1;
            for (size_t i = 0; i <= 2; ++i) {
                facet.vertex[i].x = unscale(polygon->points[i].x);
                facet.vertex[i].y = unscale(polygon->points[i].y);
                facet.vertex[i].z = z;
            }
            stl_add_facet(&lower->stl, &facet);
        }
    }
    
    // Update the bounding box / sphere of the new meshes.
    stl_get_size(&upper->stl);
    stl_get_size(&lower->stl);
}

// Generate the vertex list for a cube solid of arbitrary size in X/Y/Z.
TriangleMesh make_cube(double x, double y, double z) {
    Pointf3 pv[8] = { 
        Pointf3(x, y, 0), Pointf3(x, 0, 0), Pointf3(0, 0, 0), 
        Pointf3(0, y, 0), Pointf3(x, y, z), Pointf3(0, y, z), 
        Pointf3(0, 0, z), Pointf3(x, 0, z) 
    };
    Point3 fv[12] = { 
        Point3(0, 1, 2), Point3(0, 2, 3), Point3(4, 5, 6), 
        Point3(4, 6, 7), Point3(0, 4, 7), Point3(0, 7, 1), 
        Point3(1, 7, 6), Point3(1, 6, 2), Point3(2, 6, 5), 
        Point3(2, 5, 3), Point3(4, 0, 3), Point3(4, 3, 5) 
    };

    std::vector<Point3> facets(&fv[0], &fv[0]+12);
    Pointf3s vertices(&pv[0], &pv[0]+8);

    TriangleMesh mesh(vertices ,facets);
    return mesh;
}

// Generate the mesh for a cylinder and return it, using 
// the generated angle to calculate the top mesh triangles.
// Default is 360 sides, angle fa is in radians.
TriangleMesh make_cylinder(double r, double h, double fa) {
    Pointf3s vertices;
    std::vector<Point3> facets;

    // 2 special vertices, top and bottom center, rest are relative to this
    vertices.push_back(Pointf3(0.0, 0.0, 0.0));
    vertices.push_back(Pointf3(0.0, 0.0, h));

    // adjust via rounding to get an even multiple for any provided angle.
    double angle = (2*PI / floor(2*PI / fa));

    // for each line along the polygon approximating the top/bottom of the
    // circle, generate four points and four facets (2 for the wall, 2 for the
    // top and bottom.
    // Special case: Last line shares 2 vertices with the first line.
    unsigned id = vertices.size() - 1;
    vertices.push_back(Pointf3(sin(0) * r , cos(0) * r, 0));
    vertices.push_back(Pointf3(sin(0) * r , cos(0) * r, h));
    for (double i = 0; i < 2*PI; i+=angle) {
        Pointf3 b(0, r, 0);
        Pointf3 t(0, r, h);
        b.rotate(i, Pointf3(0,0,0)); 
        t.rotate(i, Pointf3(0,0,h));
        vertices.push_back(b);
        vertices.push_back(t);
        id = vertices.size() - 1;
        facets.push_back(Point3( 0, id - 1, id - 3)); // top
        facets.push_back(Point3(id,      1, id - 2)); // bottom
        facets.push_back(Point3(id, id - 2, id - 3)); // upper-right of side
        facets.push_back(Point3(id, id - 3, id - 1)); // bottom-left of side
    }
    // Connect the last set of vertices with the first.
    facets.push_back(Point3( 2, 0, id - 1));
    facets.push_back(Point3( 1, 3,     id));
    facets.push_back(Point3(id, 3,      2));
    facets.push_back(Point3(id, 2, id - 1));
    
    TriangleMesh mesh(vertices, facets);
    return mesh;
}

// Generates mesh for a sphere centered about the origin, using the generated angle
// to determine the granularity. 
// Default angle is 1 degree.
TriangleMesh make_sphere(double rho, double fa) {
    Pointf3s vertices;
    std::vector<Point3> facets;

    // Algorithm: 
    // Add points one-by-one to the sphere grid and form facets using relative coordinates.
    // Sphere is composed effectively of a mesh of stacked circles.

    // adjust via rounding to get an even multiple for any provided angle.
    double angle = (2*PI / floor(2*PI / fa));

    // Ring to be scaled to generate the steps of the sphere
    std::vector<double> ring;
    for (double i = 0; i < 2*PI; i+=angle) {
        ring.push_back(i);
    }
    const size_t steps = ring.size(); 
    const double increment = (double)(1.0 / (double)steps);

    // special case: first ring connects to 0,0,0
    // insert and form facets.
    vertices.push_back(Pointf3(0.0, 0.0, -rho));
    size_t id = vertices.size();
    for (size_t i = 0; i < ring.size(); i++) {
        // Fixed scaling 
        const double z = -rho + increment*rho*2.0;
        // radius of the circle for this step.
        const double r = sqrt(abs(rho*rho - z*z));
        Pointf3 b(0, r, z);
        b.rotate(ring[i], Pointf3(0,0,z)); 
        vertices.push_back(b);
        if (i == 0) {
            facets.push_back(Point3(1, 0, ring.size()));
        } else {
            facets.push_back(Point3(id, 0, id - 1));
        }
        id++;
    }

    // General case: insert and form facets for each step, joining it to the ring below it.
    for (size_t s = 2; s < steps - 1; s++) {
        const double z = -rho + increment*(double)s*2.0*rho;
        const double r = sqrt(abs(rho*rho - z*z));

        for (size_t i = 0; i < ring.size(); i++) {
            Pointf3 b(0, r, z);
            b.rotate(ring[i], Pointf3(0,0,z)); 
            vertices.push_back(b);
            if (i == 0) {
                // wrap around
                facets.push_back(Point3(id + ring.size() - 1 , id, id - 1)); 
                facets.push_back(Point3(id, id - ring.size(),  id - 1)); 
            } else {
                facets.push_back(Point3(id , id - ring.size(), (id - 1) - ring.size())); 
                facets.push_back(Point3(id, id - 1 - ring.size() ,  id - 1)); 
            }
            id++;
        } 
    }


    // special case: last ring connects to 0,0,rho*2.0
    // only form facets.
    vertices.push_back(Pointf3(0.0, 0.0, rho));
    for (size_t i = 0; i < ring.size(); i++) {
        if (i == 0) {
            // third vertex is on the other side of the ring.
            facets.push_back(Point3(id, id - ring.size(),  id - 1));
        } else {
            facets.push_back(Point3(id, id - ring.size() + i,  id - ring.size() + (i - 1)));
        }
    }
    id++;
    TriangleMesh mesh(vertices, facets);
    return mesh;
}
}