toddcox-visualize/vis/include/geometry.hpp

#pragma once

#include <tc/core.hpp>
#include <cmath>
#include <optional>
#include <numeric>
#include <iostream>
#include "combo_iterator.hpp"

/**
 * An primitive with N indices.
 * @tparam N
 */
template<unsigned N>
struct Primitive {
    static_assert(N > 0, "Primitives must contain at least one point. Primitive<0> or lower is impossible.");

    std::array<unsigned, N> inds;

    Primitive() = default;

    Primitive(const Primitive<N> &) = default;

    Primitive(const Primitive<N - 1> &sub, unsigned root) {
        std::copy(sub.inds.begin(), sub.inds.end(), inds.begin());
        inds[N - 1] = root;
    }

    ~Primitive() = default;

    inline void flip() {
        if (N > 1) std::swap(inds[0], inds[1]);
    }

    void apply(const tc::Cosets &table, int gen) {
        for (auto &ind : inds) {
            ind = table.get(ind, gen);
        }
        flip();
    }
};

/**
 * Produce a list of all generators for the group context. The range [0..group.ngens).
 */
std::vector<int> gens(const tc::Group &context) {
    std::vector<int> g_gens(context.ngens);
    std::iota(g_gens.begin(), g_gens.end(), 0);
    return g_gens;
}

/**
 * Determine which of g_gens are the correct names for sg_gens within the current context
 */
std::vector<int> recontext_gens(
    const tc::Group &context,
    std::vector<int> g_gens,
    std::vector<int> sg_gens) {

    std::sort(g_gens.begin(), g_gens.end());

    int inv_gen_map[context.ngens];
    for (size_t i = 0; i < g_gens.size(); i++) {
        inv_gen_map[g_gens[i]] = i;
    }

    std::vector<int> s_sg_gens;
    s_sg_gens.reserve(sg_gens.size());
    for (const auto gen : sg_gens) {
        s_sg_gens.push_back(inv_gen_map[gen]);
    }
    std::sort(s_sg_gens.begin(), s_sg_gens.end());

    return s_sg_gens;
}

/**
 * Determine whether the orientation of the group sg_gens is reversed from the group g_gens within group context
 */
int get_parity(
    const tc::Group &context,
    const std::vector<int> &g_gens,
    const std::vector<int> &sg_gens
) {
    if (g_gens.size() != sg_gens.size() + 1) return 0;

    const auto proper_sg_gens = recontext_gens(context, g_gens, sg_gens);

    int i = 0;
    for (; i < sg_gens.size(); ++i) {
        if (proper_sg_gens[i] != i) {
            break;
        }
    }

    return i & 1;
}

/**
 * Solve the cosets generated by sg_gens within the subgroup generated by g_gens of the group context
 */
tc::Cosets solve(
    const tc::Group &context,
    const std::vector<int> &g_gens,
    const std::vector<int> &sg_gens
) {
    const auto proper_sg_gens = recontext_gens(context, g_gens, sg_gens);
    return context.subgroup(g_gens).solve(proper_sg_gens);
}

template<unsigned N>
struct Mesh {
    std::vector<Primitive<N>> prims;

    Mesh() : prims() {}

    Mesh(const Mesh<N> &) = default;

    explicit Mesh(std::vector<Primitive<N>> &prims) : prims(prims) {}

    [[nodiscard]] size_t size() const {
        return prims.size();
    }

    /**
     * Apply some context transformation to all primitives of this mesh.
     */
    void apply(const tc::Cosets &table, int gen) {
        for (auto &prim : prims) {
            prim.apply(table, gen);
        }
    }

    /**
     * Reverse the orientation of all primitives in this mesh.
     */
    void flip() {
        for (auto &prim : prims) {
            prim.flip();
        }
    }

    /**
     * Convert the indexes of this mesh to those of a different context, using g_gens to build the parent context and sg_gens to build this context.
     */
    [[nodiscard]]
    Mesh<N> recontext(
        const tc::Group &context,
        const std::vector<int> &g_gens,
        const std::vector<int> &sg_gens
    ) const {
        const auto proper_sg_gens = recontext_gens(context, g_gens, sg_gens);
        const auto table = solve(context, g_gens, {});
        const auto path = solve(context, sg_gens, {}).path;

        auto map = path.template walk<int, int>(0, proper_sg_gens, [table](int coset, int gen) {
            return table.get(coset, gen);
        });

        Mesh<N> res = *this;
        for (Primitive<N> &prim : res.prims) {
            for (auto &ind : prim.inds) {
                ind = map[ind];
            }
        }

        if (get_parity(context, g_gens, sg_gens) == 1)
            res.flip();

        return res;
    }

    [[nodiscard]]
    Mesh<N> tile(
        const tc::Group &context,
        const std::vector<int> &g_gens,
        const std::vector<int> &sg_gens
    ) const {
        Mesh<N> base = recontext(context, g_gens, sg_gens);
        const auto proper_sg_gens = recontext_gens(context, g_gens, sg_gens);

        const auto table = solve(context, g_gens, {});
        const auto path = solve(context, g_gens, sg_gens).path;

        const auto all = path.template walk<Mesh<N>, int>(base, gens(context), [table](Mesh<N> from, int gen) {
            from.apply(table, gen);
            return from;
        });

        return merge(all);
    }

    /**
     * Produce a mesh of higher dimension by fanning a single point to all primitives in this mesh.
     */
    [[nodiscard]]
    Mesh<N + 1> fan(int root) const {
        std::vector<Primitive<N + 1>> res(prims.size());
        std::transform(prims.begin(), prims.end(), res.begin(),
            [root](const Primitive<N> &prim) {
                return Primitive<N + 1>(prim, root);
            }
        );
        return Mesh<N + 1>(res);
    }
};

/**
 * Union several meshes of the same dimension
 */
template<unsigned N>
Mesh<N> merge(const std::vector<Mesh<N>> &meshes) {
    size_t size = 0;
    for (const auto &mesh : meshes) {
        size += mesh.size();
    }

    std::vector<Primitive<N>> prims;
    prims.reserve(size);
    for (const auto &mesh : meshes) {
        prims.insert(prims.end(), mesh.prims.begin(), mesh.prims.end());
    }

    return Mesh(prims);
}

/**
 * Produce a mesh of primitives that fill out the volume of the subgroup generated by generators g_gens within the group context
 */
template<unsigned N>
Mesh<N> triangulate(
    const tc::Group &context,
    const std::vector<int> &g_gens
) {
    if (g_gens.size() + 1 != N) // todo make static assert
        throw std::logic_error("g_gens size must be one less than N");

    const auto &combos = Combos(g_gens, g_gens.size() - 1);

    std::vector<Mesh<N>> meshes;
    for (const auto &sg_gens : combos) {
        Mesh<N - 1> base = triangulate<N - 1>(context, sg_gens);
        Mesh<N - 1> raised = base.tile(context, g_gens, sg_gens);
        raised.prims.erase(raised.prims.begin(), raised.prims.begin() + base.size());
        Mesh<N> fan = raised.fan(0);
        meshes.push_back(fan);
    }

    return merge(meshes);
}

/**
 * Single-index primitives should not be further triangulated.
 */
template<>
Mesh<1> triangulate<1>(
    const tc::Group &context,
    const std::vector<int> &g_gens
) {
    if (not g_gens.empty()) // todo make static assert
        throw std::logic_error("g_gens must be empty for a trivial Mesh");

    Mesh<1> res;
    res.prims.emplace_back();
    return res;
}