bgc-c/basic-geometry/turn3.h

#ifndef _BGC_TURN3_H_INCLUDED_
#define _BGC_TURN3_H_INCLUDED_

#include <stdint.h>

#include "./utilities.h"
#include "./types.h"
#include "./angle.h"
#include "./quaternion.h"

#define BGC_ERROR_TURN3_FIRST_VECTOR_ZERO -3010
#define BGC_ERROR_TURN3_SECOND_VECTOR_ZERO -3011
#define BGC_ERROR_TURN3_VECTORS_OPPOSITE -3012

#define _BGC_ERROR_TURN3_FIRST_PAIR -3020
#define _BGC_ERROR_TURN3_SECOND_PAIR -3030
#define _BGC_ERROR_TURN3_EMPTY_MAIN -1
#define _BGC_ERROR_TURN3_EMPTY_BRANCH -2
#define _BGC_ERROR_TURN3_PAIR_PARALLEL -3

#define BGC_ERROR_TURN3_FIRST_PAIR_ZERO_MAIN -3021
#define BGC_ERROR_TURN3_FIRST_PAIR_ZERO_BRANCH -3022
#define BGC_ERROR_TURN3_FIRST_PAIR_PARALLEL -3023

#define BGC_ERROR_TURN3_SECOND_PAIR_ZERO_MAIN -3031
#define BGC_ERROR_TURN3_SECOND_PAIR_ZERO_BRANCH -3032
#define BGC_ERROR_TURN3_SECOND_PAIR_PARALLEL -3033

// ================= Constants ================== //

extern const BGC_FP32_Turn3 BGC_FP32_IDLE_TURN3;
extern const BGC_FP64_Turn3 BGC_FP64_IDLE_TURN3;

// =================== Reset ==================== //

inline void bgc_fp32_turn3_reset(BGC_FP32_Turn3* turn)
{
    bgc_fp32_quaternion_make(&turn->_versor, 1.0f, 0.0f, 0.0f, 0.0f);
}

inline void bgc_fp64_turn3_reset(BGC_FP64_Turn3* turn)
{
    bgc_fp64_quaternion_make(&turn->_versor, 1.0, 0.0, 0.0, 0.0);
}

// ============= Private: Normalize ============= //

void _bgc_fp32_turn3_normalize(BGC_FP32_Turn3* turn, const float square_modulus);

void _bgc_fp64_turn3_normalize(BGC_FP64_Turn3* turn, const double square_modulus);

// ================= Set Values ================= //

inline void bgc_fp32_turn3_set_raw_values(BGC_FP32_Turn3* turn, const float s0, const float x1, const float x2, const float x3)
{
    bgc_fp32_quaternion_make(&turn->_versor, s0, x1, x2, x3);

    const float square_modulus = (s0 * s0 + x1 * x1) + (x2 * x2 + x3 * x3);

    if (!bgc_fp32_is_square_unit(square_modulus)) {
        _bgc_fp32_turn3_normalize(turn, square_modulus);
    }
}

inline void bgc_fp64_turn3_set_raw_values(BGC_FP64_Turn3* turn, const double s0, const double x1, const double x2, const double x3)
{
    bgc_fp64_quaternion_make(&turn->_versor, s0, x1, x2, x3);

    const double square_modulus = (s0 * s0 + x1 * x1) + (x2 * x2 + x3 * x3);

    if (!bgc_fp64_is_square_unit(square_modulus)) {
        _bgc_fp64_turn3_normalize(turn, square_modulus);
    }
}

// =============== Get Quaternion =============== //

inline void bgc_fp32_turn3_get_quaternion(BGC_FP32_Quaternion* quaternion, const BGC_FP32_Turn3* turn)
{
    bgc_fp32_quaternion_copy(quaternion, &turn->_versor);
}

inline void bgc_fp64_turn3_get_quaternion(BGC_FP64_Quaternion* quaternion, const BGC_FP64_Turn3* turn)
{
    bgc_fp64_quaternion_copy(quaternion, &turn->_versor);
}

// =============== Set Quaternion =============== //

inline void bgc_fp32_turn3_set_quaternion(BGC_FP32_Turn3* turn, const BGC_FP32_Quaternion* quaternion)
{
    bgc_fp32_quaternion_copy(&turn->_versor, quaternion);

    const float square_modulus = bgc_fp32_quaternion_get_square_modulus(quaternion);

    if (!bgc_fp32_is_square_unit(square_modulus)) {
        _bgc_fp32_turn3_normalize(turn, square_modulus);
    }
}

inline void bgc_fp64_turn3_set_quaternion(BGC_FP64_Turn3* turn, const BGC_FP64_Quaternion* quaternion)
{
    bgc_fp64_quaternion_copy(&turn->_versor, quaternion);

    const double square_modulus = bgc_fp64_quaternion_get_square_modulus(quaternion);

    if (!bgc_fp64_is_square_unit(square_modulus)) {
        _bgc_fp64_turn3_normalize(turn, square_modulus);
    }
}

// ================ Get Rotation ================ //

float bgc_fp32_turn3_get_rotation(BGC_FP32_Vector3* axis, const BGC_FP32_Turn3* turn, const int angle_unit);

double bgc_fp64_turn3_get_rotation(BGC_FP64_Vector3* axis, const BGC_FP64_Turn3* turn, const int angle_unit);

// ================ Set Rotation ================ //

void bgc_fp32_turn3_set_rotation(BGC_FP32_Turn3* turn, const float x1, const float x2, const float x3, const float angle, const int angle_unit);

void bgc_fp64_turn3_set_rotation(BGC_FP64_Turn3* turn, const double x1, const double x2, const double x3, const double angle, const int angle_unit);

// ========= Find Direction Difference ========== //

int bgc_fp32_turn3_find_direction_difference(BGC_FP32_Turn3* turn, const BGC_FP32_Vector3* start, const BGC_FP32_Vector3* end);

int bgc_fp64_turn3_find_direction_difference(BGC_FP64_Turn3* turn, const BGC_FP64_Vector3* start, const BGC_FP64_Vector3* end);

// ======= Find Direction Pair Difference ======= //

int bgc_fp32_turn3_find_pair_difference(
    BGC_FP32_Turn3* turn,
    const BGC_FP32_Vector3* first_pair_main,
    const BGC_FP32_Vector3* first_pair_branch,
    const BGC_FP32_Vector3* second_pair_main,
    const BGC_FP32_Vector3* second_pair_branch
);

int bgc_fp64_turn3_find_pair_difference(
    BGC_FP64_Turn3* turn,
    const BGC_FP64_Vector3* first_pair_main,
    const BGC_FP64_Vector3* first_pair_branch,
    const BGC_FP64_Vector3* second_pair_main,
    const BGC_FP64_Vector3* second_pair_branch
);

// ==================== Copy ==================== //

inline void bgc_fp32_turn3_copy(BGC_FP32_Turn3* destination, const BGC_FP32_Turn3* source)
{
    bgc_fp32_quaternion_copy(&destination->_versor, &source->_versor);
}

inline void bgc_fp64_turn3_copy(BGC_FP64_Turn3* destination, const BGC_FP64_Turn3* source)
{
    bgc_fp64_quaternion_copy(&destination->_versor, &source->_versor);
}

// ==================== Swap ==================== //

inline void bgc_fp32_turn3_swap(BGC_FP32_Turn3* turn1, BGC_FP32_Turn3* turn2)
{
    bgc_fp32_quaternion_swap(&turn1->_versor, &turn2->_versor);
}

inline void bgc_fp64_turn3_swap(BGC_FP64_Turn3* turn1, BGC_FP64_Turn3* turn2)
{
    bgc_fp64_quaternion_swap(&turn1->_versor, &turn2->_versor);
}

// ================= Comparison ================= //

inline int bgc_fp32_turn3_is_idle(const BGC_FP32_Turn3* turn)
{
    return turn->_versor.x1 * turn->_versor.x1 + turn->_versor.x2 * turn->_versor.x2 + turn->_versor.x3 * turn->_versor.x3 <= BGC_FP32_SQUARE_EPSILON;
}

inline int bgc_fp64_turn3_is_idle(const BGC_FP64_Turn3* turn)
{
    return turn->_versor.x1 * turn->_versor.x1 + turn->_versor.x2 * turn->_versor.x2 + turn->_versor.x3 * turn->_versor.x3 <= BGC_FP64_SQUARE_EPSILON;
}

// ================== Convert =================== //

inline void bgc_fp32_turn3_convert_to_fp64(BGC_FP64_Turn3* destination, const BGC_FP32_Turn3* source)
{
    bgc_fp32_quaternion_convert_to_fp64(&destination->_versor, &source->_versor);

    const double square_modulus = bgc_fp64_quaternion_get_square_modulus(&destination->_versor);

    if (!bgc_fp64_is_square_unit(square_modulus)) {
        _bgc_fp64_turn3_normalize(destination, square_modulus);
    }
}

inline void bgc_fp64_turn3_convert_to_fp32(BGC_FP32_Turn3* destination, const BGC_FP64_Turn3* source)
{
    bgc_fp64_quaternion_convert_to_fp32(&destination->_versor, &source->_versor);

    const float square_modulus = bgc_fp32_quaternion_get_square_modulus(&destination->_versor);

    if (!bgc_fp32_is_square_unit(square_modulus)) {
        _bgc_fp32_turn3_normalize(destination, square_modulus);
    }
}

// ================== Shorten =================== //

inline void bgc_fp32_turn3_shorten(BGC_FP32_Turn3* turn)
{
    if (turn->_versor.s0 < 0.0f) {
        bgc_fp32_quaternion_revert(&turn->_versor);
    }
}

inline void bgc_fp64_turn3_shorten(BGC_FP64_Turn3* turn)
{
    if (turn->_versor.s0 < 0.0) {
        bgc_fp64_quaternion_revert(&turn->_versor);
    }
}

inline void bgc_fp32_turn3_get_shortened(BGC_FP32_Turn3* shortened, const BGC_FP32_Turn3* turn)
{
    if (turn->_versor.s0 >= 0.0f) {
        bgc_fp32_quaternion_copy(&shortened->_versor, &turn->_versor);
    }
    else {
        bgc_fp32_quaternion_get_reverse(&shortened->_versor, &turn->_versor);
    }
}

inline void bgc_fp64_turn3_get_shortened(BGC_FP64_Turn3* shortened, const BGC_FP64_Turn3* turn)
{
    if (turn->_versor.s0 >= 0.0) {
        bgc_fp64_quaternion_copy(&shortened->_versor, &turn->_versor);
    }
    else {
        bgc_fp64_quaternion_get_reverse(&shortened->_versor, &turn->_versor);
    }
}

// ================= Alternate ================== //

inline void bgc_fp32_turn3_alternate(BGC_FP32_Turn3* turn)
{
    bgc_fp32_quaternion_revert(&turn->_versor);
}

inline void bgc_fp64_turn3_alternate(BGC_FP64_Turn3* turn)
{
    bgc_fp64_quaternion_revert(&turn->_versor);
}

inline void bgc_fp32_turn3_get_alternative(BGC_FP32_Turn3* alternative, const BGC_FP32_Turn3* turn)
{
    bgc_fp32_quaternion_get_reverse(&alternative->_versor, &turn->_versor);
}

inline void bgc_fp64_turn3_get_alternative(BGC_FP64_Turn3* alternative, const BGC_FP64_Turn3* turn)
{
    bgc_fp64_quaternion_get_reverse(&alternative->_versor, &turn->_versor);
}

// =================== Revert =================== //

inline void bgc_fp32_turn3_revert(BGC_FP32_Turn3* turn)
{
    bgc_fp32_quaternion_conjugate(&turn->_versor);
}

inline void bgc_fp64_turn3_revert(BGC_FP64_Turn3* turn)
{
    bgc_fp64_quaternion_conjugate(&turn->_versor);
}

inline void bgc_fp32_turn3_get_reverse(BGC_FP32_Turn3* inverse, const BGC_FP32_Turn3* turn)
{
    bgc_fp32_quaternion_get_conjugate(&inverse->_versor, &turn->_versor);
}

inline void bgc_fp64_turn3_get_reverse(BGC_FP64_Turn3* inverse, const BGC_FP64_Turn3* turn)
{
    bgc_fp64_quaternion_get_conjugate(&inverse->_versor, &turn->_versor);
}

// =============== Get Exponation =============== //

void bgc_fp32_turn3_get_exponation(BGC_FP32_Turn3* power, const BGC_FP32_Turn3* base, const float exponent);

void bgc_fp64_turn3_get_exponation(BGC_FP64_Turn3* power, const BGC_FP64_Turn3* base, const double exponent);

// ================ Combination ================= //

inline void bgc_fp32_turn3_combine(BGC_FP32_Turn3* combination, const BGC_FP32_Turn3* first, const BGC_FP32_Turn3* second)
{
    bgc_fp32_quaternion_multiply_by_quaternion(&combination->_versor, &second->_versor, &first->_versor);

    const float square_modulus = bgc_fp32_quaternion_get_square_modulus(&combination->_versor);

    if (!bgc_fp32_is_square_unit(square_modulus)) {
        _bgc_fp32_turn3_normalize(combination, square_modulus);
    }
}

inline void bgc_fp64_turn3_combine(BGC_FP64_Turn3* combination, const BGC_FP64_Turn3* first, const BGC_FP64_Turn3* second)
{
    bgc_fp64_quaternion_multiply_by_quaternion(&combination->_versor, &second->_versor, &first->_versor);

    const double square_modulus = bgc_fp64_quaternion_get_square_modulus(&combination->_versor);

    if (!bgc_fp64_is_square_unit(square_modulus)) {
        _bgc_fp64_turn3_normalize(combination, square_modulus);
    }
}

// ============ Combination of three ============ //

inline void bgc_fp32_turn3_combine3(BGC_FP32_Turn3* combination, const BGC_FP32_Turn3* first, const BGC_FP32_Turn3* second, const BGC_FP32_Turn3* third)
{
    BGC_FP32_Quaternion product;

    bgc_fp32_quaternion_multiply_by_quaternion(&product, &second->_versor, &first->_versor);

    bgc_fp32_quaternion_multiply_by_quaternion(&combination->_versor, &third->_versor, &product);

    const float square_modulus = bgc_fp32_quaternion_get_square_modulus(&combination->_versor);

    if (!bgc_fp32_is_square_unit(square_modulus)) {
        _bgc_fp32_turn3_normalize(combination, square_modulus);
    }
}

inline void bgc_fp64_turn3_combine3(BGC_FP64_Turn3* combination, const BGC_FP64_Turn3* first, const BGC_FP64_Turn3* second, const BGC_FP64_Turn3* third)
{
    BGC_FP64_Quaternion product;

    bgc_fp64_quaternion_multiply_by_quaternion(&product, &second->_versor, &first->_versor);

    bgc_fp64_quaternion_multiply_by_quaternion(&combination->_versor, &third->_versor, &product);

    const double square_modulus = bgc_fp64_quaternion_get_square_modulus(&combination->_versor);

    if (!bgc_fp64_is_square_unit(square_modulus)) {
        _bgc_fp64_turn3_normalize(combination, square_modulus);
    }
}

// ================= Exclusion ================== //

inline void bgc_fp32_turn3_exclude(BGC_FP32_Turn3* difference, const BGC_FP32_Turn3* base, const BGC_FP32_Turn3* excludant)
{
    bgc_fp32_quaternion_multiply_by_conjugate(&difference->_versor, &base->_versor, &excludant->_versor);

    const float square_modulus = bgc_fp32_quaternion_get_square_modulus(&difference->_versor);

    if (!bgc_fp32_is_square_unit(square_modulus)) {
        _bgc_fp32_turn3_normalize(difference, square_modulus);
    }
}

inline void bgc_fp64_turn3_exclude(BGC_FP64_Turn3* difference, const BGC_FP64_Turn3* base, const BGC_FP64_Turn3* excludant)
{
    bgc_fp64_quaternion_multiply_by_conjugate(&difference->_versor, &base->_versor, &excludant->_versor);

    const double square_modulus = bgc_fp64_quaternion_get_square_modulus(&difference->_versor);

    if (!bgc_fp64_is_square_unit(square_modulus)) {
        _bgc_fp64_turn3_normalize(difference, square_modulus);
    }
}

// ============ Sphere Interpolation ============ //

void bgc_fp32_turn3_spherically_interpolate(BGC_FP32_Turn3* interpolation, const BGC_FP32_Turn3* start, const BGC_FP32_Turn3* end, const float phase);

void bgc_fp64_turn3_spherically_interpolate(BGC_FP64_Turn3* interpolation, const BGC_FP64_Turn3* start, const BGC_FP64_Turn3* end, const double phase);

// ============ Get Rotation Matrix ============= //

inline void bgc_fp32_turn3_get_rotation_matrix(BGC_FP32_Matrix3x3* matrix, const BGC_FP32_Turn3* turn)
{
    const float s0s0 = turn->_versor.s0 * turn->_versor.s0;
    const float x1x1 = turn->_versor.x1 * turn->_versor.x1;
    const float x2x2 = turn->_versor.x2 * turn->_versor.x2;
    const float x3x3 = turn->_versor.x3 * turn->_versor.x3;

    const float s0x1 = turn->_versor.s0 * turn->_versor.x1;
    const float s0x2 = turn->_versor.s0 * turn->_versor.x2;
    const float s0x3 = turn->_versor.s0 * turn->_versor.x3;

    const float x1x2 = turn->_versor.x1 * turn->_versor.x2;
    const float x1x3 = turn->_versor.x1 * turn->_versor.x3;

    const float x2x3 = turn->_versor.x2 * turn->_versor.x3;

    matrix->r1c1 = ((s0s0 + x1x1) - (x2x2 + x3x3));
    matrix->r2c2 = ((s0s0 + x2x2) - (x1x1 + x3x3));
    matrix->r3c3 = ((s0s0 + x3x3) - (x1x1 + x2x2));

    matrix->r1c2 = 2.0f * (x1x2 - s0x3);
    matrix->r2c3 = 2.0f * (x2x3 - s0x1);
    matrix->r3c1 = 2.0f * (x1x3 - s0x2);

    matrix->r2c1 = 2.0f * (x1x2 + s0x3);
    matrix->r3c2 = 2.0f * (x2x3 + s0x1);
    matrix->r1c3 = 2.0f * (x1x3 + s0x2);
}

inline void bgc_fp64_turn3_get_rotation_matrix(BGC_FP64_Matrix3x3* matrix, const BGC_FP64_Turn3* turn)
{
    const double s0s0 = turn->_versor.s0 * turn->_versor.s0;
    const double x1x1 = turn->_versor.x1 * turn->_versor.x1;
    const double x2x2 = turn->_versor.x2 * turn->_versor.x2;
    const double x3x3 = turn->_versor.x3 * turn->_versor.x3;

    const double s0x1 = turn->_versor.s0 * turn->_versor.x1;
    const double s0x2 = turn->_versor.s0 * turn->_versor.x2;
    const double s0x3 = turn->_versor.s0 * turn->_versor.x3;

    const double x1x2 = turn->_versor.x1 * turn->_versor.x2;
    const double x1x3 = turn->_versor.x1 * turn->_versor.x3;

    const double x2x3 = turn->_versor.x2 * turn->_versor.x3;

    matrix->r1c1 = ((s0s0 + x1x1) - (x2x2 + x3x3));
    matrix->r2c2 = ((s0s0 + x2x2) - (x1x1 + x3x3));
    matrix->r3c3 = ((s0s0 + x3x3) - (x1x1 + x2x2));

    matrix->r1c2 = 2.0 * (x1x2 - s0x3);
    matrix->r2c3 = 2.0 * (x2x3 - s0x1);
    matrix->r3c1 = 2.0 * (x1x3 - s0x2);

    matrix->r2c1 = 2.0 * (x1x2 + s0x3);
    matrix->r3c2 = 2.0 * (x2x3 + s0x1);
    matrix->r1c3 = 2.0 * (x1x3 + s0x2);
}

// ============= Get Reverse Matrix ============= //

inline void bgc_fp32_turn3_get_reverse_matrix(BGC_FP32_Matrix3x3* matrix, const BGC_FP32_Turn3* turn)
{
    const float s0s0 = turn->_versor.s0 * turn->_versor.s0;
    const float x1x1 = turn->_versor.x1 * turn->_versor.x1;
    const float x2x2 = turn->_versor.x2 * turn->_versor.x2;
    const float x3x3 = turn->_versor.x3 * turn->_versor.x3;

    const float s0x1 = turn->_versor.s0 * turn->_versor.x1;
    const float s0x2 = turn->_versor.s0 * turn->_versor.x2;
    const float s0x3 = turn->_versor.s0 * turn->_versor.x3;

    const float x1x2 = turn->_versor.x1 * turn->_versor.x2;
    const float x1x3 = turn->_versor.x1 * turn->_versor.x3;

    const float x2x3 = turn->_versor.x2 * turn->_versor.x3;

    matrix->r1c1 = ((s0s0 + x1x1) - (x2x2 + x3x3));
    matrix->r2c2 = ((s0s0 + x2x2) - (x1x1 + x3x3));
    matrix->r3c3 = ((s0s0 + x3x3) - (x1x1 + x2x2));

    matrix->r1c2 = 2.0f * (x1x2 + s0x3);
    matrix->r2c3 = 2.0f * (x2x3 + s0x1);
    matrix->r3c1 = 2.0f * (x1x3 + s0x2);

    matrix->r2c1 = 2.0f * (x1x2 - s0x3);
    matrix->r3c2 = 2.0f * (x2x3 - s0x1);
    matrix->r1c3 = 2.0f * (x1x3 - s0x2);
}

inline void bgc_fp64_turn3_get_reverse_matrix(BGC_FP64_Matrix3x3* matrix, const BGC_FP64_Turn3* turn)
{
    const double s0s0 = turn->_versor.s0 * turn->_versor.s0;
    const double x1x1 = turn->_versor.x1 * turn->_versor.x1;
    const double x2x2 = turn->_versor.x2 * turn->_versor.x2;
    const double x3x3 = turn->_versor.x3 * turn->_versor.x3;

    const double s0x1 = turn->_versor.s0 * turn->_versor.x1;
    const double s0x2 = turn->_versor.s0 * turn->_versor.x2;
    const double s0x3 = turn->_versor.s0 * turn->_versor.x3;

    const double x1x2 = turn->_versor.x1 * turn->_versor.x2;
    const double x1x3 = turn->_versor.x1 * turn->_versor.x3;

    const double x2x3 = turn->_versor.x2 * turn->_versor.x3;

    matrix->r1c1 = ((s0s0 + x1x1) - (x2x2 + x3x3));
    matrix->r2c2 = ((s0s0 + x2x2) - (x1x1 + x3x3));
    matrix->r3c3 = ((s0s0 + x3x3) - (x1x1 + x2x2));

    matrix->r1c2 = 2.0 * (x1x2 + s0x3);
    matrix->r2c3 = 2.0 * (x2x3 + s0x1);
    matrix->r3c1 = 2.0 * (x1x3 + s0x2);

    matrix->r2c1 = 2.0 * (x1x2 - s0x3);
    matrix->r3c2 = 2.0 * (x2x3 - s0x1);
    matrix->r1c3 = 2.0 * (x1x3 - s0x2);
}

// ============= Get Both Matrixes ============== //

inline void bgc_fp32_turn3_get_both_matrices(BGC_FP32_Matrix3x3* rotation, BGC_FP32_Matrix3x3* reverse, const BGC_FP32_Turn3* turn)
{
    bgc_fp32_turn3_get_reverse_matrix(reverse, turn);
    bgc_fp32_matrix3x3_get_transposed(rotation, reverse);
}

inline void bgc_fp64_turn3_get_both_matrices(BGC_FP64_Matrix3x3* rotation, BGC_FP64_Matrix3x3* reverse, const BGC_FP64_Turn3* turn)
{
    bgc_fp64_turn3_get_reverse_matrix(reverse, turn);
    bgc_fp64_matrix3x3_get_transposed(rotation, reverse);
}

// ================ Turn Vector ================= //

inline void bgc_fp32_turn3_vector(BGC_FP32_Vector3* turned_vector, const BGC_FP32_Turn3* turn, const BGC_FP32_Vector3* original_vector)
{
    _bgc_fp32_quaternion_turn_vector_roughly(turned_vector, &turn->_versor, original_vector);
}

inline void bgc_fp64_turn3_vector(BGC_FP64_Vector3* turned_vector, const BGC_FP64_Turn3* turn, const BGC_FP64_Vector3* original_vector)
{
    _bgc_fp64_quaternion_turn_vector_roughly(turned_vector, &turn->_versor, original_vector);
}

// ============== Turn Vector Back ============== //

inline void bgc_fp32_turn3_vector_back(BGC_FP32_Vector3* turned_vector, const BGC_FP32_Turn3* turn, const BGC_FP32_Vector3* original_vector)
{
    _bgc_fp32_quaternion_turn_vector_back_roughly(turned_vector, &turn->_versor, original_vector);
}

inline void bgc_fp64_turn3_vector_back(BGC_FP64_Vector3* turned_vector, const BGC_FP64_Turn3* turn, const BGC_FP64_Vector3* original_vector)
{
    _bgc_fp64_quaternion_turn_vector_back_roughly(turned_vector, &turn->_versor, original_vector);
}

// ================== Are Close ================= //

inline int bgc_fp32_turn3_are_close(const BGC_FP32_Turn3* turn1, const BGC_FP32_Turn3* turn2)
{
    BGC_FP32_Quaternion difference;
    bgc_fp32_quaternion_subtract(&difference, &turn1->_versor, &turn2->_versor);
    return bgc_fp32_quaternion_get_square_modulus(&difference) <= BGC_FP32_SQUARE_EPSILON;
}

inline int bgc_fp64_turn3_are_close(const BGC_FP64_Turn3* turn1, const BGC_FP64_Turn3* turn2)
{
    BGC_FP64_Quaternion difference;
    bgc_fp64_quaternion_subtract(&difference, &turn1->_versor, &turn2->_versor);
    return bgc_fp64_quaternion_get_square_modulus(&difference) <= BGC_FP64_SQUARE_EPSILON;
}

#endif