bgc-c/basic-geometry/versor.h

#ifndef _GEOMETRY_VERSOR_H_
#define _GEOMETRY_VERSOR_H_

#include <stdint.h>

#include "basis.h"
#include "angle.h"
#include "vector3.h"
#include "rotation3.h"
#include "matrix3x3.h"

// =================== Types ==================== //

typedef struct {
    const float s0, x1, x2, x3;
} BgFP32Versor;

typedef struct {
    const double s0, x1, x2, x3;
} BgFP64Versor;

// ================= Dark Twins ================= //

typedef struct {
    float s0, x1, x2, x3;
} __BgFP32DarkTwinVersor;

typedef struct {
    double s0, x1, x2, x3;
} __BgFP64DarkTwinVersor;

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

extern const BgFP32Versor BG_FP32_IDLE_VERSOR;
extern const BgFP64Versor BG_FP64_IDLE_VERSOR;

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

static inline void bg_fp32_versor_reset(BgFP32Versor* versor)
{
    __BgFP32DarkTwinVersor* twin = (__BgFP32DarkTwinVersor*)versor;

    twin->s0 = 1.0f;
    twin->x1 = 0.0f;
    twin->x2 = 0.0f;
    twin->x3 = 0.0f;
}

static inline void bg_fp64_versor_reset(BgFP64Versor* versor)
{
    __BgFP64DarkTwinVersor* twin = (__BgFP64DarkTwinVersor*)versor;

    twin->s0 = 1.0;
    twin->x1 = 0.0;
    twin->x2 = 0.0;
    twin->x3 = 0.0;
}

// ==================== Set ===================== //

static inline void bg_fp32_versor_set_values(const float s0, const float x1, const float x2, const float x3, BgFP32Versor* versor)
{
    __BgFP32DarkTwinVersor* twin = (__BgFP32DarkTwinVersor*)versor;

    twin->s0 = s0;
    twin->x1 = x1;
    twin->x2 = x2;
    twin->x3 = x3;

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

    if (1.0f - BG_FP32_TWO_EPSYLON <= square_modulus && square_modulus <= 1.0f + BG_FP32_TWO_EPSYLON) {
        return;
    }

    if (square_modulus <= BG_FP32_SQUARE_EPSYLON) {
        twin->s0 = 1.0f;
        twin->x1 = 0.0f;
        twin->x2 = 0.0f;
        twin->x3 = 0.0f;
        return;
    }

    const float multiplier = sqrtf(1.0f / square_modulus);

    twin->s0 *= multiplier;
    twin->x1 *= multiplier;
    twin->x2 *= multiplier;
    twin->x3 *= multiplier;
}

static inline void bg_fp64_versor_set_values(const double s0, const double x1, const double x2, const double x3, BgFP64Versor* versor)
{
    __BgFP64DarkTwinVersor* twin = (__BgFP64DarkTwinVersor*)versor;

    twin->s0 = s0;
    twin->x1 = x1;
    twin->x2 = x2;
    twin->x3 = x3;

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

    if (1.0 - BG_FP64_TWO_EPSYLON <= square_modulus && square_modulus <= 1.0 + BG_FP64_TWO_EPSYLON) {
        return;
    }

    if (square_modulus <= BG_FP64_SQUARE_EPSYLON) {
        twin->s0 = 1.0;
        twin->x1 = 0.0;
        twin->x2 = 0.0;
        twin->x3 = 0.0;
        return;
    }

    const double multiplier = sqrt(1.0 / square_modulus);

    twin->s0 *= multiplier;
    twin->x1 *= multiplier;
    twin->x2 *= multiplier;
    twin->x3 *= multiplier;
}

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

static inline void bg_fp32_versor_copy(const BgFP32Versor* from, BgFP32Versor* to)
{
    __BgFP32DarkTwinVersor* twin = (__BgFP32DarkTwinVersor*)to;

    twin->s0 = from->s0;
    twin->x1 = from->x1;
    twin->x2 = from->x2;
    twin->x3 = from->x3;
}

static inline void bg_fp64_versor_copy(const BgFP64Versor* from, BgFP64Versor* to)
{
    __BgFP64DarkTwinVersor* twin = (__BgFP64DarkTwinVersor*)to;

    twin->s0 = from->s0;
    twin->x1 = from->x1;
    twin->x2 = from->x2;
    twin->x3 = from->x3;
}

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

static inline void bg_fp32_versor_swap(BgFP32Versor* versor1, BgFP32Versor* versor2)
{
    const float s0 = versor1->s0;
    const float x1 = versor1->x1;
    const float x2 = versor1->x2;
    const float x3 = versor1->x3;

    __BgFP32DarkTwinVersor* twin1 = (__BgFP32DarkTwinVersor*)versor1;

    twin1->s0 = versor2->s0;
    twin1->x1 = versor2->x1;
    twin1->x2 = versor2->x2;
    twin1->x3 = versor2->x3;

    __BgFP32DarkTwinVersor* twin2 = (__BgFP32DarkTwinVersor*)versor2;

    twin2->s0 = s0;
    twin2->x1 = x1;
    twin2->x2 = x2;
    twin2->x3 = x3;
}

static inline void bg_fp64_versor_swap(BgFP64Versor* versor1, BgFP64Versor* versor2)
{
    const double s0 = versor1->s0;
    const double x1 = versor1->x1;
    const double x2 = versor1->x2;
    const double x3 = versor1->x3;

    __BgFP64DarkTwinVersor* twin1 = (__BgFP64DarkTwinVersor*)versor1;

    twin1->s0 = versor2->s0;
    twin1->x1 = versor2->x1;
    twin1->x2 = versor2->x2;
    twin1->x3 = versor2->x3;

    __BgFP64DarkTwinVersor* twin2 = (__BgFP64DarkTwinVersor*)versor2;

    twin2->s0 = s0;
    twin2->x1 = x1;
    twin2->x2 = x2;
    twin2->x3 = x3;
}

// =============== Set Crude Turn =============== //

void bg_fp32_versor_set_crude_turn(const float x1, const float x2, const float x3, const float angle, const angle_unit_t unit, BgFP32Versor* result);

void bg_fp64_versor_set_crude_turn(const double x1, const double x2, const double x3, const double angle, const angle_unit_t unit, BgFP64Versor* result);

// ================== Set Turn ================== //

static inline void bg_fp32_versor_set_turn(const BgFP32Vector3* axis, const float angle, const angle_unit_t unit, BgFP32Versor* result)
{
    bg_fp32_versor_set_crude_turn(axis->x1, axis->x2, axis->x3, angle, unit, result);
}

static inline void bg_fp64_versor_set_turn(const BgFP32Vector3* axis, const double angle, const angle_unit_t unit, BgFP64Versor* result)
{
    bg_fp64_versor_set_crude_turn(axis->x1, axis->x2, axis->x3, angle, unit, result);
}

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

static inline void bg_fp32_versor_set_rotation(const BgFP32Rotation3* rotation, BgFP32Versor* result)
{
    bg_fp32_versor_set_crude_turn(rotation->axis.x1, rotation->axis.x2, rotation->axis.x3, rotation->radians, BG_ANGLE_UNIT_RADIANS, result);
}

static inline void bg_fp64_versor_set_rotation(const BgFP64Rotation3* rotation, BgFP64Versor* result)
{
    bg_fp64_versor_set_crude_turn(rotation->axis.x1, rotation->axis.x2, rotation->axis.x3, rotation->radians, BG_ANGLE_UNIT_RADIANS, result);
}

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

static inline int bg_fp32_versor_is_idle(const BgFP32Versor* versor)
{
    return 1.0f - BG_FP32_EPSYLON <= versor->s0 || versor->s0 <= -(1.0 - BG_FP32_EPSYLON);
}

static inline int bg_fp64_versor_is_idle(const BgFP64Versor* versor)
{
    return 1.0 - BG_FP64_EPSYLON <= versor->s0 || versor->s0 <= -(1.0 - BG_FP64_EPSYLON);
}

// ============= Copy to twin type ============== //

static inline void bg_fp32_versor_set_from_fp64(const BgFP64Versor* versor, BgFP32Versor* result)
{
    bg_fp32_versor_set_values(
        (float) versor->s0,
        (float) versor->x1,
        (float) versor->x2,
        (float) versor->x3,
        result
    );
}

static inline void bg_fp64_versor_set_from_fp32(const BgFP32Versor* versor, BgFP64Versor* result)
{
    bg_fp64_versor_set_values(
        versor->s0,
        versor->x1,
        versor->x2,
        versor->x3,
        result
    );
}

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

static inline void bg_fp32_versor_shorten(BgFP32Versor* versor)
{
    if (versor->s0 >= 0.0f) {
        return;
    }

    __BgFP32DarkTwinVersor* twin = (__BgFP32DarkTwinVersor*)versor;
    twin->s0 = -versor->s0;
    twin->x1 = -versor->x1;
    twin->x2 = -versor->x2;
    twin->x3 = -versor->x3;
}

static inline void bg_fp64_versor_shorten(BgFP64Versor* versor)
{
    if (versor->s0 >= 0.0f) {
        return;
    }

    __BgFP64DarkTwinVersor* twin = (__BgFP64DarkTwinVersor*)versor;
    twin->s0 = -versor->s0;
    twin->x1 = -versor->x1;
    twin->x2 = -versor->x2;
    twin->x3 = -versor->x3;
}

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

static inline void bg_fp32_versor_set_shortened(const BgFP32Versor* versor, BgFP32Versor* shortened)
{
    __BgFP32DarkTwinVersor* twin = (__BgFP32DarkTwinVersor*)shortened;

    if (versor->s0 >= 0.0f) {
        twin->x1 = versor->s0;
        twin->x1 = versor->x1;
        twin->x2 = versor->x2;
        twin->x3 = versor->x3;
        return;
    }

    twin->x1 = -versor->s0;
    twin->x1 = -versor->x1;
    twin->x2 = -versor->x2;
    twin->x3 = -versor->x3;
}

static inline void bg_fp64_versor_set_shortened(const BgFP64Versor* versor, BgFP64Versor* shortened)
{
    __BgFP64DarkTwinVersor* twin = (__BgFP64DarkTwinVersor*)shortened;

    if (versor->s0 >= 0.0) {
        twin->x1 = versor->s0;
        twin->x1 = versor->x1;
        twin->x2 = versor->x2;
        twin->x3 = versor->x3;
        return;
    }

    twin->x1 = -versor->s0;
    twin->x1 = -versor->x1;
    twin->x2 = -versor->x2;
    twin->x3 = -versor->x3;
    }

// ================= Inversion ================== //

static inline void bg_fp32_versor_invert(BgFP32Versor* versor)
{
    __BgFP32DarkTwinVersor* twin = (__BgFP32DarkTwinVersor*)versor;
    twin->x1 = -versor->x1;
    twin->x2 = -versor->x2;
    twin->x3 = -versor->x3;
}

static inline void bg_fp64_versor_invert(BgFP64Versor* versor)
{
    __BgFP64DarkTwinVersor* twin = (__BgFP64DarkTwinVersor*)versor;
    twin->x1 = -versor->x1;
    twin->x2 = -versor->x2;
    twin->x3 = -versor->x3;
}

// ================ Set Inverted ================ //

static inline void bg_fp32_versor_set_inverted(const BgFP32Versor* versor, BgFP32Versor* to)
{
    __BgFP32DarkTwinVersor* twin = (__BgFP32DarkTwinVersor*)to;
    twin->s0 = versor->s0;
    twin->x1 = -versor->x1;
    twin->x2 = -versor->x2;
    twin->x3 = -versor->x3;
}

static inline void bg_fp64_versor_set_inverted(const BgFP64Versor* versor, BgFP64Versor* to)
{
    __BgFP64DarkTwinVersor* twin = (__BgFP64DarkTwinVersor*)to;
    twin->s0 = versor->s0;
    twin->x1 = -versor->x1;
    twin->x2 = -versor->x2;
    twin->x3 = -versor->x3;
}

// ================ Set Inverted ================ //

static inline void bg_fp32_versor_set_inverted_fp64(const BgFP64Versor* versor, BgFP32Versor* to)
{
    bg_fp32_versor_set_values(
        (float) versor->s0,
        (float) -versor->x1,
        (float) -versor->x2,
        (float) -versor->x3,
        to
    );
}

static inline void bg_fp64_versor_set_inverted_fp32(const BgFP32Versor* versor, BgFP64Versor* to)
{
    bg_fp64_versor_set_values(
        versor->s0,
        -versor->x1,
        -versor->x2,
        -versor->x3,
        to
    );
}

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

static inline void bg_fp32_versor_combine(const BgFP32Versor* second, const BgFP32Versor* first, BgFP32Versor* result)
{
    const float s0 = (second->s0 * first->s0 - second->x1 * first->x1) - (second->x2 * first->x2 + second->x3 * first->x3);
    const float x1 = (second->x1 * first->s0 + second->s0 * first->x1) - (second->x3 * first->x2 - second->x2 * first->x3);
    const float x2 = (second->x2 * first->s0 + second->s0 * first->x2) - (second->x1 * first->x3 - second->x3 * first->x1);
    const float x3 = (second->x3 * first->s0 + second->s0 * first->x3) - (second->x2 * first->x1 - second->x1 * first->x2);

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

    __BgFP32DarkTwinVersor* twin = (__BgFP32DarkTwinVersor*)result;

    twin->s0 = s0;
    twin->x1 = x1;
    twin->x2 = x2;
    twin->x3 = x3;

    if (1.0f - BG_FP32_TWO_EPSYLON <= square_modulus && square_modulus <= 1.0f + BG_FP32_TWO_EPSYLON) {
        return;
    }

    const float multiplier = sqrtf(1.0f / square_modulus);

    twin->s0 *= multiplier;
    twin->x1 *= multiplier;
    twin->x2 *= multiplier;
    twin->x3 *= multiplier;
}

static inline void bg_fp64_versor_combine(const BgFP64Versor* second, const BgFP64Versor* first, BgFP64Versor* result)
{
    const double s0 = (second->s0 * first->s0 - second->x1 * first->x1) - (second->x2 * first->x2 + second->x3 * first->x3);
    const double x1 = (second->x1 * first->s0 + second->s0 * first->x1) - (second->x3 * first->x2 - second->x2 * first->x3);
    const double x2 = (second->x2 * first->s0 + second->s0 * first->x2) - (second->x1 * first->x3 - second->x3 * first->x1);
    const double x3 = (second->x3 * first->s0 + second->s0 * first->x3) - (second->x2 * first->x1 - second->x1 * first->x2);

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

    __BgFP64DarkTwinVersor* twin = (__BgFP64DarkTwinVersor*)result;

    twin->s0 = s0;
    twin->x1 = x1;
    twin->x2 = x2;
    twin->x3 = x3;

    if (1.0 - BG_FP64_TWO_EPSYLON <= square_modulus && square_modulus <= 1.0 + BG_FP64_TWO_EPSYLON) {
        return;
    }

    const double multiplier = sqrt(1.0 / square_modulus);

    twin->s0 *= multiplier;
    twin->x1 *= multiplier;
    twin->x2 *= multiplier;
    twin->x3 *= multiplier;
}

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

static inline void bg_fp32_versor_combine3(const BgFP32Versor* third, const BgFP32Versor* second, const BgFP32Versor* first, BgFP32Versor* result)
{
    const float s0a = (second->s0 * first->s0 - second->x1 * first->x1) - (second->x2 * first->x2 + second->x3 * first->x3);
    const float x1a = (second->x1 * first->s0 + second->s0 * first->x1) - (second->x3 * first->x2 - second->x2 * first->x3);
    const float x2a = (second->x2 * first->s0 + second->s0 * first->x2) - (second->x1 * first->x3 - second->x3 * first->x1);
    const float x3a = (second->x3 * first->s0 + second->s0 * first->x3) - (second->x2 * first->x1 - second->x1 * first->x2);

    const float s0b = (third->s0 * s0a - third->x1 * x1a) - (third->x2 * x2a + third->x3 * x3a);
    const float x1b = (third->x1 * s0a + third->s0 * x1a) - (third->x3 * x2a - third->x2 * x3a);
    const float x2b = (third->x2 * s0a + third->s0 * x2a) - (third->x1 * x3a - third->x3 * x1a);
    const float x3b = (third->x3 * s0a + third->s0 * x3a) - (third->x2 * x1a - third->x1 * x2a);

    const float square_modulus = (s0b * s0b + x1b * x1b) + (x2b * x2b + x3b * x3b);

    __BgFP32DarkTwinVersor* twin = (__BgFP32DarkTwinVersor*)result;

    twin->s0 = s0b;
    twin->x1 = x1b;
    twin->x2 = x2b;
    twin->x3 = x3b;

    if (1.0f - BG_FP32_TWO_EPSYLON <= square_modulus && square_modulus <= 1.0f + BG_FP32_TWO_EPSYLON) {
        return;
    }

    const float multiplier = sqrtf(1.0f / square_modulus);

    twin->s0 *= multiplier;
    twin->x1 *= multiplier;
    twin->x2 *= multiplier;
    twin->x3 *= multiplier;
}

static inline void bg_fp64_versor_combine3(const BgFP64Versor* third, const BgFP64Versor* second, const BgFP64Versor* first, BgFP64Versor* result)
{
    const double s0a = (second->s0 * first->s0 - second->x1 * first->x1) - (second->x2 * first->x2 + second->x3 * first->x3);
    const double x1a = (second->x1 * first->s0 + second->s0 * first->x1) - (second->x3 * first->x2 - second->x2 * first->x3);
    const double x2a = (second->x2 * first->s0 + second->s0 * first->x2) - (second->x1 * first->x3 - second->x3 * first->x1);
    const double x3a = (second->x3 * first->s0 + second->s0 * first->x3) - (second->x2 * first->x1 - second->x1 * first->x2);

    const double s0b = (third->s0 * s0a - third->x1 * x1a) - (third->x2 * x2a + third->x3 * x3a);
    const double x1b = (third->x1 * s0a + third->s0 * x1a) - (third->x3 * x2a - third->x2 * x3a);
    const double x2b = (third->x2 * s0a + third->s0 * x2a) - (third->x1 * x3a - third->x3 * x1a);
    const double x3b = (third->x3 * s0a + third->s0 * x3a) - (third->x2 * x1a - third->x1 * x2a);

    const double square_modulus = (s0b * s0b + x1b * x1b) + (x2b * x2b + x3b * x3b);

    __BgFP64DarkTwinVersor* twin = (__BgFP64DarkTwinVersor*)result;

    twin->s0 = s0b;
    twin->x1 = x1b;
    twin->x2 = x2b;
    twin->x3 = x3b;

    if (1.0 - BG_FP64_TWO_EPSYLON <= square_modulus && square_modulus <= 1.0 + BG_FP64_TWO_EPSYLON) {
        return;
    }

    const double multiplier = sqrt(1.0 / square_modulus);

    twin->s0 *= multiplier;
    twin->x1 *= multiplier;
    twin->x2 *= multiplier;
    twin->x3 *= multiplier;
}

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

static inline void bg_fp32_versor_exclude(const BgFP32Versor* basic, const BgFP32Versor* exclusion, BgFP32Versor* result)
{
    const float s0 = (basic->s0 * exclusion->s0 + basic->x1 * exclusion->x1) + (basic->x2 * exclusion->x2 + basic->x3 * exclusion->x3);
    const float x1 = (basic->x1 * exclusion->s0 - basic->s0 * exclusion->x1) + (basic->x3 * exclusion->x2 - basic->x2 * exclusion->x3);
    const float x2 = (basic->x2 * exclusion->s0 - basic->s0 * exclusion->x2) + (basic->x1 * exclusion->x3 - basic->x3 * exclusion->x1);
    const float x3 = (basic->x3 * exclusion->s0 - basic->s0 * exclusion->x3) + (basic->x2 * exclusion->x1 - basic->x1 * exclusion->x2);

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

    __BgFP32DarkTwinVersor* twin = (__BgFP32DarkTwinVersor*)result;

    twin->s0 = s0;
    twin->x1 = x1;
    twin->x2 = x2;
    twin->x3 = x3;

    if (1.0f - BG_FP32_TWO_EPSYLON <= square_modulus && square_modulus <= 1.0f + BG_FP32_TWO_EPSYLON) {
        return;
    }

    const float multiplier = sqrtf(1.0f / square_modulus);

    twin->s0 *= multiplier;
    twin->x1 *= multiplier;
    twin->x2 *= multiplier;
    twin->x3 *= multiplier;
}

static inline void bg_fp64_versor_exclude(const BgFP64Versor* basic, const BgFP64Versor* exclusion, BgFP64Versor* result)
{
    const double s0 = (basic->s0 * exclusion->s0 + basic->x1 * exclusion->x1) + (basic->x2 * exclusion->x2 + basic->x3 * exclusion->x3);
    const double x1 = (basic->x1 * exclusion->s0 - basic->s0 * exclusion->x1) + (basic->x3 * exclusion->x2 - basic->x2 * exclusion->x3);
    const double x2 = (basic->x2 * exclusion->s0 - basic->s0 * exclusion->x2) + (basic->x1 * exclusion->x3 - basic->x3 * exclusion->x1);
    const double x3 = (basic->x3 * exclusion->s0 - basic->s0 * exclusion->x3) + (basic->x2 * exclusion->x1 - basic->x1 * exclusion->x2);

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

    __BgFP64DarkTwinVersor* twin = (__BgFP64DarkTwinVersor*)result;

    twin->s0 = s0;
    twin->x1 = x1;
    twin->x2 = x2;
    twin->x3 = x3;

    if (1.0 - BG_FP64_TWO_EPSYLON <= square_modulus && square_modulus <= 1.0 + BG_FP64_TWO_EPSYLON) {
        return;
    }

    const double multiplier = sqrt(1.0 / square_modulus);

    twin->s0 *= multiplier;
    twin->x1 *= multiplier;
    twin->x2 *= multiplier;
    twin->x3 *= multiplier;
}

// ================= Rotation3 ================== //

void bg_fp32_versor_get_rotation(const BgFP32Versor* versor, BgFP32Rotation3* result);

void bg_fp64_versor_get_rotation(const BgFP64Versor* versor, BgFP64Rotation3* result);

// =========== Make Rotation Matrix3x3 ========== //

static inline void bg_fp32_versor_make_rotation_matrix(const BgFP32Versor* versor, BgFP32Matrix3x3* matrix)
{
    const float s0s0 = versor->s0 * versor->s0;
    const float x1x1 = versor->x1 * versor->x1;
    const float x2x2 = versor->x2 * versor->x2;
    const float x3x3 = versor->x3 * versor->x3;

    const float s0x1 = 2.0f * versor->s0 * versor->x1;
    const float s0x2 = 2.0f * versor->s0 * versor->x2;
    const float s0x3 = 2.0f * versor->s0 * versor->x3;

    const float x1x2 = 2.0f * versor->x1 * versor->x2;
    const float x1x3 = 2.0f * versor->x1 * versor->x3;
    const float x2x3 = 2.0f * versor->x2 * versor->x3;

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

    matrix->r1c2 = x1x2 - s0x3;
    matrix->r2c3 = x2x3 - s0x1;
    matrix->r3c1 = x1x3 - s0x2;

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

static inline void bg_fp64_versor_make_rotation_matrix(const BgFP64Versor* versor, BgFP64Matrix3x3* matrix)
{
    const double s0s0 = versor->s0 * versor->s0;
    const double x1x1 = versor->x1 * versor->x1;
    const double x2x2 = versor->x2 * versor->x2;
    const double x3x3 = versor->x3 * versor->x3;

    const double s0x1 = 2.0 * versor->s0 * versor->x1;
    const double s0x2 = 2.0 * versor->s0 * versor->x2;
    const double s0x3 = 2.0 * versor->s0 * versor->x3;

    const double x1x2 = 2.0 * versor->x1 * versor->x2;
    const double x1x3 = 2.0 * versor->x1 * versor->x3;
    const double x2x3 = 2.0 * versor->x2 * versor->x3;

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

    matrix->r1c2 = x1x2 - s0x3;
    matrix->r2c3 = x2x3 - s0x1;
    matrix->r3c1 = x1x3 - s0x2;

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

// =========== Make Reverse Matrix3x3 =========== //

static inline void bg_fp32_versor_make_reverse_matrix(const BgFP32Versor* versor, BgFP32Matrix3x3* matrix)
{
    const float s0s0 = versor->s0 * versor->s0;
    const float x1x1 = versor->x1 * versor->x1;
    const float x2x2 = versor->x2 * versor->x2;
    const float x3x3 = versor->x3 * versor->x3;

    const float s0x1 = 2.0f * versor->s0 * versor->x1;
    const float s0x2 = 2.0f * versor->s0 * versor->x2;
    const float s0x3 = 2.0f * versor->s0 * versor->x3;

    const float x1x2 = 2.0f * versor->x1 * versor->x2;
    const float x1x3 = 2.0f * versor->x1 * versor->x3;
    const float x2x3 = 2.0f * versor->x2 * versor->x3;

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

    matrix->r1c2 = x1x2 + s0x3;
    matrix->r2c3 = x2x3 + s0x1;
    matrix->r3c1 = x1x3 + s0x2;

    matrix->r2c1 = x1x2 - s0x3;
    matrix->r3c2 = x2x3 - s0x1;
    matrix->r1c3 = x1x3 - s0x2;
}

static inline void bg_fp64_versor_make_reverse_matrix(const BgFP64Versor* versor, BgFP64Matrix3x3* matrix)
{
    const double s0s0 = versor->s0 * versor->s0;
    const double x1x1 = versor->x1 * versor->x1;
    const double x2x2 = versor->x2 * versor->x2;
    const double x3x3 = versor->x3 * versor->x3;

    const double s0x1 = 2.0 * versor->s0 * versor->x1;
    const double s0x2 = 2.0 * versor->s0 * versor->x2;
    const double s0x3 = 2.0 * versor->s0 * versor->x3;

    const double x1x2 = 2.0 * versor->x1 * versor->x2;
    const double x1x3 = 2.0 * versor->x1 * versor->x3;
    const double x2x3 = 2.0 * versor->x2 * versor->x3;

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

    matrix->r1c2 = x1x2 + s0x3;
    matrix->r2c3 = x2x3 + s0x1;
    matrix->r3c1 = x1x3 + s0x2;

    matrix->r2c1 = x1x2 - s0x3;
    matrix->r3c2 = x2x3 - s0x1;
    matrix->r1c3 = x1x3 - s0x2;
}

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

static inline void bg_fp32_versor_turn(const BgFP32Versor* versor, const BgFP32Vector3* vector, BgFP32Vector3* result)
{
    const float tx1 = 2.0f * (versor->x2 * vector->x3 - versor->x3 * vector->x2);
    const float tx2 = 2.0f * (versor->x3 * vector->x1 - versor->x1 * vector->x3);
    const float tx3 = 2.0f * (versor->x1 * vector->x2 - versor->x2 * vector->x1);

    const float x1 = (vector->x1 + tx1 * versor->s0) + (versor->x2 * tx3 - versor->x3 * tx2);
    const float x2 = (vector->x2 + tx2 * versor->s0) + (versor->x3 * tx1 - versor->x1 * tx3);
    const float x3 = (vector->x3 + tx3 * versor->s0) + (versor->x1 * tx2 - versor->x2 * tx1);

    result->x1 = x1;
    result->x2 = x2;
    result->x3 = x3;
}

static inline void bg_fp64_versor_turn(const BgFP64Versor* versor, const BgFP64Vector3* vector, BgFP64Vector3* result)
{
    const double tx1 = 2.0 * (versor->x2 * vector->x3 - versor->x3 * vector->x2);
    const double tx2 = 2.0 * (versor->x3 * vector->x1 - versor->x1 * vector->x3);
    const double tx3 = 2.0 * (versor->x1 * vector->x2 - versor->x2 * vector->x1);

    const double x1 = (vector->x1 + tx1 * versor->s0) + (versor->x2 * tx3 - versor->x3 * tx2);
    const double x2 = (vector->x2 + tx2 * versor->s0) + (versor->x3 * tx1 - versor->x1 * tx3);
    const double x3 = (vector->x3 + tx3 * versor->s0) + (versor->x1 * tx2 - versor->x2 * tx1);

    result->x1 = x1;
    result->x2 = x2;
    result->x3 = x3;
}

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

static inline void bg_fp32_versor_turn_back(const BgFP32Versor* versor, const BgFP32Vector3* vector, BgFP32Vector3* result)
{
    const float tx1 = 2.0f * (versor->x2 * vector->x3 - versor->x3 * vector->x2);
    const float tx2 = 2.0f * (versor->x3 * vector->x1 - versor->x1 * vector->x3);
    const float tx3 = 2.0f * (versor->x1 * vector->x2 - versor->x2 * vector->x1);

    const float x1 = (vector->x1 - tx1 * versor->s0) + (versor->x2 * tx3 - versor->x3 * tx2);
    const float x2 = (vector->x2 - tx2 * versor->s0) + (versor->x3 * tx1 - versor->x1 * tx3);
    const float x3 = (vector->x3 - tx3 * versor->s0) + (versor->x1 * tx2 - versor->x2 * tx1);

    result->x1 = x1;
    result->x2 = x2;
    result->x3 = x3;
}

static inline void bg_fp64_versor_turn_back(const BgFP64Versor* versor, const BgFP64Vector3* vector, BgFP64Vector3* result)
{
    const double tx1 = 2.0 * (versor->x2 * vector->x3 - versor->x3 * vector->x2);
    const double tx2 = 2.0 * (versor->x3 * vector->x1 - versor->x1 * vector->x3);
    const double tx3 = 2.0 * (versor->x1 * vector->x2 - versor->x2 * vector->x1);

    const double x1 = (vector->x1 - tx1 * versor->s0) + (versor->x2 * tx3 - versor->x3 * tx2);
    const double x2 = (vector->x2 - tx2 * versor->s0) + (versor->x3 * tx1 - versor->x1 * tx3);
    const double x3 = (vector->x3 - tx3 * versor->s0) + (versor->x1 * tx2 - versor->x2 * tx1);

    result->x1 = x1;
    result->x2 = x2;
    result->x3 = x3;
}

#endif