SymmetryHelper.cs

// Copyright (C) 2022 Maxim Gumin, The MIT License (MIT)

using System;
using System.Linq;
using System.Collections.Generic;

/// <summary>
/// Helper functions for generating symmetries of patterns.
/// </summary>
static class SymmetryHelper
{
    /// <summary>Lookup table for subgroups of 2D symmetries.</summary>
    public static Dictionary<string, bool[]> squareSubgroups = new()
    {
        ["()"] = new bool[8] { true, false, false, false, false, false, false, false },
        ["(x)"] = new bool[8] { true, true, false, false, false, false, false, false },
        ["(y)"] = new bool[8] { true, false, false, false, false, true, false, false },
        ["(x)(y)"] = new bool[8] { true, true, false, false, true, true, false, false },
        ["(xy+)"] = new bool[8] { true, false, true, false, true, false, true, false },
        ["(xy)"] = new bool[8] { true, true, true, true, true, true, true, true }
    };

    /// <summary>
    /// Returns an enumerable of symmetries of a 2D <c>thing</c>.
    /// </summary>
    /// <param name="thing">The original thing to generate symmetries of.</param>
    /// <param name="rotation">A function which applies a 90-degree rotation.</param>
    /// <param name="reflection">A function which applies a reflection.</param>
    /// <param name="same">A predicate which will be used to deduplicate symmetries; use <c>(q1, q2) => false</c> to prevent deduplication.</param>
    /// <param name="subgroup"><inheritdoc cref="SymmetryHelper.GetSymmetry(bool, string, bool[])" path="/returns"/> If <c>null</c>, the full symmetry group is used.</param>
    public static IEnumerable<T> SquareSymmetries<T>(T thing, Func<T, T> rotation, Func<T, T> reflection, Func<T, T, bool> same, bool[] subgroup = null)
    {
        T[] things = new T[8];

        things[0] = thing;                  // e
        things[1] = reflection(things[0]);  // b
        things[2] = rotation(things[0]);    // a
        things[3] = reflection(things[2]);  // ba
        things[4] = rotation(things[2]);    // a2
        things[5] = reflection(things[4]);  // ba2
        things[6] = rotation(things[4]);    // a3
        things[7] = reflection(things[6]);  // ba3

        List<T> result = new();
        for (int i = 0; i < 8; i++) if ((subgroup == null || subgroup[i]) && !result.Where(s => same(s, things[i])).Any()) result.Add(things[i]);
        return result;
    }

    /// <summary>Lookup table for subgroups of 3D symmetries.</summary>
    public static Dictionary<string, bool[]> cubeSubgroups = new()
    {
        ["()"] = AH.Array1D(48, l => l == 0),
        ["(x)"] = AH.Array1D(48, l => l == 0 || l == 1),
        ["(z)"] = AH.Array1D(48, l => l == 0 || l == 17),
        ["(xy)"] = AH.Array1D(48, l => l < 8),
        ["(xyz+)"] = AH.Array1D(48, l => l % 2 == 0),
        ["(xyz)"] = AH.Array1D(48, true),
        //["(xy)(z)"] = AH.Array1D(48, l => l < 8 || l == 17 || ...),
    };

    /// <summary>
    /// Returns an enumerable of symmetries of a 3D <c>thing</c>.
    /// </summary>
    /// <param name="thing"><inheritdoc cref="SymmetryHelper.SquareSymmetries{T}(T, Func{T, T}, Func{T, T}, Func{T, T, bool}, bool[])" path="/param[@name='thing']"/></param>
    /// <param name="a">A function which applies a 90-degree rotation about the z axis.</param>
    /// <param name="b">A function which applies a 90-degree rotation about the y axis.</param>
    /// <param name="r"><inheritdoc cref="SymmetryHelper.SquareSymmetries{T}(T, Func{T, T}, Func{T, T}, Func{T, T, bool}, bool[])" path="/param[@name='reflection']"/></param>
    /// <param name="same"><inheritdoc cref="SymmetryHelper.SquareSymmetries{T}(T, Func{T, T}, Func{T, T}, Func{T, T, bool}, bool[])" path="/param[@name='same']"/></param>
    /// <param name="subgroup"><inheritdoc cref="SymmetryHelper.SquareSymmetries{T}(T, Func{T, T}, Func{T, T}, Func{T, T, bool}, bool[])" path="/param[@name='subgroup']"/></param>
    public static IEnumerable<T> CubeSymmetries<T>(T thing, Func<T, T> a, Func<T, T> b, Func<T, T> r, Func<T, T, bool> same, bool[] subgroup = null)
    {
        T[] s = new T[48];

        s[0] = thing;        // e
        s[1] = r(s[0]);
        s[2] = a(s[0]);      // a
        s[3] = r(s[2]);
        s[4] = a(s[2]);      // a2
        s[5] = r(s[4]);
        s[6] = a(s[4]);      // a3
        s[7] = r(s[6]);
        s[8] = b(s[0]);      // b
        s[9] = r(s[8]);
        s[10] = b(s[2]);     // b a
        s[11] = r(s[10]);
        s[12] = b(s[4]);     // b a2
        s[13] = r(s[12]);
        s[14] = b(s[6]);     // b a3
        s[15] = r(s[14]);
        s[16] = b(s[8]);     // b2
        s[17] = r(s[16]);
        s[18] = b(s[10]);    // b2 a
        s[19] = r(s[18]);
        s[20] = b(s[12]);    // b2 a2
        s[21] = r(s[20]);
        s[22] = b(s[14]);    // b2 a3
        s[23] = r(s[22]);
        s[24] = b(s[16]);    // b3
        s[25] = r(s[24]);
        s[26] = b(s[18]);    // b3 a
        s[27] = r(s[26]);
        s[28] = b(s[20]);    // b3 a2
        s[29] = r(s[28]);
        s[30] = b(s[22]);    // b3 a3
        s[31] = r(s[30]);
        s[32] = a(s[8]);     // a b
        s[33] = r(s[32]);
        s[34] = a(s[10]);    // a b a
        s[35] = r(s[34]);
        s[36] = a(s[12]);    // a b a2
        s[37] = r(s[36]);
        s[38] = a(s[14]);    // a b a3
        s[39] = r(s[38]);
        s[40] = a(s[24]);    // a3 b a2 = a b3
        s[41] = r(s[40]);
        s[42] = a(s[26]);    // a3 b a3 = a b3 a
        s[43] = r(s[42]);
        s[44] = a(s[28]);    // a3 b = a b3 a2
        s[45] = r(s[44]);
        s[46] = a(s[30]);    // a3 b a = a b3 a3
        s[47] = r(s[46]);

        List<T> result = new();
        for (int i = 0; i < 48; i++) if ((subgroup == null || subgroup[i]) && !result.Where(t => same(t, s[i])).Any()) result.Add(s[i]);
        return result;
    }
    
    /// <summary>
    /// Finds the subgroup of symmetries associated with the lookup key <c>s</c>.
    /// </summary>
    /// <param name="d2">If <c>true</c>, a 2D symmetry group is found; otherwise, a 3D symmetry group is found.</param>
    /// <param name="s">The lookup key for the symmetry subgroup.</param>
    /// <param name="dflt">The default subgroup, which is returned if <c>s</c> is <c>null</c>.</param>
    /// <returns>An array of flags defining a subgroup of symmetries.</returns>
    public static bool[] GetSymmetry(bool d2, string s, bool[] dflt)
    {
        if (s == null) return dflt;
        bool success = d2 ? squareSubgroups.TryGetValue(s, out bool[] result) : cubeSubgroups.TryGetValue(s, out result);
        return success ? result : null;
    }
}