From 542520640124e86f262f0614204fb3beda41991b Mon Sep 17 00:00:00 2001
From: Andrey Pokidov <andrey.pokidov@gmail.com>
Date: Fri, 3 Apr 2026 15:10:57 +0700
Subject: [PATCH] =?UTF-8?q?=D0=A3=D0=B4=D0=B0=D0=BB=D0=B5=D0=BD=D0=B8?=
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---
 basic-geometry-dev/basic-geometry-dev.cbp |   4 +
 basic-geometry-dev/main.c                 | 130 +------------
 basic-geometry-dev/turn3_combination.c    | 218 ++++++++++++++++++++++
 basic-geometry-dev/turn3_combination.h    |   9 +
 basic-geometry/rigid-pose3.c              |   8 +-
 basic-geometry/rigid-pose3.h              |  24 +--
 basic-geometry/turn2.c                    |   7 +-
 basic-geometry/turn2.h                    |  42 +----
 basic-geometry/turn3.c                    |  13 +-
 basic-geometry/turn3.h                    |  42 +----
 10 files changed, 277 insertions(+), 220 deletions(-)
 create mode 100644 basic-geometry-dev/turn3_combination.c
 create mode 100644 basic-geometry-dev/turn3_combination.h

diff --git a/basic-geometry-dev/basic-geometry-dev.cbp b/basic-geometry-dev/basic-geometry-dev.cbp
index 0e7489f..35770f1 100644
--- a/basic-geometry-dev/basic-geometry-dev.cbp
+++ b/basic-geometry-dev/basic-geometry-dev.cbp
@@ -53,6 +53,10 @@
 			<Option compilerVar="CC" />
 		</Unit>
 		<Unit filename="printing_utils.h" />
+		<Unit filename="turn3_combination.c">
+			<Option compilerVar="CC" />
+		</Unit>
+		<Unit filename="turn3_combination.h" />
 		<Unit filename="vector3_pair_difference.c">
 			<Option compilerVar="CC" />
 		</Unit>
diff --git a/basic-geometry-dev/main.c b/basic-geometry-dev/main.c
index 13ca36a..491f855 100644
--- a/basic-geometry-dev/main.c
+++ b/basic-geometry-dev/main.c
@@ -1,128 +1,12 @@
 #include <stdio.h>
-#include <stdlib.h>
 #include <math.h>
 #include <basic-geometry.h>
 
-#ifdef _WIN64
-#include <windows.h>
-#else
-#include <time.h>
-#endif // _WINDOWS_
-
 #include "printing_utils.h"
 #include "vector3_pair_difference.h"
+#include "turn3_combination.h"
 #include "affine3.h"
 
-typedef struct {
-    BGC_FP32_Turn3 versor1, versor2, result;
-} structure_fp32_t;
-
-structure_fp32_t* allocate_structures(const unsigned int amount)
-{
-    return calloc(amount, sizeof(structure_fp32_t));
-}
-
-structure_fp32_t* make_structures(const unsigned int amount)
-{
-    structure_fp32_t* list = allocate_structures(amount);
-
-    if (list == 0) {
-        return 0;
-    }
-
-    const float multiplier = 2.0f / RAND_MAX;
-
-    for (unsigned int i = 0; i < amount; i++) {
-        bgc_fp32_turn3_set_values(
-            &list[i].versor1,
-            rand() * multiplier - 1.0f,
-            rand() * multiplier - 1.0f,
-            rand() * multiplier - 1.0f,
-            rand() * multiplier - 1.0f
-        );
-
-        bgc_fp32_turn3_set_values(
-            &list[i].versor2,
-            rand() * multiplier - 1.0f,
-            rand() * multiplier - 1.0f,
-            rand() * multiplier - 1.0f,
-            rand() * multiplier - 1.0f
-        );
-
-        bgc_fp32_turn3_reset(&list[i].result);
-    }
-
-    return list;
-}
-
-
-void list_work(const uint_fast32_t amount, structure_fp32_t* list)
-{
-    for (uint_fast32_t j = 0; j < 1000; j++) {
-        for (uint_fast32_t i = 0; i < amount; i++) {
-            bgc_fp32_turn3_combine(&list[i].result, &list[i].versor1, &list[i].versor2);
-        }
-    }
-}
-/*
-int main()
-{
-    const unsigned int amount = 1000000;
-    structure_fp32_t* list = make_structures(amount);
-
-#ifdef _WIN64
-    ULONGLONG start, end;
-
-    start = GetTickCount64();
-
-    srand((unsigned int)(start & 0xfffffff));
-
-    start = GetTickCount64();
-#else
-    struct timespec start, end;
-    clock_gettime(0, &start);
-    srand((unsigned int)(start.tv_nsec & 0xfffffff));
-
-    clock_gettime(CLOCK_REALTIME, &start);
-#endif // _WIN64
-
-    list_work(amount, list);
-
-#ifdef _WIN64
-    end = GetTickCount64();
-
-    printf("Time: %lld\n", end - start);
-#else
-    clock_gettime(CLOCK_REALTIME, &end);
-
-    printf("Time: %lf\n", (end.tv_sec - start.tv_sec) * 1000.0 + (end.tv_nsec - start.tv_nsec) * 0.000001);
-#endif // _WIN64
-
-    print_fp32_quaternion(&list[10].versor1._versor);
-    print_fp32_quaternion(&list[10].versor2._versor);
-    print_fp32_quaternion(&list[10].result._versor);
-
-    free(list);
-
-    return 0;
-}
-*/
-
-/*
-int main() {
-    BGC_FP32_Complex complex, exponent, result;
-
-    bgc_fp32_complex_set_values(0, 1, &complex);
-
-    bgc_fp32_complex_set_values(4, 0, &exponent);
-
-    bgc_fp32_complex_get_exponation(&complex, exponent.real, exponent.imaginary, &result);
-
-    printf("(%f, %f) ^ (%f, %f) = (%f, %f)\n", complex.real, complex.imaginary, exponent.real, exponent.imaginary, result.real, result.imaginary);
-
-    return 0;
-}
-*/
 /*
 int main() {
     BGC_FP32_Turn3 start = { 1.0f, 0.0f, 0.0f, 0.0f };
@@ -139,7 +23,7 @@ void test_fp32_quaternion_set_matrix()
     BGC_FP32_Turn3 turn;
     BGC_FP32_Matrix3x3 rotation;
     BGC_FP32_Quaternion quaternion;
-    
+
     bgc_fp32_turn3_set_rotation(&turn, 1.0f, 1.0f, 1.0f, 120.0f, BGC_ANGLE_UNIT_DEGREES);
 
     bgc_fp32_turn3_get_rotation_matrix(&rotation, &turn);
@@ -177,13 +61,11 @@ int main()
     //test_fp32_pair_difference();
     //test_fp64_pair_difference();
 
-    //printf("Affine3 performance test: %f\n", test_bgc_affine3_performance(10000000, 10));
+    //test_fp32_quaternion_set_matrix();
+    //test_fp64_quaternion_set_matrix();
 
-    //printf("sizeof(BGC_FP32_Affine3)   = %zu\n", sizeof(BGC_FP32_Affine3));
-    //printf("offsetof(shift)         = %zu\n", offsetof(BGC_FP32_Affine3, shift));
-    //printf("sizeof(BGC_FP32_Matrix3x3) = %zu\n", sizeof(BGC_FP32_Matrix3x3));
-    
-    test_fp32_quaternion_set_matrix();
+    //test_fp32_turn3_combination();
+    //test_fp64_turn3_combination();
 
     return 0;
 }
diff --git a/basic-geometry-dev/turn3_combination.c b/basic-geometry-dev/turn3_combination.c
new file mode 100644
index 0000000..d611bd3
--- /dev/null
+++ b/basic-geometry-dev/turn3_combination.c
@@ -0,0 +1,218 @@
+#include <stdlib.h>
+#include <math.h>
+#include <stdio.h>
+#include <stdint.h>
+
+#ifdef _WIN64
+#include <windows.h>
+#else
+#include <time.h>
+#endif // _WINDOWS_
+
+#include "printing_utils.h"
+#include "turn3_combination.h"
+
+// =================== Types ==================== //
+
+typedef struct {
+    BGC_FP32_Turn3 versor1, versor2, result;
+} Testing_FP32_Turn3Structure;
+
+typedef struct {
+    BGC_FP64_Turn3 versor1, versor2, result;
+} Testing_FP64_Turn3Structure;
+
+// ================= Allocation ================= //
+
+static Testing_FP32_Turn3Structure* _fp32_allocate_structures(const unsigned int amount)
+{
+    return malloc((size_t)amount * sizeof(Testing_FP32_Turn3Structure));
+}
+
+static Testing_FP64_Turn3Structure* _fp64_allocate_structures(const unsigned int amount)
+{
+    return malloc((size_t)amount * sizeof(Testing_FP64_Turn3Structure));
+}
+
+// =============== Randomization ================ //
+
+static Testing_FP32_Turn3Structure* _fp32_make_structures(const unsigned int amount)
+{
+    Testing_FP32_Turn3Structure* list = _fp32_allocate_structures(amount);
+
+    if (list == 0) {
+        return 0;
+    }
+
+    const float multiplier = 2.0f / RAND_MAX;
+
+    for (unsigned int i = 0; i < amount; i++) {
+        bgc_fp32_turn3_set_values(
+            &list[i].versor1,
+            rand() * multiplier - 1.0f,
+            rand() * multiplier - 1.0f,
+            rand() * multiplier - 1.0f,
+            rand() * multiplier - 1.0f
+        );
+
+        bgc_fp32_turn3_set_values(
+            &list[i].versor2,
+            rand() * multiplier - 1.0f,
+            rand() * multiplier - 1.0f,
+            rand() * multiplier - 1.0f,
+            rand() * multiplier - 1.0f
+        );
+
+        bgc_fp32_turn3_reset(&list[i].result);
+    }
+
+    return list;
+}
+
+static Testing_FP64_Turn3Structure* _fp64_make_structures(const unsigned int amount)
+{
+    Testing_FP64_Turn3Structure* list = _fp64_allocate_structures(amount);
+
+    if (list == 0) {
+        return 0;
+    }
+
+    const double multiplier = 2.0 / RAND_MAX;
+
+    for (unsigned int i = 0; i < amount; i++) {
+        bgc_fp64_turn3_set_values(
+            &list[i].versor1,
+            rand() * multiplier - 1.0,
+            rand() * multiplier - 1.0,
+            rand() * multiplier - 1.0,
+            rand() * multiplier - 1.0
+        );
+
+        bgc_fp64_turn3_set_values(
+            &list[i].versor2,
+            rand() * multiplier - 1.0,
+            rand() * multiplier - 1.0,
+            rand() * multiplier - 1.0,
+            rand() * multiplier - 1.0
+        );
+
+        bgc_fp64_turn3_reset(&list[i].result);
+    }
+
+    return list;
+}
+
+// ================ Circle Work ================= //
+
+static int_fast64_t _fp32_circle_work(const uint_fast32_t amount, Testing_FP32_Turn3Structure* list)
+{
+    int_fast64_t total = 0;
+
+    for (uint_fast32_t j = 0; j < 1000; j++) {
+        for (uint_fast32_t i = 0; i < amount; i++) {
+            bgc_fp32_turn3_combine(&list[i].result, &list[i].versor1, &list[i].versor2);
+            total++;
+        }
+    }
+
+    return total;
+}
+
+static int_fast64_t _fp64_circle_work(const uint_fast32_t amount, Testing_FP64_Turn3Structure* list)
+{
+    int_fast64_t total = 0;
+
+    for (uint_fast32_t j = 0; j < 1000; j++) {
+        for (uint_fast32_t i = 0; i < amount; i++) {
+            bgc_fp64_turn3_combine(&list[i].result, &list[i].versor1, &list[i].versor2);
+            total++;
+        }
+    }
+
+    return total;
+}
+
+// ==================== Test ==================== //
+
+void test_fp32_turn3_combination()
+{
+    const unsigned int amount = 1000000;
+    int_fast64_t total = 0;
+    Testing_FP32_Turn3Structure* list = _fp32_make_structures(amount);
+
+#ifdef _WIN64
+    ULONGLONG start, end;
+
+    start = GetTickCount64();
+
+    srand((unsigned int)(start & 0xfffffff));
+
+    start = GetTickCount64();
+#else
+    struct timespec start, end;
+    clock_gettime(0, &start);
+    srand((unsigned int)(start.tv_nsec & 0xfffffff));
+
+    clock_gettime(CLOCK_REALTIME, &start);
+#endif // _WIN64
+
+    total = _fp32_circle_work(amount, list);
+
+#ifdef _WIN64
+    end = GetTickCount64();
+
+    printf("Time: %lld, Total iterations: %ld\n", end - start, total);
+#else
+    clock_gettime(CLOCK_REALTIME, &end);
+
+    printf("Time: %lf, Total iterations: %ld\n", (end.tv_sec - start.tv_sec) * 1000.0 + (end.tv_nsec - start.tv_nsec) * 0.000001, total);
+#endif // _WIN64
+
+    print_fp32_quaternion(&list[10].versor1._versor);
+    print_fp32_quaternion(&list[10].versor2._versor);
+    print_fp32_quaternion(&list[10].result._versor);
+
+    free(list);
+}
+
+
+void test_fp64_turn3_combination()
+{
+    const unsigned int amount = 1000000;
+    int_fast64_t total = 0;
+    Testing_FP64_Turn3Structure* list = _fp64_make_structures(amount);
+
+#ifdef _WIN64
+    ULONGLONG start, end;
+
+    start = GetTickCount64();
+
+    srand((unsigned int)(start & 0xfffffff));
+
+    start = GetTickCount64();
+#else
+    struct timespec start, end;
+    clock_gettime(0, &start);
+    srand((unsigned int)(start.tv_nsec & 0xfffffff));
+
+    clock_gettime(CLOCK_REALTIME, &start);
+#endif // _WIN64
+
+    total = _fp64_circle_work(amount, list);
+
+#ifdef _WIN64
+    end = GetTickCount64();
+
+    printf("Time: %lld, Total iterations: %ld\n", end - start, total);
+#else
+    clock_gettime(CLOCK_REALTIME, &end);
+
+    printf("Time: %lf, Total iterations: %ld\n", (end.tv_sec - start.tv_sec) * 1000.0 + (end.tv_nsec - start.tv_nsec) * 0.000001, total);
+#endif // _WIN64
+
+    print_fp64_quaternion(&list[10].versor1._versor);
+    print_fp64_quaternion(&list[10].versor2._versor);
+    print_fp64_quaternion(&list[10].result._versor);
+
+    free(list);
+}
diff --git a/basic-geometry-dev/turn3_combination.h b/basic-geometry-dev/turn3_combination.h
new file mode 100644
index 0000000..7260df5
--- /dev/null
+++ b/basic-geometry-dev/turn3_combination.h
@@ -0,0 +1,9 @@
+#ifndef TURN3_COMBINATION_H_INCLUDED
+#define TURN3_COMBINATION_H_INCLUDED
+
+#include <basic-geometry.h>
+
+void test_fp32_turn3_combination();
+void test_fp64_turn3_combination();
+
+#endif // TURN3_COMBINATION_H_INCLUDED
diff --git a/basic-geometry/rigid-pose3.c b/basic-geometry/rigid-pose3.c
index 1ec9dbd..0af8a81 100644
--- a/basic-geometry/rigid-pose3.c
+++ b/basic-geometry/rigid-pose3.c
@@ -48,11 +48,11 @@ extern inline void bgc_fp64_rigid_pose3_revert(BGC_FP64_RigidPose3* const pose);
 extern inline void bgc_fp32_rigid_pose3_get_reverse_pose(BGC_FP32_RigidPose3* const reverse, const BGC_FP32_RigidPose3* const pose);
 extern inline void bgc_fp64_rigid_pose3_get_reverse_pose(BGC_FP64_RigidPose3* const reverse, const BGC_FP64_RigidPose3* const pose);
 
-extern inline void bgc_fp32_rigid_pose3_combine(BGC_FP32_RigidPose3* const combination, const BGC_FP32_RigidPose3* const first, const BGC_FP32_RigidPose3* const second);
-extern inline void bgc_fp64_rigid_pose3_combine(BGC_FP64_RigidPose3* const combination, const BGC_FP64_RigidPose3* const first, const BGC_FP64_RigidPose3* const second);
+extern inline void bgc_fp32_rigid_pose3_combine(BGC_FP32_RigidPose3* const combination, const BGC_FP32_RigidPose3* const external_pose, const BGC_FP32_RigidPose3* const internal_pose);
+extern inline void bgc_fp64_rigid_pose3_combine(BGC_FP64_RigidPose3* const combination, const BGC_FP64_RigidPose3* const external_pose, const BGC_FP64_RigidPose3* const internal_pose);
 
-extern inline void bgc_fp32_rigid_pose3_exclude(BGC_FP32_RigidPose3* const difference, const BGC_FP32_RigidPose3* const base, const BGC_FP32_RigidPose3* const excludant);
-extern inline void bgc_fp64_rigid_pose3_exclude(BGC_FP64_RigidPose3* const difference, const BGC_FP64_RigidPose3* const base, const BGC_FP64_RigidPose3* const excludant);
+extern inline void bgc_fp32_rigid_pose3_exclude(BGC_FP32_RigidPose3* const difference, const BGC_FP32_RigidPose3* const pose, const BGC_FP32_RigidPose3* const excludant);
+extern inline void bgc_fp64_rigid_pose3_exclude(BGC_FP64_RigidPose3* const difference, const BGC_FP64_RigidPose3* const pose, const BGC_FP64_RigidPose3* const excludant);
 
 extern inline void bgc_fp32_rigid_pose3_get_matrix(BGC_FP32_Matrix3x3* const matrix, const BGC_FP32_RigidPose3* const pose);
 extern inline void bgc_fp64_rigid_pose3_get_matrix(BGC_FP64_Matrix3x3* const matrix, const BGC_FP64_RigidPose3* const pose);
diff --git a/basic-geometry/rigid-pose3.h b/basic-geometry/rigid-pose3.h
index 7486eaf..8bcac50 100644
--- a/basic-geometry/rigid-pose3.h
+++ b/basic-geometry/rigid-pose3.h
@@ -280,41 +280,41 @@ inline void bgc_fp64_rigid_pose3_get_reverse_pose(BGC_FP64_RigidPose3* const rev
 
 // ================== Combine =================== //
 
-inline void bgc_fp32_rigid_pose3_combine(BGC_FP32_RigidPose3* const combination, const BGC_FP32_RigidPose3* const first, const BGC_FP32_RigidPose3* const second)
+inline void bgc_fp32_rigid_pose3_combine(BGC_FP32_RigidPose3* const combination, const BGC_FP32_RigidPose3* const external_pose, const BGC_FP32_RigidPose3* const internal_pose)
 {
-    bgc_fp32_dual_quaternion_multiply_by_dual_quaternion(&combination->_versor, &second->_versor, &first->_versor);
+    bgc_fp32_dual_quaternion_multiply_by_dual_quaternion(&combination->_versor, &external_pose->_versor, &internal_pose->_versor);
     _bgc_fp32_rigid_pose3_normalize(combination);
 }
 
-inline void bgc_fp64_rigid_pose3_combine(BGC_FP64_RigidPose3* const combination, const BGC_FP64_RigidPose3* const first, const BGC_FP64_RigidPose3* const second)
+inline void bgc_fp64_rigid_pose3_combine(BGC_FP64_RigidPose3* const combination, const BGC_FP64_RigidPose3* const external_pose, const BGC_FP64_RigidPose3* const internal_pose)
 {
-    bgc_fp64_dual_quaternion_multiply_by_dual_quaternion(&combination->_versor, &second->_versor, &first->_versor);
+    bgc_fp64_dual_quaternion_multiply_by_dual_quaternion(&combination->_versor, &external_pose->_versor, &internal_pose->_versor);
     _bgc_fp64_rigid_pose3_normalize(combination);
 }
 
 // ================== Exclude =================== //
 
-inline void bgc_fp32_rigid_pose3_exclude(BGC_FP32_RigidPose3* const difference, const BGC_FP32_RigidPose3* const base, const BGC_FP32_RigidPose3* const excludant)
+inline void bgc_fp32_rigid_pose3_exclude(BGC_FP32_RigidPose3* const difference, const BGC_FP32_RigidPose3* const pose, const BGC_FP32_RigidPose3* const excludant)
 {
     BGC_FP32_Quaternion dual_part1, dual_part2;
 
-    bgc_fp32_quaternion_multiply_by_conjugate(&dual_part1, &base->_versor.real_part, &excludant->_versor.dual_part);
-    bgc_fp32_quaternion_multiply_by_conjugate(&dual_part2, &base->_versor.dual_part, &excludant->_versor.real_part);
+    bgc_fp32_quaternion_multiply_by_conjugate(&dual_part1, &pose->_versor.real_part, &excludant->_versor.dual_part);
+    bgc_fp32_quaternion_multiply_by_conjugate(&dual_part2, &pose->_versor.dual_part, &excludant->_versor.real_part);
 
-    bgc_fp32_quaternion_multiply_by_conjugate(&difference->_versor.real_part, &base->_versor.real_part, &excludant->_versor.real_part);
+    bgc_fp32_quaternion_multiply_by_conjugate(&difference->_versor.real_part, &pose->_versor.real_part, &excludant->_versor.real_part);
     bgc_fp32_quaternion_add(&difference->_versor.dual_part, &dual_part1, &dual_part2);
 
     _bgc_fp32_rigid_pose3_normalize(difference);
 }
 
-inline void bgc_fp64_rigid_pose3_exclude(BGC_FP64_RigidPose3* const difference, const BGC_FP64_RigidPose3* const base, const BGC_FP64_RigidPose3* const excludant)
+inline void bgc_fp64_rigid_pose3_exclude(BGC_FP64_RigidPose3* const difference, const BGC_FP64_RigidPose3* const pose, const BGC_FP64_RigidPose3* const excludant)
 {
     BGC_FP64_Quaternion dual_part1, dual_part2;
 
-    bgc_fp64_quaternion_multiply_by_conjugate(&dual_part1, &base->_versor.real_part, &excludant->_versor.dual_part);
-    bgc_fp64_quaternion_multiply_by_conjugate(&dual_part2, &base->_versor.dual_part, &excludant->_versor.real_part);
+    bgc_fp64_quaternion_multiply_by_conjugate(&dual_part1, &pose->_versor.real_part, &excludant->_versor.dual_part);
+    bgc_fp64_quaternion_multiply_by_conjugate(&dual_part2, &pose->_versor.dual_part, &excludant->_versor.real_part);
 
-    bgc_fp64_quaternion_multiply_by_conjugate(&difference->_versor.real_part, &base->_versor.real_part, &excludant->_versor.real_part);
+    bgc_fp64_quaternion_multiply_by_conjugate(&difference->_versor.real_part, &pose->_versor.real_part, &excludant->_versor.real_part);
     bgc_fp64_quaternion_add(&difference->_versor.dual_part, &dual_part1, &dual_part2);
 
     _bgc_fp64_rigid_pose3_normalize(difference);
diff --git a/basic-geometry/turn2.c b/basic-geometry/turn2.c
index db0a5db..f867b2d 100644
--- a/basic-geometry/turn2.c
+++ b/basic-geometry/turn2.c
@@ -43,11 +43,8 @@ extern inline void bgc_fp64_turn2_get_power(BGC_FP64_Turn2* const power, const B
 extern inline void bgc_fp32_turn2_combine(BGC_FP32_Turn2* const combination, const BGC_FP32_Turn2* const turn1, const BGC_FP32_Turn2* const turn2);
 extern inline void bgc_fp64_turn2_combine(BGC_FP64_Turn2* const combination, const BGC_FP64_Turn2* const turn1, const BGC_FP64_Turn2* const turn2);
 
-extern inline void bgc_fp32_turn2_combine3(BGC_FP32_Turn2* const combination, const BGC_FP32_Turn2* const turn1, const BGC_FP32_Turn2* const turn2, const BGC_FP32_Turn2* const turn3);
-extern inline void bgc_fp64_turn2_combine3(BGC_FP64_Turn2* const combination, const BGC_FP64_Turn2* const turn1, const BGC_FP64_Turn2* const turn2, const BGC_FP64_Turn2* const turn3);
-
-extern inline void bgc_fp32_turn2_exclude(BGC_FP32_Turn2* const difference, const BGC_FP32_Turn2* const base, const BGC_FP32_Turn2* const excludant);
-extern inline void bgc_fp64_turn2_exclude(BGC_FP64_Turn2* const difference, const BGC_FP64_Turn2* const base, const BGC_FP64_Turn2* const excludant);
+extern inline void bgc_fp32_turn2_exclude(BGC_FP32_Turn2* const difference, const BGC_FP32_Turn2* const turn, const BGC_FP32_Turn2* const excludant);
+extern inline void bgc_fp64_turn2_exclude(BGC_FP64_Turn2* const difference, const BGC_FP64_Turn2* const turn, const BGC_FP64_Turn2* const excludant);
 
 extern inline void bgc_fp32_turn2_get_rotation_matrix(BGC_FP32_Matrix2x2* const matrix, const BGC_FP32_Turn2* const turn);
 extern inline void bgc_fp64_turn2_get_rotation_matrix(BGC_FP64_Matrix2x2* const matrix, const BGC_FP64_Turn2* const turn);
diff --git a/basic-geometry/turn2.h b/basic-geometry/turn2.h
index 9dc23d9..a5d139d 100644
--- a/basic-geometry/turn2.h
+++ b/basic-geometry/turn2.h
@@ -253,42 +253,12 @@ inline void bgc_fp64_turn2_combine(BGC_FP64_Turn2* const combination, const BGC_
     _bgc_fp64_turn2_normalize(combination);
 }
 
-// ============ Combination of three ============ //
-
-inline void bgc_fp32_turn2_combine3(BGC_FP32_Turn2* const combination, const BGC_FP32_Turn2* const turn1, const BGC_FP32_Turn2* const turn2, const BGC_FP32_Turn2* const turn3)
-{
-    const float _cos_of_two = turn1->_cos * turn2->_cos - turn1->_sin * turn2->_sin;
-    const float _sin_of_two = turn1->_cos * turn2->_sin + turn1->_sin * turn2->_cos;
-
-    const float _cos_of_three = _cos_of_two * turn3->_cos - _sin_of_two * turn3->_sin;
-    const float _sin_of_three = _cos_of_two * turn3->_sin + _sin_of_two * turn3->_cos;
-
-    combination->_cos = _cos_of_three;
-    combination->_sin = _sin_of_three;
-
-    _bgc_fp32_turn2_normalize(combination);
-}
-
-inline void bgc_fp64_turn2_combine3(BGC_FP64_Turn2* const combination, const BGC_FP64_Turn2* const turn1, const BGC_FP64_Turn2* const turn2, const BGC_FP64_Turn2* const turn3)
-{
-    const double _cos_of_two = turn1->_cos * turn2->_cos - turn1->_sin * turn2->_sin;
-    const double _sin_of_two = turn1->_cos * turn2->_sin + turn1->_sin * turn2->_cos;
-
-    const double _cos_of_three = _cos_of_two * turn3->_cos - _sin_of_two * turn3->_sin;
-    const double _sin_of_three = _cos_of_two * turn3->_sin + _sin_of_two * turn3->_cos;
-
-    combination->_cos = _cos_of_three;
-    combination->_sin = _sin_of_three;
-
-    _bgc_fp64_turn2_normalize(combination);
-}
-
 // ================= Exclusion ================== //
 
-inline void bgc_fp32_turn2_exclude(BGC_FP32_Turn2* const difference, const BGC_FP32_Turn2* const base, const BGC_FP32_Turn2* const excludant)
+inline void bgc_fp32_turn2_exclude(BGC_FP32_Turn2* const difference, const BGC_FP32_Turn2* const turn, const BGC_FP32_Turn2* const excludant)
 {
-    const float _cos = base->_cos * excludant->_cos + base->_sin * excludant->_sin;
-    const float _sin = base->_sin * excludant->_cos - base->_cos * excludant->_sin;
+    const float _cos = turn->_cos * excludant->_cos + turn->_sin * excludant->_sin;
+    const float _sin = turn->_sin * excludant->_cos - turn->_cos * excludant->_sin;
 
     difference->_cos = _cos;
     difference->_sin = _sin;
@@ -296,10 +266,10 @@ inline void bgc_fp32_turn2_exclude(BGC_FP32_Turn2* const difference, const BGC_F
     _bgc_fp32_turn2_normalize(difference);
 }
 
-inline void bgc_fp64_turn2_exclude(BGC_FP64_Turn2* const difference, const BGC_FP64_Turn2* const base, const BGC_FP64_Turn2* const excludant)
+inline void bgc_fp64_turn2_exclude(BGC_FP64_Turn2* const difference, const BGC_FP64_Turn2* const turn, const BGC_FP64_Turn2* const excludant)
 {
-    const double _cos = base->_cos * excludant->_cos + base->_sin * excludant->_sin;
-    const double _sin = base->_sin * excludant->_cos - base->_cos * excludant->_sin;
+    const double _cos = turn->_cos * excludant->_cos + turn->_sin * excludant->_sin;
+    const double _sin = turn->_sin * excludant->_cos - turn->_cos * excludant->_sin;
 
     difference->_cos = _cos;
     difference->_sin = _sin;
diff --git a/basic-geometry/turn3.c b/basic-geometry/turn3.c
index 68f3104..98d78f5 100644
--- a/basic-geometry/turn3.c
+++ b/basic-geometry/turn3.c
@@ -6,6 +6,8 @@ const BGC_FP32_Turn3 BGC_FP32_IDLE_TURN3 = {{ 1.0f, 0.0f, 0.0f, 0.0f }};
 
 const BGC_FP64_Turn3 BGC_FP64_IDLE_TURN3 = {{ 1.0, 0.0, 0.0, 0.0 }};
 
+int64_t turn3_normalize_counter = 0;
+
 extern inline void bgc_fp32_turn3_reset(BGC_FP32_Turn3* const turn);
 extern inline void bgc_fp64_turn3_reset(BGC_FP64_Turn3* const turn);
 
@@ -51,14 +53,11 @@ extern inline void bgc_fp64_turn3_revert(BGC_FP64_Turn3* const turn);
 extern inline void bgc_fp32_turn3_get_reverse(BGC_FP32_Turn3* const inverse, const BGC_FP32_Turn3* const turn);
 extern inline void bgc_fp64_turn3_get_reverse(BGC_FP64_Turn3* const inverse, const BGC_FP64_Turn3* const turn);
 
-extern inline void bgc_fp32_turn3_combine(BGC_FP32_Turn3* const combination, const BGC_FP32_Turn3* const first, const BGC_FP32_Turn3* const second);
-extern inline void bgc_fp64_turn3_combine(BGC_FP64_Turn3* const combination, const BGC_FP64_Turn3* const first, const BGC_FP64_Turn3* const second);
+extern inline void bgc_fp32_turn3_combine(BGC_FP32_Turn3* const combination, const BGC_FP32_Turn3* const external_turn, const BGC_FP32_Turn3* const internal_turn);
+extern inline void bgc_fp64_turn3_combine(BGC_FP64_Turn3* const combination, const BGC_FP64_Turn3* const external_turn, const BGC_FP64_Turn3* const internal_turn);
 
-extern inline void bgc_fp32_turn3_combine3(BGC_FP32_Turn3* const combination, const BGC_FP32_Turn3* const first, const BGC_FP32_Turn3* const second, const BGC_FP32_Turn3* const third);
-extern inline void bgc_fp64_turn3_combine3(BGC_FP64_Turn3* const combination, const BGC_FP64_Turn3* const first, const BGC_FP64_Turn3* const second, const BGC_FP64_Turn3* const third);
-
-extern inline void bgc_fp32_turn3_exclude(BGC_FP32_Turn3* const difference, const BGC_FP32_Turn3* const base, const BGC_FP32_Turn3* const excludant);
-extern inline void bgc_fp64_turn3_exclude(BGC_FP64_Turn3* const difference, const BGC_FP64_Turn3* const base, const BGC_FP64_Turn3* const excludant);
+extern inline void bgc_fp32_turn3_exclude(BGC_FP32_Turn3* const difference, const BGC_FP32_Turn3* const turn, const BGC_FP32_Turn3* const excludant);
+extern inline void bgc_fp64_turn3_exclude(BGC_FP64_Turn3* const difference, const BGC_FP64_Turn3* const turn, const BGC_FP64_Turn3* const excludant);
 
 extern inline void bgc_fp32_turn3_get_rotation_matrix(BGC_FP32_Matrix3x3* const matrix, const BGC_FP32_Turn3* const turn);
 extern inline void bgc_fp64_turn3_get_rotation_matrix(BGC_FP64_Matrix3x3* const matrix, const BGC_FP64_Turn3* const turn);
diff --git a/basic-geometry/turn3.h b/basic-geometry/turn3.h
index d8c0599..5a2ca98 100644
--- a/basic-geometry/turn3.h
+++ b/basic-geometry/turn3.h
@@ -52,6 +52,8 @@ inline void bgc_fp64_turn3_reset(BGC_FP64_Turn3* const turn)
 
 // ============= Private: Normalize ============= //
 
+extern int64_t turn3_normalize_counter;
+
 inline void _bgc_fp32_turn3_normalize(BGC_FP32_Turn3* const turn)
 {
     const float square_magnitude = bgc_fp32_quaternion_get_square_magnitude(&turn->_versor);
@@ -323,54 +325,30 @@ void bgc_fp64_turn3_get_power(BGC_FP64_Turn3* const power, const BGC_FP64_Turn3*
 
 // ================ Combination ================= //
 
-inline void bgc_fp32_turn3_combine(BGC_FP32_Turn3* const combination, const BGC_FP32_Turn3* const first, const BGC_FP32_Turn3* const second)
+inline void bgc_fp32_turn3_combine(BGC_FP32_Turn3* const combination, const BGC_FP32_Turn3* const external_turn, const BGC_FP32_Turn3* const internal_turn)
 {
-    bgc_fp32_quaternion_multiply_by_quaternion(&combination->_versor, &second->_versor, &first->_versor);
+    bgc_fp32_quaternion_multiply_by_quaternion(&combination->_versor, &external_turn->_versor, &internal_turn->_versor);
     _bgc_fp32_turn3_normalize(combination);
 }
 
-inline void bgc_fp64_turn3_combine(BGC_FP64_Turn3* const combination, const BGC_FP64_Turn3* const first, const BGC_FP64_Turn3* const second)
+inline void bgc_fp64_turn3_combine(BGC_FP64_Turn3* const combination, const BGC_FP64_Turn3* const external_turn, const BGC_FP64_Turn3* const internal_turn)
 {
-    bgc_fp64_quaternion_multiply_by_quaternion(&combination->_versor, &second->_versor, &first->_versor);
-    _bgc_fp64_turn3_normalize(combination);
-}
-
-// ============ Combination of three ============ //
-
-inline void bgc_fp32_turn3_combine3(BGC_FP32_Turn3* const combination, const BGC_FP32_Turn3* const first, const BGC_FP32_Turn3* const second, const BGC_FP32_Turn3* const 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);
-
-    _bgc_fp32_turn3_normalize(combination);
-}
-
-inline void bgc_fp64_turn3_combine3(BGC_FP64_Turn3* const combination, const BGC_FP64_Turn3* const first, const BGC_FP64_Turn3* const second, const BGC_FP64_Turn3* const 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);
-
+    bgc_fp64_quaternion_multiply_by_quaternion(&combination->_versor, &external_turn->_versor, &internal_turn->_versor);
     _bgc_fp64_turn3_normalize(combination);
 }
 
 // ================= Exclusion ================== //
 
-inline void bgc_fp32_turn3_exclude(BGC_FP32_Turn3* const difference, const BGC_FP32_Turn3* const base, const BGC_FP32_Turn3* const excludant)
+inline void bgc_fp32_turn3_exclude(BGC_FP32_Turn3* const difference, const BGC_FP32_Turn3* const turn, const BGC_FP32_Turn3* const excludant)
 {
-    bgc_fp32_quaternion_multiply_by_conjugate(&difference->_versor, &base->_versor, &excludant->_versor);
+    bgc_fp32_quaternion_multiply_by_conjugate(&difference->_versor, &turn->_versor, &excludant->_versor);
 
     _bgc_fp32_turn3_normalize(difference);
 }
 
-inline void bgc_fp64_turn3_exclude(BGC_FP64_Turn3* const difference, const BGC_FP64_Turn3* const base, const BGC_FP64_Turn3* const excludant)
+inline void bgc_fp64_turn3_exclude(BGC_FP64_Turn3* const difference, const BGC_FP64_Turn3* const turn, const BGC_FP64_Turn3* const excludant)
 {
-    bgc_fp64_quaternion_multiply_by_conjugate(&difference->_versor, &base->_versor, &excludant->_versor);
+    bgc_fp64_quaternion_multiply_by_conjugate(&difference->_versor, &turn->_versor, &excludant->_versor);
 
     _bgc_fp64_turn3_normalize(difference);
 }