Исправление функции, которая находит трёхмерных поворот между двумя парами векторов

basic-geometry-dev/main.c

@@ -79,7 +79,7 @@ void list_work(const uint_fast32_t amount, structure_fp32_t* list)
         }
     }
 }
-
+/*
 int main()
 {
     const unsigned int amount = 1000000;
@@ -121,7 +121,7 @@ int main()
 
     return 0;
 }
-
+*/
 
 /*
 int main() {
@@ -149,84 +149,84 @@ int main() {
 }
 */
 
-void test_basis_difference_fp32()
+void test_pair_difference_fp32()
 {
 
-    BGC_FP32_Vector3 initial_primary, initial_auxiliary;
+    BGC_FP32_Vector3 initial_main, initial_branch;
-    BGC_FP32_Vector3 final_primary, final_auxiliary;
+    BGC_FP32_Vector3 final_main, final_branch;
     BGC_FP32_Turn3 turn;
 
     // No turn
-    bgc_fp32_vector3_make(&initial_primary,   1.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&initial_main,   1.0f, 0.0f, 0.0f);
-    bgc_fp32_vector3_make(&initial_auxiliary, 0.0f, 1.0f, 0.0f);
+    bgc_fp32_vector3_make(&initial_branch, 0.0f, 1.0f, 0.0f);
 
-    bgc_fp32_vector3_make(&final_primary,   1.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&final_main,   1.0f, 0.0f, 0.0f);
-    bgc_fp32_vector3_make(&final_auxiliary, 0.0f, 1.0f, 0.0f);
+    bgc_fp32_vector3_make(&final_branch, 0.0f, 1.0f, 0.0f);
 
-    bgc_fp32_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    bgc_fp32_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
     printf("\nNo turn:\n");
     print_quaternion_fp32(&turn._versor);
 
     // Turn around (1, 1, 0) axis on 180 degrees
-    bgc_fp32_vector3_make(&initial_primary,   1.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&initial_main,   1.0f, 0.0f, 0.0f);
-    bgc_fp32_vector3_make(&initial_auxiliary, 0.0f, 1.0f, 0.0f);
+    bgc_fp32_vector3_make(&initial_branch, 0.0f, 1.0f, 0.0f);
 
-    bgc_fp32_vector3_make(&final_primary,   0.0f, 1.0f, 0.0f);
+    bgc_fp32_vector3_make(&final_main,   0.0f, 1.0f, 0.0f);
-    bgc_fp32_vector3_make(&final_auxiliary, 1.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&final_branch, 1.0f, 0.0f, 0.0f);
 
-    bgc_fp32_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    bgc_fp32_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
     printf("\nTurn around (1, 1, 0) axis on 180 degrees:\n");
     print_quaternion_fp32(&turn._versor);
 
     // 180 degree turn
-    bgc_fp32_vector3_make(&initial_primary,   1.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&initial_main,   1.0f, 0.0f, 0.0f);
-    bgc_fp32_vector3_make(&initial_auxiliary, 0.0f, 1.0f, 0.0f);
+    bgc_fp32_vector3_make(&initial_branch, 0.0f, 1.0f, 0.0f);
 
-    bgc_fp32_vector3_make(&final_primary,  -1.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&final_main,  -1.0f, 0.0f, 0.0f);
-    bgc_fp32_vector3_make(&final_auxiliary, 0.0f, 1.0f, 0.0f);
+    bgc_fp32_vector3_make(&final_branch, 0.0f, 1.0f, 0.0f);
 
-    bgc_fp32_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    bgc_fp32_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
     printf("\n180 degree turn around (0, 1, 0):\n");
     print_quaternion_fp32(&turn._versor);
 
     // 90 degree turn around x3 axis
-    bgc_fp32_vector3_make(&initial_primary,   2.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&initial_main,   2.0f, 0.0f, 0.0f);
-    bgc_fp32_vector3_make(&initial_auxiliary, 0.0f, 3.1f, 0.0f);
+    bgc_fp32_vector3_make(&initial_branch, 0.0f, 3.1f, 0.0f);
 
-    bgc_fp32_vector3_make(&final_primary,  0.0f, 10.0f, 0.0f);
+    bgc_fp32_vector3_make(&final_main,  0.0f, 10.0f, 0.0f);
-    bgc_fp32_vector3_make(&final_auxiliary,-1.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&final_branch,-1.0f, 0.0f, 0.0f);
 
-    bgc_fp32_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    bgc_fp32_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
     printf("\n90 degree turn around (0, 0, 1):\n");
     print_quaternion_fp32(&turn._versor);
 
     // Unorthogonal pairs turn at 90 degrees around x3 axis
-    bgc_fp32_vector3_make(&initial_primary,    2.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&initial_main,    2.0f, 0.0f, 0.0f);
-    bgc_fp32_vector3_make(&initial_auxiliary, -2.0f, 3.1f, 0.0f);
+    bgc_fp32_vector3_make(&initial_branch, -2.0f, 3.1f, 0.0f);
 
-    bgc_fp32_vector3_make(&final_primary, 0.0f, 10.0f, 0.0f);
+    bgc_fp32_vector3_make(&final_main, 0.0f, 10.0f, 0.0f);
-    bgc_fp32_vector3_make(&final_auxiliary, -1.0f, 5.0f, 0.0f);
+    bgc_fp32_vector3_make(&final_branch, -1.0f, 5.0f, 0.0f);
 
-    bgc_fp32_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    bgc_fp32_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
     printf("\nUnorthogonal pairs turn at 90 degrees around (0, 0, 1):\n");
     print_quaternion_fp32(&turn._versor);
 
     // Zero vectors
-    bgc_fp32_vector3_make(&initial_primary,   0.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&initial_main,   0.0f, 0.0f, 0.0f);
-    bgc_fp32_vector3_make(&initial_auxiliary, 0.0f, 1.0f, 0.0f);
+    bgc_fp32_vector3_make(&initial_branch, 0.0f, 1.0f, 0.0f);
-    bgc_fp32_vector3_make(&final_primary,     1.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&final_main,     1.0f, 0.0f, 0.0f);
-    bgc_fp32_vector3_make(&final_auxiliary,   0.0f, 1.0f, 0.0f);
+    bgc_fp32_vector3_make(&final_branch,   0.0f, 1.0f, 0.0f);
 
     int code;
 
-    code = bgc_fp32_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    code = bgc_fp32_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
-    if (code >= 0) {
+    if (code == BGC_SUCCESS) {
         printf("\nZero vectors: this cannot be!\n");
         print_quaternion_fp32(&turn._versor);
     }
@@ -235,14 +235,14 @@ void test_basis_difference_fp32()
     }
 
     // Parallel vectors
-    bgc_fp32_vector3_make(&initial_primary,   1.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&initial_main,   1.0f, 0.0f, 0.0f);
-    bgc_fp32_vector3_make(&initial_auxiliary, 2.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&initial_branch, 2.0f, 0.0f, 0.0f);
-    bgc_fp32_vector3_make(&final_primary,     1.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&final_main,     1.0f, 0.0f, 0.0f);
-    bgc_fp32_vector3_make(&final_auxiliary,   0.0f, 1.0f, 0.0f);
+    bgc_fp32_vector3_make(&final_branch,   0.0f, 1.0f, 0.0f);
 
-    code = bgc_fp32_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    code = bgc_fp32_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
-    if (code >= 0) {
+    if (code == BGC_SUCCESS) {
         printf("\nParallel vectors: this cannot be!\n");
         print_quaternion_fp32(&turn._versor);
     }
@@ -251,141 +251,171 @@ void test_basis_difference_fp32()
     }
 
     // Small angle turn (about 1 degree):
-    bgc_fp32_vector3_make(&initial_primary,   1.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&initial_main,   1.0f, 0.0f, 0.0f);
-    bgc_fp32_vector3_make(&initial_auxiliary, 0.0f, 1.0f, 0.0f);
+    bgc_fp32_vector3_make(&initial_branch, 0.0f, 1.0f, 0.0f);
 
-    bgc_fp32_vector3_make(&final_primary,    0.999848f, 0.017452f, 0.0f);
+    bgc_fp32_vector3_make(&final_main,    0.999848f, 0.017452f, 0.0f);
-    bgc_fp32_vector3_make(&final_auxiliary, -0.017452f, 0.999848f, 0.0f);
+    bgc_fp32_vector3_make(&final_branch, -0.017452f, 0.999848f, 0.0f);
 
-    bgc_fp32_turn3_make_basis_difference(&turn , &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    bgc_fp32_turn3_find_pair_difference(&turn , &initial_main, &initial_branch, &final_main, &final_branch);
 
     printf("\nSmall angle turn (about 1 degree):\n");
     print_quaternion_fp32(&turn._versor);
 
     // About 179 degrees turn
-    bgc_fp32_vector3_make(&initial_primary,   1.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&initial_main,   1.0f, 0.0f, 0.0f);
-    bgc_fp32_vector3_make(&initial_auxiliary, 0.0f, 1.0f, 0.0f);
+    bgc_fp32_vector3_make(&initial_branch, 0.0f, 1.0f, 0.0f);
 
-    bgc_fp32_vector3_make(&final_primary,  -0.999848f, -0.017452f, 0.0f);
+    bgc_fp32_vector3_make(&final_main,  -0.999848f, -0.017452f, 0.0f);
-    bgc_fp32_vector3_make(&final_auxiliary, 0.017452f, -0.999848f, 0.0f);
+    bgc_fp32_vector3_make(&final_branch, 0.017452f, -0.999848f, 0.0f);
 
-    bgc_fp32_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    bgc_fp32_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
     printf("\nAbout 179 degrees turn:\n");
     print_quaternion_fp32(&turn._versor);
 
     // 120 degrees around (-1, -1, 1)
-    bgc_fp32_vector3_make(&initial_primary,   1.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&initial_main,   1.0f, 0.0f, 0.0f);
-    bgc_fp32_vector3_make(&initial_auxiliary, 0.0f, 1.0f, 0.0f);
+    bgc_fp32_vector3_make(&initial_branch, 0.0f, 1.0f, 0.0f);
 
-    bgc_fp32_vector3_make(&final_primary,   0.0f, 1.0f, 0.0f);
+    bgc_fp32_vector3_make(&final_main,   0.0f, 1.0f, 0.0f);
-    bgc_fp32_vector3_make(&final_auxiliary, 0.0f, 0.0f, -1.0f);
+    bgc_fp32_vector3_make(&final_branch, 0.0f, 0.0f, -1.0f);
 
-    bgc_fp32_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    bgc_fp32_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
     printf("\n120 degees turn:\n");
     print_quaternion_fp32(&turn._versor);
 
 
-    // About 1 degree turn difference between initial_primary and initial_auxiliary directions
+    // About 1 degree turn difference between initial_main and initial_branch directions
-    bgc_fp32_vector3_make(&initial_primary,   1.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&initial_main,   1.0f, 0.0f, 0.0f);
-    bgc_fp32_vector3_make(&initial_auxiliary, 0.999848f, 0.017452f, 0.0f);
+    bgc_fp32_vector3_make(&initial_branch, 0.999848f, 0.017452f, 0.0f);
-    bgc_fp32_vector3_make(&final_primary, 0.0f, 1.0f, 0.0f);
+    bgc_fp32_vector3_make(&final_main, 0.0f, 1.0f, 0.0f);
-    bgc_fp32_vector3_make(&final_auxiliary, -1.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&final_branch, -1.0f, 0.0f, 0.0f);
 
-    bgc_fp32_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    bgc_fp32_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
-    printf("\nAbout 1 degree turn difference between initial_primary and initial_auxiliary directions:\n");
+    printf("\nAbout 1 degree turn difference between initial_main and initial_branch directions:\n");
     print_quaternion_fp32(&turn._versor);
 
-    // About 0.01 degree turn difference between initial_primary and initial_auxiliary directions
+    // About 0.01 degree turn difference between initial_main and initial_branch directions
-    bgc_fp32_vector3_make(&initial_primary,   1.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&initial_main,   1.0f, 0.0f, 0.0f);
-    bgc_fp32_vector3_make(&initial_auxiliary, 1.0f, 0.000001f, 0.0f);
+    bgc_fp32_vector3_make(&initial_branch, 1.0f, 0.000001f, 0.0f);
-    bgc_fp32_vector3_make(&final_primary,     0.0f, -1.0f, 0.0f);
+    bgc_fp32_vector3_make(&final_main,     0.0f, -1.0f, 0.0f);
-    bgc_fp32_vector3_make(&final_auxiliary,   1.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&final_branch,   1.0f, 0.0f, 0.0f);
 
-    bgc_fp32_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    bgc_fp32_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
-    printf("\nAbout 0.01 degree turn difference between initial_primary and initial_auxiliary directions:\n");
+    printf("\nAbout 0.01 degree turn difference between initial_main and initial_branch directions:\n");
     print_quaternion_fp32(&turn._versor);
+
+    bgc_fp32_vector3_make(&initial_main,   1.0f, 0.0f, 0.0f);
+    bgc_fp32_vector3_make(&initial_branch, 0.0f, 0.999999f, 0.00014142f); // почти (0,1,0), но крошечный z
+
+    bgc_fp32_vector3_make(&final_main,    -0.999999f, 0.0f, 0.00014142f);
+    bgc_fp32_vector3_make(&final_branch,   0.0f, 0.999999f, -0.00014142f);
+
+    bgc_fp32_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
+    printf("\nNear 180° with tiny branch deviation:\n");
+    print_quaternion_fp32(&turn._versor);
+
+    bgc_fp32_vector3_make(&initial_main,   1.0f, 0.2f, 0.1f);
+    bgc_fp32_vector3_make(&initial_branch, 0.1f, 1.0f, 0.3f); // почти (0,1,0), но крошечный z
+
+    BGC_FP32_Turn3 known;
+
+    bgc_fp32_turn3_set_rotation(&known, 0.0f, 0.0f, 1.0f, 90.0f, BGC_ANGLE_UNIT_DEGREES);
+
+    bgc_fp32_vector3_make(&initial_main,    -0.999999f, 0.0f, 0.00014142f);
+    bgc_fp32_vector3_make(&initial_branch,   0.0f, 0.999999f, -0.00014142f);
+
+    bgc_fp32_turn3_vector(&final_main,   &known, &initial_main);
+    bgc_fp32_turn3_vector(&final_branch, &known, &initial_branch);
+
+    bgc_fp32_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
+    printf("\nRecover known 90° Z rotation:\n");
+    print_quaternion_fp32(&turn._versor);
+
+    printf("Known was: ");
+    print_quaternion_fp32(&known._versor);
 }
 
-void test_basis_difference_fp64()
+void test_pair_difference_fp64()
 {
 
-    BGC_FP64_Vector3 initial_primary, initial_auxiliary;
+    BGC_FP64_Vector3 initial_main, initial_branch;
-    BGC_FP64_Vector3 final_primary, final_auxiliary;
+    BGC_FP64_Vector3 final_main, final_branch;
     BGC_FP64_Turn3 turn;
 
     // No turn
-    bgc_fp64_vector3_make(&initial_primary,   1.0, 0.0, 0.0);
+    bgc_fp64_vector3_make(&initial_main,   1.0, 0.0, 0.0);
-    bgc_fp64_vector3_make(&initial_auxiliary, 0.0, 1.0, 0.0);
+    bgc_fp64_vector3_make(&initial_branch, 0.0, 1.0, 0.0);
-    bgc_fp64_vector3_make(&final_primary,     1.0, 0.0, 0.0);
+    bgc_fp64_vector3_make(&final_main,     1.0, 0.0, 0.0);
-    bgc_fp64_vector3_make(&final_auxiliary,   0.0, 1.0, 0.0);
+    bgc_fp64_vector3_make(&final_branch,   0.0, 1.0, 0.0);
 
-    bgc_fp64_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    bgc_fp64_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
     printf("\nNo turn:\n");
     print_quaternion_fp64(&turn._versor);
 
     // Turn around (1, 1, 0) axis on 180 degrees
-    bgc_fp64_vector3_make(&initial_primary,   1.0, 0.0, 0.0);
+    bgc_fp64_vector3_make(&initial_main,   1.0, 0.0, 0.0);
-    bgc_fp64_vector3_make(&initial_auxiliary, 0.0, 1.0, 0.0);
+    bgc_fp64_vector3_make(&initial_branch, 0.0, 1.0, 0.0);
-    bgc_fp64_vector3_make(&final_primary,     0.0, 1.0, 0.0);
+    bgc_fp64_vector3_make(&final_main,     0.0, 1.0, 0.0);
-    bgc_fp64_vector3_make(&final_auxiliary,   1.0, 0.0, 0.0);
+    bgc_fp64_vector3_make(&final_branch,   1.0, 0.0, 0.0);
 
-    bgc_fp64_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    bgc_fp64_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
     printf("\nTurn around (1, 1, 0) axis on 180 degrees:\n");
     print_quaternion_fp64(&turn._versor);
 
     // 180 degree turn
-    bgc_fp64_vector3_make(&initial_primary,   1.0, 0.0, 0.0);
+    bgc_fp64_vector3_make(&initial_main,   1.0, 0.0, 0.0);
-    bgc_fp64_vector3_make(&initial_auxiliary, 0.0, 1.0, 0.0);
+    bgc_fp64_vector3_make(&initial_branch, 0.0, 1.0, 0.0);
 
-    bgc_fp64_vector3_make(&initial_auxiliary, -1.0, 0.0, 0.0);
+    bgc_fp64_vector3_make(&initial_branch, -1.0, 0.0, 0.0);
-    bgc_fp64_vector3_make(&final_auxiliary,    0.0, 1.0, 0.0);
+    bgc_fp64_vector3_make(&final_branch,    0.0, 1.0, 0.0);
 
-    bgc_fp64_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    bgc_fp64_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
     printf("\n180 degree turn around (0, 1, 0):\n");
     print_quaternion_fp64(&turn._versor);
 
     // 90 degree turn around x3 axis
-    bgc_fp64_vector3_make(&initial_primary,   2.0, 0.0, 0.0);
+    bgc_fp64_vector3_make(&initial_main,   2.0, 0.0, 0.0);
-    bgc_fp64_vector3_make(&initial_auxiliary, 0.0, 3.1, 0.0);
+    bgc_fp64_vector3_make(&initial_branch, 0.0, 3.1, 0.0);
 
-    bgc_fp64_vector3_make(&final_primary,   0.0, 10.0, 0.0);
+    bgc_fp64_vector3_make(&final_main,   0.0, 10.0, 0.0);
-    bgc_fp64_vector3_make(&final_auxiliary, -1.0, 0.0, 0.0);
+    bgc_fp64_vector3_make(&final_branch, -1.0, 0.0, 0.0);
 
-    bgc_fp64_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    bgc_fp64_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
     printf("\n90 degree turn around (0, 0, 1):\n");
     print_quaternion_fp64(&turn._versor);
 
     // Unorthogonal pairs turn at 90 degrees around x3 axis
-    bgc_fp64_vector3_make(&initial_primary,    2.0, 0.0, 0.0);
+    bgc_fp64_vector3_make(&initial_main,    2.0, 0.0, 0.0);
-    bgc_fp64_vector3_make(&initial_auxiliary, -2.0, 3.1, 0.0);
+    bgc_fp64_vector3_make(&initial_branch, -2.0, 3.1, 0.0);
 
-    bgc_fp64_vector3_make(&final_primary,   0.0, 10.0, 0.0);
+    bgc_fp64_vector3_make(&final_main,   0.0, 10.0, 0.0);
-    bgc_fp64_vector3_make(&final_auxiliary, -1.0, 5.0, 0.0);
+    bgc_fp64_vector3_make(&final_branch, -1.0, 5.0, 0.0);
 
-    bgc_fp64_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    bgc_fp64_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
     printf("\nUnorthogonal pairs turn at 90 degrees around (0, 0, 1):\n");
     print_quaternion_fp64(&turn._versor);
 
     // Zero vectors
-    bgc_fp64_vector3_make(&initial_primary,   0.0, 0.0, 0.0);
+    bgc_fp64_vector3_make(&initial_main,   0.0, 0.0, 0.0);
-    bgc_fp64_vector3_make(&initial_auxiliary, 0.0, 1.0, 0.0);
+    bgc_fp64_vector3_make(&initial_branch, 0.0, 1.0, 0.0);
-    bgc_fp64_vector3_make(&final_primary,     1.0, 0.0, 0.0);
+    bgc_fp64_vector3_make(&final_main,     1.0, 0.0, 0.0);
-    bgc_fp64_vector3_make(&final_auxiliary,   0.0, 1.0, 0.0);
+    bgc_fp64_vector3_make(&final_branch,   0.0, 1.0, 0.0);
 
     int code;
 
-    code = bgc_fp64_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    code = bgc_fp64_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
-    if (code >= 0) {
+    if (code == BGC_SUCCESS) {
         printf("\nZero vectors: this cannot be!\n");
         print_quaternion_fp64(&turn._versor);
     }
@@ -394,14 +424,14 @@ void test_basis_difference_fp64()
     }
 
     // Parallel vectors
-    bgc_fp64_vector3_make(&initial_primary,   1.0, 0.0, 0.0);
+    bgc_fp64_vector3_make(&initial_main,   1.0, 0.0, 0.0);
-    bgc_fp64_vector3_make(&initial_auxiliary, 2.0, 0.0, 0.0);
+    bgc_fp64_vector3_make(&initial_branch, 2.0, 0.0, 0.0);
-    bgc_fp64_vector3_make(&final_primary,     1.0, 0.0, 0.0);
+    bgc_fp64_vector3_make(&final_main,     1.0, 0.0, 0.0);
-    bgc_fp64_vector3_make(&final_auxiliary,   0.0, 1.0, 0.0);
+    bgc_fp64_vector3_make(&final_branch,   0.0, 1.0, 0.0);
 
-    code = bgc_fp64_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    code = bgc_fp64_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
-    if (code >= 0) {
+    if (code == BGC_SUCCESS) {
         printf("\nParallel vectors: this cannot be!\n");
         print_quaternion_fp64(&turn._versor);
     }
@@ -410,82 +440,71 @@ void test_basis_difference_fp64()
     }
 
     // Small angle turn (about 1 degree):
-    bgc_fp64_vector3_make(&initial_primary,   1.0, 0.0, 0.0);
+    bgc_fp64_vector3_make(&initial_main,   1.0, 0.0, 0.0);
-    bgc_fp64_vector3_make(&initial_auxiliary, 0.0, 1.0, 0.0);
+    bgc_fp64_vector3_make(&initial_branch, 0.0, 1.0, 0.0);
 
-    bgc_fp64_vector3_make(&final_primary,    0.999848, 0.017452, 0.0);
+    bgc_fp64_vector3_make(&final_main,    0.999848, 0.017452, 0.0);
-    bgc_fp64_vector3_make(&final_auxiliary, -0.017452, 0.999848, 0.0);
+    bgc_fp64_vector3_make(&final_branch, -0.017452, 0.999848, 0.0);
 
-    bgc_fp64_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    bgc_fp64_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
     printf("\nSmall angle turn (about 1 degree):\n");
     print_quaternion_fp64(&turn._versor);
 
     // About 179 degrees turn
-    bgc_fp64_vector3_make(&initial_primary,   1.0, 0.0, 0.0);
+    bgc_fp64_vector3_make(&initial_main,   1.0, 0.0, 0.0);
-    bgc_fp64_vector3_make(&initial_auxiliary, 0.0, 1.0, 0.0);
+    bgc_fp64_vector3_make(&initial_branch, 0.0, 1.0, 0.0);
 
-    bgc_fp64_vector3_make(&final_primary,  -0.999848, -0.017452, 0.0);
+    bgc_fp64_vector3_make(&final_main,  -0.999848, -0.017452, 0.0);
-    bgc_fp64_vector3_make(&final_auxiliary, 0.017452, -0.999848, 0.0);
+    bgc_fp64_vector3_make(&final_branch, 0.017452, -0.999848, 0.0);
 
-    bgc_fp64_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    bgc_fp64_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
     printf("\nAbout 179 degrees turn:\n");
     print_quaternion_fp64(&turn._versor);
 
     // 120 degrees around (-1, -1, 1)
-    bgc_fp64_vector3_make(&initial_primary,   1.0, 0.0, 0.0);
+    bgc_fp64_vector3_make(&initial_main,   1.0, 0.0, 0.0);
-    bgc_fp64_vector3_make(&initial_auxiliary, 0.0, 1.0, 0.0);
+    bgc_fp64_vector3_make(&initial_branch, 0.0, 1.0, 0.0);
 
-    bgc_fp64_vector3_make(&final_primary,   0.0, 1.0, 0.0);
+    bgc_fp64_vector3_make(&final_main,   0.0, 1.0, 0.0);
-    bgc_fp64_vector3_make(&final_auxiliary, 0.0, 0.0, -1.0);
+    bgc_fp64_vector3_make(&final_branch, 0.0, 0.0, -1.0);
 
-    bgc_fp64_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    bgc_fp64_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
     printf("\n120 degees turn:\n");
     print_quaternion_fp64(&turn._versor);
 
 
-    // About 1 degree turn difference between initial_primary and initial_auxiliary directions
+    // About 1 degree turn difference between initial_main and initial_branch directions
-    bgc_fp64_vector3_make(&initial_primary,   1.0, 0.0, 0.0);
+    bgc_fp64_vector3_make(&initial_main,   1.0, 0.0, 0.0);
-    bgc_fp64_vector3_make(&initial_auxiliary, 0.999848, 0.017452, 0.0);
+    bgc_fp64_vector3_make(&initial_branch, 0.999848, 0.017452, 0.0);
-    bgc_fp64_vector3_make(&final_primary,     0.0, 1.0, 0.0);
+    bgc_fp64_vector3_make(&final_main,     0.0, 1.0, 0.0);
-    bgc_fp64_vector3_make(&final_auxiliary,  -1.0, 0.0, 0.0);
+    bgc_fp64_vector3_make(&final_branch,  -1.0, 0.0, 0.0);
 
-    bgc_fp64_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    bgc_fp64_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
-    printf("\nAbout 1 degree turn difference between initial_primary and initial_auxiliary directions:\n");
+    printf("\nAbout 1 degree turn difference between initial_main and initial_branch directions:\n");
     print_quaternion_fp64(&turn._versor);
 
-    // About 0.001 degree turn difference between initial_primary and initial_auxiliary directions
+    // About 0.001 degree turn difference between initial_main and initial_branch directions
-    bgc_fp64_vector3_make(&initial_primary,   1.0, 0.0, 0.0);
+    bgc_fp64_vector3_make(&initial_main,   1.0, 0.0, 0.0);
-    bgc_fp64_vector3_make(&initial_auxiliary, 1.0, 0.000001, 0.0);
+    bgc_fp64_vector3_make(&initial_branch, 1.0, 0.000001, 0.0);
-    bgc_fp64_vector3_make(&final_primary,     0.0, -1.0, 0.0);
+    bgc_fp64_vector3_make(&final_main,     0.0, -1.0, 0.0);
-    bgc_fp64_vector3_make(&final_auxiliary,   1.0, 0.0, 0.0);
+    bgc_fp64_vector3_make(&final_branch,   1.0, 0.0, 0.0);
 
-    bgc_fp64_turn3_make_basis_difference(&turn, &initial_primary, &initial_auxiliary, &final_primary, &final_auxiliary);
+    bgc_fp64_turn3_find_pair_difference(&turn, &initial_main, &initial_branch, &final_main, &final_branch);
 
-    printf("\nAbout 0.01 degree turn difference between initial_primary and initial_auxiliary directions:\n");
+    printf("\nAbout 0.01 degree turn difference between initial_main and initial_branch directions:\n");
     print_quaternion_fp64(&turn._versor);
 }
-/*
+
 #include "affine3.h"
 
 int main()
 {
-    //BGC_FP32_Turn3 start = { 1.0f, 0.0f, 0.0f, 0.0f };
+    //test_pair_difference_fp32();
-    //BGC_FP32_Turn3 end = { 0.0f, 1.0f, 0.0f, 0.0f };
+    test_pair_difference_fp64();
-
-    BGC_FP32_Turn3 start = { 1.0f, 0.0f, 0.0f, 0.0f };
-    BGC_FP32_Turn3 end = { 0.9999f, 0.01414f, 0.0f, 0.0f };
-    BGC_FP32_Slerp slerp;
-    BGC_FP32_Turn3 result;
-    bgc_fp32_slerp_make_full(&slerp, &start, &end);
-    bgc_fp32_slerp_get_phase_versor(&result, &slerp, 0.5f);
-
-    //print_quaternion_fp32(&result);
-
-    test_basis_difference_fp64();
 
     //printf("Affine3 performance test: %f\n", test_bgc_affine3_performance(10000000, 10));
 
@@ -495,4 +514,4 @@ int main()
 
     return 0;
 }
-*/
+

basic-geometry/dual-quaternion.c

@@ -21,11 +21,11 @@ extern inline void bgc_fp64_dual_quaternion_add_scaled(BGC_FP64_DualQuaternion*
 extern inline void bgc_fp32_dual_quaternion_subtract(BGC_FP32_DualQuaternion* difference, const BGC_FP32_DualQuaternion* minuend, const BGC_FP32_DualQuaternion* subtrahend);
 extern inline void bgc_fp64_dual_quaternion_subtract(BGC_FP64_DualQuaternion* difference, const BGC_FP64_DualQuaternion* minuend, const BGC_FP64_DualQuaternion* subtrahend);
 
-extern inline void bgc_fp32_dual_quaternion_multiply(BGC_FP32_DualQuaternion* product, const BGC_FP32_DualQuaternion* multiplicand, const float multipier);
+extern inline void bgc_fp32_dual_quaternion_multiply_by_number(BGC_FP32_DualQuaternion* product, const BGC_FP32_DualQuaternion* multiplicand, const float multipier);
-extern inline void bgc_fp64_dual_quaternion_multiply(BGC_FP64_DualQuaternion* product, const BGC_FP64_DualQuaternion* multiplicand, const double multipier);
+extern inline void bgc_fp64_dual_quaternion_multiply_by_number(BGC_FP64_DualQuaternion* product, const BGC_FP64_DualQuaternion* multiplicand, const double multipier);
 
-extern inline void bgc_fp32_dual_quaternion_divide(BGC_FP32_DualQuaternion* quotient, const BGC_FP32_DualQuaternion* divident, const float divisor);
+extern inline void bgc_fp32_dual_quaternion_divide_by_number(BGC_FP32_DualQuaternion* quotient, const BGC_FP32_DualQuaternion* divident, const float divisor);
-extern inline void bgc_fp64_dual_quaternion_divide(BGC_FP64_DualQuaternion* quotient, const BGC_FP64_DualQuaternion* divident, const double divisor);
+extern inline void bgc_fp64_dual_quaternion_divide_by_number(BGC_FP64_DualQuaternion* quotient, const BGC_FP64_DualQuaternion* divident, const double divisor);
 
 extern inline void bgc_fp32_dual_quaternion_get_mean2(BGC_FP32_DualQuaternion* mean, const BGC_FP32_DualQuaternion* quaternion1, const BGC_FP32_DualQuaternion* quaternion2);
 extern inline void bgc_fp64_dual_quaternion_get_mean2(BGC_FP64_DualQuaternion* mean, const BGC_FP64_DualQuaternion* quaternion1, const BGC_FP64_DualQuaternion* quaternion2);

basic-geometry/dual-quaternion.h

@@ -113,28 +113,28 @@ inline void bgc_fp64_dual_quaternion_subtract(BGC_FP64_DualQuaternion* differenc
 
 // ================== Multiply ================== //
 
-inline void bgc_fp32_dual_quaternion_multiply(BGC_FP32_DualQuaternion* product, const BGC_FP32_DualQuaternion* multiplicand, const float multipier)
+inline void bgc_fp32_dual_quaternion_multiply_by_number(BGC_FP32_DualQuaternion* product, const BGC_FP32_DualQuaternion* multiplicand, const float multipier)
 {
-    bgc_fp32_quaternion_multiply(&product->real, &multiplicand->real, multipier);
+    bgc_fp32_quaternion_multiply_by_number(&product->real, &multiplicand->real, multipier);
-    bgc_fp32_quaternion_multiply(&product->dual, &multiplicand->dual, multipier);
+    bgc_fp32_quaternion_multiply_by_number(&product->dual, &multiplicand->dual, multipier);
 }
 
-inline void bgc_fp64_dual_quaternion_multiply(BGC_FP64_DualQuaternion* product, const BGC_FP64_DualQuaternion* multiplicand, const double multipier)
+inline void bgc_fp64_dual_quaternion_multiply_by_number(BGC_FP64_DualQuaternion* product, const BGC_FP64_DualQuaternion* multiplicand, const double multipier)
 {
-    bgc_fp64_quaternion_multiply(&product->real, &multiplicand->real, multipier);
+    bgc_fp64_quaternion_multiply_by_number(&product->real, &multiplicand->real, multipier);
-    bgc_fp64_quaternion_multiply(&product->dual, &multiplicand->dual, multipier);
+    bgc_fp64_quaternion_multiply_by_number(&product->dual, &multiplicand->dual, multipier);
 }
 
 // =================== Divide =================== //
 
-inline void bgc_fp32_dual_quaternion_divide(BGC_FP32_DualQuaternion* quotient, const BGC_FP32_DualQuaternion* divident, const float divisor)
+inline void bgc_fp32_dual_quaternion_divide_by_number(BGC_FP32_DualQuaternion* quotient, const BGC_FP32_DualQuaternion* divident, const float divisor)
 {
-    bgc_fp32_dual_quaternion_multiply(quotient, divident, 1.0f / divisor);
+    bgc_fp32_dual_quaternion_multiply_by_number(quotient, divident, 1.0f / divisor);
 }
 
-inline void bgc_fp64_dual_quaternion_divide(BGC_FP64_DualQuaternion* quotient, const BGC_FP64_DualQuaternion* divident, const double divisor)
+inline void bgc_fp64_dual_quaternion_divide_by_number(BGC_FP64_DualQuaternion* quotient, const BGC_FP64_DualQuaternion* divident, const double divisor)
 {
-    bgc_fp64_dual_quaternion_multiply(quotient, divident, 1.0 / divisor);
+    bgc_fp64_dual_quaternion_multiply_by_number(quotient, divident, 1.0 / divisor);
 }
 
 // ================ Mean of Two ================= //

basic-geometry/quaternion.c

@@ -40,20 +40,20 @@ extern inline void bgc_fp64_quaternion_add_scaled(BGC_FP64_Quaternion* sum, cons
 extern inline void bgc_fp32_quaternion_subtract(BGC_FP32_Quaternion* difference, const BGC_FP32_Quaternion* minuend, const BGC_FP32_Quaternion* subtrahend);
 extern inline void bgc_fp64_quaternion_subtract(BGC_FP64_Quaternion* difference, const BGC_FP64_Quaternion* minuend, const BGC_FP64_Quaternion* subtrahend);
 
-extern inline void bgc_fp32_quaternion_get_product(BGC_FP32_Quaternion* product, const BGC_FP32_Quaternion* left, const BGC_FP32_Quaternion* right);
+extern inline void bgc_fp32_quaternion_multiply_by_quaternion(BGC_FP32_Quaternion* product, const BGC_FP32_Quaternion* left, const BGC_FP32_Quaternion* right);
-extern inline void bgc_fp64_quaternion_get_product(BGC_FP64_Quaternion* product, const BGC_FP64_Quaternion* left, const BGC_FP64_Quaternion* right);
+extern inline void bgc_fp64_quaternion_multiply_by_quaternion(BGC_FP64_Quaternion* product, const BGC_FP64_Quaternion* left, const BGC_FP64_Quaternion* right);
 
-extern inline void bgc_fp32_quaternion_get_product_by_conjugate(BGC_FP32_Quaternion* product, const BGC_FP32_Quaternion* left, const BGC_FP32_Quaternion* right);
+extern inline void bgc_fp32_quaternion_multiply_by_conjugate(BGC_FP32_Quaternion* product, const BGC_FP32_Quaternion* left, const BGC_FP32_Quaternion* right);
-extern inline void bgc_fp64_quaternion_get_product_by_conjugate(BGC_FP64_Quaternion* product, const BGC_FP64_Quaternion* left, const BGC_FP64_Quaternion* right);
+extern inline void bgc_fp64_quaternion_multiply_by_conjugate(BGC_FP64_Quaternion* product, const BGC_FP64_Quaternion* left, const BGC_FP64_Quaternion* right);
 
-extern inline void bgc_fp32_quaternion_multiply(BGC_FP32_Quaternion* product, const BGC_FP32_Quaternion* multiplicand, const float multipier);
+extern inline void bgc_fp32_quaternion_multiply_by_number(BGC_FP32_Quaternion* product, const BGC_FP32_Quaternion* multiplicand, const float multipier);
-extern inline void bgc_fp64_quaternion_multiply(BGC_FP64_Quaternion* product, const BGC_FP64_Quaternion* multiplicand, const double multipier);
+extern inline void bgc_fp64_quaternion_multiply_by_number(BGC_FP64_Quaternion* product, const BGC_FP64_Quaternion* multiplicand, const double multipier);
 
-extern inline int bgc_fp32_quaternion_get_ratio(BGC_FP32_Quaternion* quotient, const BGC_FP32_Quaternion* divident, const BGC_FP32_Quaternion* divisor);
+extern inline int bgc_fp32_quaternion_divide_by_quaternion(BGC_FP32_Quaternion* quotient, const BGC_FP32_Quaternion* divident, const BGC_FP32_Quaternion* divisor);
-extern inline int bgc_fp64_quaternion_get_ratio(BGC_FP64_Quaternion* quotient, const BGC_FP64_Quaternion* divident, const BGC_FP64_Quaternion* divisor);
+extern inline int bgc_fp64_quaternion_divide_by_quaternion(BGC_FP64_Quaternion* quotient, const BGC_FP64_Quaternion* divident, const BGC_FP64_Quaternion* divisor);
 
-extern inline void bgc_fp32_quaternion_divide(BGC_FP32_Quaternion* quotient, const BGC_FP32_Quaternion* dividend, const float divisor);
+extern inline void bgc_fp32_quaternion_divide_by_number(BGC_FP32_Quaternion* quotient, const BGC_FP32_Quaternion* dividend, const float divisor);
-extern inline void bgc_fp64_quaternion_divide(BGC_FP64_Quaternion* quotient, const BGC_FP64_Quaternion* dividend, const double divisor);
+extern inline void bgc_fp64_quaternion_divide_by_number(BGC_FP64_Quaternion* quotient, const BGC_FP64_Quaternion* dividend, const double divisor);
 
 extern inline void bgc_fp32_quaternion_get_mean2(BGC_FP32_Quaternion* mean, const BGC_FP32_Quaternion* quaternion1, const BGC_FP32_Quaternion* quaternion2);
 extern inline void bgc_fp64_quaternion_get_mean2(BGC_FP64_Quaternion* mean, const BGC_FP64_Quaternion* quaternion1, const BGC_FP64_Quaternion* quaternion2);
@@ -88,6 +88,18 @@ extern inline int bgc_fp64_quaternion_normalize(BGC_FP64_Quaternion* quaternion)
 extern inline int bgc_fp32_quaternion_get_normalized(BGC_FP32_Quaternion* normalized, const BGC_FP32_Quaternion* quaternion);
 extern inline int bgc_fp64_quaternion_get_normalized(BGC_FP64_Quaternion* normalized, const BGC_FP64_Quaternion* quaternion);
 
+extern inline void _bgc_fp32_quaternion_turn_vector_roughly(BGC_FP32_Vector3* turned_vector, const BGC_FP32_Quaternion* quaternion, const BGC_FP32_Vector3* original_vector);
+extern inline void _bgc_fp64_quaternion_turn_vector_roughly(BGC_FP64_Vector3* turned_vector, const BGC_FP64_Quaternion* quaternion, const BGC_FP64_Vector3* original_vector);
+
+extern inline void _bgc_fp32_quaternion_turn_vector_back_roughly(BGC_FP32_Vector3* turned_vector, const BGC_FP32_Quaternion* quaternion, const BGC_FP32_Vector3* original_vector);
+extern inline void _bgc_fp64_quaternion_turn_vector_back_roughly(BGC_FP64_Vector3* turned_vector, const BGC_FP64_Quaternion* quaternion, const BGC_FP64_Vector3* original_vector);
+
+extern inline int bgc_fp32_quaternion_turn_vector(BGC_FP32_Vector3* turned_vector, const BGC_FP32_Quaternion* quaternion, const BGC_FP32_Vector3* original_vector);
+extern inline int bgc_fp64_quaternion_turn_vector(BGC_FP64_Vector3* turned_vector, const BGC_FP64_Quaternion* quaternion, const BGC_FP64_Vector3* original_vector);
+
+extern inline int bgc_fp32_quaternion_turn_vector_back(BGC_FP32_Vector3* turned_vector, const BGC_FP32_Quaternion* quaternion, const BGC_FP32_Vector3* original_vector);
+extern inline int bgc_fp64_quaternion_turn_vector_back(BGC_FP64_Vector3* turned_vector, const BGC_FP64_Quaternion* quaternion, const BGC_FP64_Vector3* original_vector);
+
 extern inline int bgc_fp32_quaternion_get_rotation_matrix(BGC_FP32_Matrix3x3* rotation, const BGC_FP32_Quaternion* quaternion);
 extern inline int bgc_fp64_quaternion_get_rotation_matrix(BGC_FP64_Matrix3x3* rotation, const BGC_FP64_Quaternion* quaternion);
 
@@ -114,12 +126,12 @@ int bgc_fp32_quaternion_get_exponation(BGC_FP32_Quaternion* power, const BGC_FP3
 
     // isnan(square_modulus) means checking for NaN value at square_modulus
     if (isnan(square_modulus)) {
-        return 0;
+        return BGC_FAILED;
     }
 
     if (square_vector <= BGC_FP32_SQUARE_EPSILON) {
         if (base->s0 < 0.0f) {
-            return 0;
+            return BGC_FAILED;
         }
 
         power->s0 = powf(base->s0, exponent);
@@ -127,7 +139,7 @@ int bgc_fp32_quaternion_get_exponation(BGC_FP32_Quaternion* power, const BGC_FP3
         power->x2 = 0.0f;
         power->x3 = 0.0f;
 
-        return 1;
+        return BGC_SUCCESS;
     }
 
     const float vector_modulus = sqrtf(square_vector);
@@ -140,7 +152,7 @@ int bgc_fp32_quaternion_get_exponation(BGC_FP32_Quaternion* power, const BGC_FP3
     power->x2 = base->x2 * multiplier;
     power->x3 = base->x3 * multiplier;
 
-    return 1;
+    return BGC_SUCCESS;
 }
 
 int bgc_fp64_quaternion_get_exponation(BGC_FP64_Quaternion* power, const BGC_FP64_Quaternion* base, const double exponent)
@@ -155,12 +167,12 @@ int bgc_fp64_quaternion_get_exponation(BGC_FP64_Quaternion* power, const BGC_FP6
 
     // isnan(square_modulus) means checking for NaN value at square_modulus
     if (isnan(square_modulus)) {
-        return 0;
+        return BGC_FAILED;
     }
 
     if (square_vector <= BGC_FP64_SQUARE_EPSILON) {
         if (base->s0 < 0.0) {
-            return 0;
+            return BGC_FAILED;
         }
 
         power->s0 = pow(base->s0, exponent);
@@ -168,7 +180,7 @@ int bgc_fp64_quaternion_get_exponation(BGC_FP64_Quaternion* power, const BGC_FP6
         power->x2 = 0.0;
         power->x3 = 0.0;
 
-        return 1;
+        return BGC_SUCCESS;
     }
 
     const double vector_modulus = sqrt(square_vector);
@@ -181,5 +193,5 @@ int bgc_fp64_quaternion_get_exponation(BGC_FP64_Quaternion* power, const BGC_FP6
     power->x2 = base->x2 * multiplier;
     power->x3 = base->x3 * multiplier;
 
-    return 1;
+    return BGC_SUCCESS;
 }

basic-geometry/quaternion.h

@@ -5,6 +5,7 @@
 
 #include "utilities.h"
 #include "angle.h"
+#include "vector3.h"
 #include "matrix3x3.h"
 
 typedef struct {
@@ -247,7 +248,7 @@ inline void bgc_fp64_quaternion_subtract(BGC_FP64_Quaternion* difference, const
 
 // ================== Multiply ================== //
 
-inline void bgc_fp32_quaternion_get_product(BGC_FP32_Quaternion* product, const BGC_FP32_Quaternion* left, const BGC_FP32_Quaternion* right)
+inline void bgc_fp32_quaternion_multiply_by_quaternion(BGC_FP32_Quaternion* product, const BGC_FP32_Quaternion* left, const BGC_FP32_Quaternion* right)
 {
     const float s0 = (left->s0 * right->s0 - left->x1 * right->x1) - (left->x2 * right->x2 + left->x3 * right->x3);
     const float x1 = (left->x1 * right->s0 + left->s0 * right->x1) - (left->x3 * right->x2 - left->x2 * right->x3);
@@ -260,7 +261,7 @@ inline void bgc_fp32_quaternion_get_product(BGC_FP32_Quaternion* product, const
     product->x3 = x3;
 }
 
-inline void bgc_fp64_quaternion_get_product(BGC_FP64_Quaternion* product, const BGC_FP64_Quaternion* left, const BGC_FP64_Quaternion* right)
+inline void bgc_fp64_quaternion_multiply_by_quaternion(BGC_FP64_Quaternion* product, const BGC_FP64_Quaternion* left, const BGC_FP64_Quaternion* right)
 {
     const double s0 = (left->s0 * right->s0 - left->x1 * right->x1) - (left->x2 * right->x2 + left->x3 * right->x3);
     const double x1 = (left->x1 * right->s0 + left->s0 * right->x1) - (left->x3 * right->x2 - left->x2 * right->x3);
@@ -273,7 +274,7 @@ inline void bgc_fp64_quaternion_get_product(BGC_FP64_Quaternion* product, const
     product->x3 = x3;
 }
 
-inline void bgc_fp32_quaternion_get_product_by_conjugate(BGC_FP32_Quaternion* product, const BGC_FP32_Quaternion* left, const BGC_FP32_Quaternion* right)
+inline void bgc_fp32_quaternion_multiply_by_conjugate(BGC_FP32_Quaternion* product, const BGC_FP32_Quaternion* left, const BGC_FP32_Quaternion* right)
 {
     const float s0 = (left->s0 * right->s0 + left->x1 * right->x1) + (left->x2 * right->x2 + left->x3 * right->x3);
     const float x1 = (left->x1 * right->s0 + left->x3 * right->x2) - (left->s0 * right->x1 + left->x2 * right->x3);
@@ -286,7 +287,7 @@ inline void bgc_fp32_quaternion_get_product_by_conjugate(BGC_FP32_Quaternion* pr
     product->x3 = x3;
 }
 
-inline void bgc_fp64_quaternion_get_product_by_conjugate(BGC_FP64_Quaternion* product, const BGC_FP64_Quaternion* left, const BGC_FP64_Quaternion* right)
+inline void bgc_fp64_quaternion_multiply_by_conjugate(BGC_FP64_Quaternion* product, const BGC_FP64_Quaternion* left, const BGC_FP64_Quaternion* right)
 {
     const double s0 = (left->s0 * right->s0 + left->x1 * right->x1) + (left->x2 * right->x2 + left->x3 * right->x3);
     const double x1 = (left->x1 * right->s0 + left->x3 * right->x2) - (left->s0 * right->x1 + left->x2 * right->x3);
@@ -299,7 +300,7 @@ inline void bgc_fp64_quaternion_get_product_by_conjugate(BGC_FP64_Quaternion* pr
     product->x3 = x3;
 }
 
-inline void bgc_fp32_quaternion_multiply(BGC_FP32_Quaternion* product, const BGC_FP32_Quaternion* multiplicand, const float multipier)
+inline void bgc_fp32_quaternion_multiply_by_number(BGC_FP32_Quaternion* product, const BGC_FP32_Quaternion* multiplicand, const float multipier)
 {
     product->s0 = multiplicand->s0 * multipier;
     product->x1 = multiplicand->x1 * multipier;
@@ -307,7 +308,7 @@ inline void bgc_fp32_quaternion_multiply(BGC_FP32_Quaternion* product, const BGC
     product->x3 = multiplicand->x3 * multipier;
 }
 
-inline void bgc_fp64_quaternion_multiply(BGC_FP64_Quaternion* product, const BGC_FP64_Quaternion* multiplicand, const double multipier)
+inline void bgc_fp64_quaternion_multiply_by_number(BGC_FP64_Quaternion* product, const BGC_FP64_Quaternion* multiplicand, const double multipier)
 {
     product->s0 = multiplicand->s0 * multipier;
     product->x1 = multiplicand->x1 * multipier;
@@ -317,12 +318,12 @@ inline void bgc_fp64_quaternion_multiply(BGC_FP64_Quaternion* product, const BGC
 
 // =================== Divide =================== //
 
-inline int bgc_fp32_quaternion_get_ratio(BGC_FP32_Quaternion* quotient, const BGC_FP32_Quaternion* divident, const BGC_FP32_Quaternion* divisor)
+inline int bgc_fp32_quaternion_divide_by_quaternion(BGC_FP32_Quaternion* quotient, const BGC_FP32_Quaternion* divident, const BGC_FP32_Quaternion* divisor)
 {
     const float square_modulus = bgc_fp32_quaternion_get_square_modulus(divisor);
 
     if (square_modulus <= BGC_FP32_SQUARE_EPSILON || isnan(square_modulus)) {
-        return 0;
+        return BGC_FAILED;
     }
 
     const float s0 = (divident->s0 * divisor->s0 + divident->x1 * divisor->x1) + (divident->x2 * divisor->x2 + divident->x3 * divisor->x3);
@@ -337,15 +338,15 @@ inline int bgc_fp32_quaternion_get_ratio(BGC_FP32_Quaternion* quotient, const BG
     quotient->x2 = x2 * multiplicand;
     quotient->x3 = x3 * multiplicand;
 
-    return 1;
+    return BGC_SUCCESS;
 }
 
-inline int bgc_fp64_quaternion_get_ratio(BGC_FP64_Quaternion* quotient, const BGC_FP64_Quaternion* divident, const BGC_FP64_Quaternion* divisor)
+inline int bgc_fp64_quaternion_divide_by_quaternion(BGC_FP64_Quaternion* quotient, const BGC_FP64_Quaternion* divident, const BGC_FP64_Quaternion* divisor)
 {
     const double square_modulus = bgc_fp64_quaternion_get_square_modulus(divisor);
 
     if (square_modulus <= BGC_FP64_SQUARE_EPSILON || isnan(square_modulus)) {
-        return 0;
+        return BGC_FAILED;
     }
 
     const double s0 = (divident->s0 * divisor->s0 + divident->x1 * divisor->x1) + (divident->x2 * divisor->x2 + divident->x3 * divisor->x3);
@@ -360,17 +361,17 @@ inline int bgc_fp64_quaternion_get_ratio(BGC_FP64_Quaternion* quotient, const BG
     quotient->x2 = x2 * multiplicand;
     quotient->x3 = x3 * multiplicand;
 
-    return 1;
+    return BGC_SUCCESS;
 }
 
-inline void bgc_fp32_quaternion_divide(BGC_FP32_Quaternion* quotient, const BGC_FP32_Quaternion* dividend, const float divisor)
+inline void bgc_fp32_quaternion_divide_by_number(BGC_FP32_Quaternion* quotient, const BGC_FP32_Quaternion* dividend, const float divisor)
 {
-    bgc_fp32_quaternion_multiply(quotient, dividend, 1.0f / divisor);
+    bgc_fp32_quaternion_multiply_by_number(quotient, dividend, 1.0f / divisor);
 }
 
-inline void bgc_fp64_quaternion_divide(BGC_FP64_Quaternion* quotient, const BGC_FP64_Quaternion* dividend, const double divisor)
+inline void bgc_fp64_quaternion_divide_by_number(BGC_FP64_Quaternion* quotient, const BGC_FP64_Quaternion* dividend, const double divisor)
 {
-    bgc_fp64_quaternion_multiply(quotient, dividend, 1.0 / divisor);
+    bgc_fp64_quaternion_multiply_by_number(quotient, dividend, 1.0 / divisor);
 }
 
 // ================ Mean of Two ================= //
@@ -504,7 +505,7 @@ inline int bgc_fp32_quaternion_get_inverse(BGC_FP32_Quaternion* inverse, const B
     const float square_modulus = bgc_fp32_quaternion_get_square_modulus(quaternion);
 
     if (square_modulus <= BGC_FP32_SQUARE_EPSILON || isnan(square_modulus)) {
-        return 0;
+        return BGC_FAILED;
     }
 
     const float multiplicand = 1.0f / square_modulus;
@@ -514,7 +515,7 @@ inline int bgc_fp32_quaternion_get_inverse(BGC_FP32_Quaternion* inverse, const B
     inverse->x2 = -quaternion->x2 * multiplicand;
     inverse->x3 = -quaternion->x3 * multiplicand;
 
-    return 1;
+    return BGC_SUCCESS;
 }
 
 inline int bgc_fp64_quaternion_get_inverse(BGC_FP64_Quaternion* inverse, const BGC_FP64_Quaternion* quaternion)
@@ -522,7 +523,7 @@ inline int bgc_fp64_quaternion_get_inverse(BGC_FP64_Quaternion* inverse, const B
     const double square_modulus = bgc_fp64_quaternion_get_square_modulus(quaternion);
 
     if (square_modulus <= BGC_FP64_SQUARE_EPSILON || isnan(square_modulus)) {
-        return 0;
+        return BGC_FAILED;
     }
 
     const double multiplicand = 1.0 / square_modulus;
@@ -532,7 +533,7 @@ inline int bgc_fp64_quaternion_get_inverse(BGC_FP64_Quaternion* inverse, const B
     inverse->x2 = -quaternion->x2 * multiplicand;
     inverse->x3 = -quaternion->x3 * multiplicand;
 
-    return 1;
+    return BGC_SUCCESS;
 }
 
 inline int bgc_fp32_quaternion_invert(BGC_FP32_Quaternion* quaternion)
@@ -552,11 +553,11 @@ inline int bgc_fp32_quaternion_normalize(BGC_FP32_Quaternion* quaternion)
     const float square_modulus = bgc_fp32_quaternion_get_square_modulus(quaternion);
 
     if (bgc_fp32_is_square_unit(square_modulus)) {
-        return 1;
+        return BGC_SUCCESS;
     }
 
     if (square_modulus <= BGC_FP32_SQUARE_EPSILON || isnan(square_modulus)) {
-        return 0;
+        return BGC_FAILED;
     }
 
     const float multiplier = sqrtf(1.0f / square_modulus);
@@ -566,7 +567,7 @@ inline int bgc_fp32_quaternion_normalize(BGC_FP32_Quaternion* quaternion)
     quaternion->x2 *= multiplier;
     quaternion->x3 *= multiplier;
 
-    return 1;
+    return BGC_SUCCESS;
 }
 
 inline int bgc_fp64_quaternion_normalize(BGC_FP64_Quaternion* quaternion)
@@ -574,11 +575,11 @@ inline int bgc_fp64_quaternion_normalize(BGC_FP64_Quaternion* quaternion)
     const double square_modulus = bgc_fp64_quaternion_get_square_modulus(quaternion);
 
     if (bgc_fp64_is_square_unit(square_modulus)) {
-        return 1;
+        return BGC_SUCCESS;
     }
 
     if (square_modulus <= BGC_FP64_SQUARE_EPSILON || isnan(square_modulus)) {
-        return 0;
+        return BGC_FAILED;
     }
 
     const double multiplier = sqrt(1.0 / square_modulus);
@@ -588,7 +589,7 @@ inline int bgc_fp64_quaternion_normalize(BGC_FP64_Quaternion* quaternion)
     quaternion->x2 *= multiplier;
     quaternion->x3 *= multiplier;
 
-    return 1;
+    return BGC_SUCCESS;
 }
 
 inline int bgc_fp32_quaternion_get_normalized(BGC_FP32_Quaternion* normalized, const BGC_FP32_Quaternion* quaternion)
@@ -597,16 +598,16 @@ inline int bgc_fp32_quaternion_get_normalized(BGC_FP32_Quaternion* normalized, c
 
     if (bgc_fp32_is_square_unit(square_modulus)) {
         bgc_fp32_quaternion_copy(normalized, quaternion);
-        return 1;
+        return BGC_SUCCESS;
     }
 
     if (square_modulus <= BGC_FP32_SQUARE_EPSILON || isnan(square_modulus)) {
         bgc_fp32_quaternion_reset(normalized);
-        return 0;
+        return BGC_FAILED;
     }
 
-    bgc_fp32_quaternion_multiply(normalized, quaternion, sqrtf(1.0f / square_modulus));
+    bgc_fp32_quaternion_multiply_by_number(normalized, quaternion, sqrtf(1.0f / square_modulus));
-    return 1;
+    return BGC_SUCCESS;
 }
 
 inline int bgc_fp64_quaternion_get_normalized(BGC_FP64_Quaternion* normalized, const BGC_FP64_Quaternion* quaternion)
@@ -615,16 +616,16 @@ inline int bgc_fp64_quaternion_get_normalized(BGC_FP64_Quaternion* normalized, c
 
     if (bgc_fp64_is_square_unit(square_modulus)) {
         bgc_fp64_quaternion_copy(normalized, quaternion);
-        return 1;
+        return BGC_SUCCESS;
     }
 
     if (square_modulus <= BGC_FP64_SQUARE_EPSILON || isnan(square_modulus)) {
         bgc_fp64_quaternion_reset(normalized);
-        return 0;
+        return BGC_FAILED;
     }
 
-    bgc_fp64_quaternion_multiply(normalized, quaternion, sqrt(1.0 / square_modulus));
+    bgc_fp64_quaternion_multiply_by_number(normalized, quaternion, sqrt(1.0 / square_modulus));
-    return 1;
+    return BGC_SUCCESS;
 }
 
 // =============== Get Exponation =============== //
@@ -633,6 +634,182 @@ int bgc_fp32_quaternion_get_exponation(BGC_FP32_Quaternion* power, const BGC_FP3
 
 int bgc_fp64_quaternion_get_exponation(BGC_FP64_Quaternion* power, const BGC_FP64_Quaternion* base, const double exponent);
 
+// ============== Raw Turn Vector3 ============== //
+
+// An internal function
+inline void _bgc_fp32_quaternion_turn_vector_roughly(BGC_FP32_Vector3* turned_vector, const BGC_FP32_Quaternion* quaternion, const BGC_FP32_Vector3* original_vector)
+{
+    const float tx1 = 2.0f * (quaternion->x2 * original_vector->x3 - quaternion->x3 * original_vector->x2);
+    const float tx2 = 2.0f * (quaternion->x3 * original_vector->x1 - quaternion->x1 * original_vector->x3);
+    const float tx3 = 2.0f * (quaternion->x1 * original_vector->x2 - quaternion->x2 * original_vector->x1);
+
+    const float x1 = (original_vector->x1 + tx1 * quaternion->s0) + (quaternion->x2 * tx3 - quaternion->x3 * tx2);
+    const float x2 = (original_vector->x2 + tx2 * quaternion->s0) + (quaternion->x3 * tx1 - quaternion->x1 * tx3);
+    const float x3 = (original_vector->x3 + tx3 * quaternion->s0) + (quaternion->x1 * tx2 - quaternion->x2 * tx1);
+
+    turned_vector->x1 = x1;
+    turned_vector->x2 = x2;
+    turned_vector->x3 = x3;
+}
+
+// An internal function
+inline void _bgc_fp64_quaternion_turn_vector_roughly(BGC_FP64_Vector3* turned_vector, const BGC_FP64_Quaternion* quaternion, const BGC_FP64_Vector3* original_vector)
+{
+    const double tx1 = 2.0f * (quaternion->x2 * original_vector->x3 - quaternion->x3 * original_vector->x2);
+    const double tx2 = 2.0f * (quaternion->x3 * original_vector->x1 - quaternion->x1 * original_vector->x3);
+    const double tx3 = 2.0f * (quaternion->x1 * original_vector->x2 - quaternion->x2 * original_vector->x1);
+
+    const double x1 = (original_vector->x1 + tx1 * quaternion->s0) + (quaternion->x2 * tx3 - quaternion->x3 * tx2);
+    const double x2 = (original_vector->x2 + tx2 * quaternion->s0) + (quaternion->x3 * tx1 - quaternion->x1 * tx3);
+    const double x3 = (original_vector->x3 + tx3 * quaternion->s0) + (quaternion->x1 * tx2 - quaternion->x2 * tx1);
+
+    turned_vector->x1 = x1;
+    turned_vector->x2 = x2;
+    turned_vector->x3 = x3;
+}
+
+// ========= Raw Turn Vector3 Backwards ========= //
+
+// An internal function
+inline void _bgc_fp32_quaternion_turn_vector_back_roughly(BGC_FP32_Vector3* turned_vector, const BGC_FP32_Quaternion* quaternion, const BGC_FP32_Vector3* original_vector)
+{
+    const float tx1 = 2.0f * (quaternion->x2 * original_vector->x3 - quaternion->x3 * original_vector->x2);
+    const float tx2 = 2.0f * (quaternion->x3 * original_vector->x1 - quaternion->x1 * original_vector->x3);
+    const float tx3 = 2.0f * (quaternion->x1 * original_vector->x2 - quaternion->x2 * original_vector->x1);
+
+    const float x1 = (original_vector->x1 + tx1 * quaternion->s0) + (quaternion->x2 * tx3 - quaternion->x3 * tx2);
+    const float x2 = (original_vector->x2 + tx2 * quaternion->s0) + (quaternion->x3 * tx1 - quaternion->x1 * tx3);
+    const float x3 = (original_vector->x3 + tx3 * quaternion->s0) + (quaternion->x1 * tx2 - quaternion->x2 * tx1);
+
+    turned_vector->x1 = x1;
+    turned_vector->x2 = x2;
+    turned_vector->x3 = x3;
+}
+
+// An internal function
+inline void _bgc_fp64_quaternion_turn_vector_back_roughly(BGC_FP64_Vector3* turned_vector, const BGC_FP64_Quaternion* quaternion, const BGC_FP64_Vector3* original_vector)
+{
+    const double tx1 = 2.0f * (quaternion->x2 * original_vector->x3 - quaternion->x3 * original_vector->x2);
+    const double tx2 = 2.0f * (quaternion->x3 * original_vector->x1 - quaternion->x1 * original_vector->x3);
+    const double tx3 = 2.0f * (quaternion->x1 * original_vector->x2 - quaternion->x2 * original_vector->x1);
+
+    const double x1 = (original_vector->x1 + tx1 * quaternion->s0) + (quaternion->x2 * tx3 - quaternion->x3 * tx2);
+    const double x2 = (original_vector->x2 + tx2 * quaternion->s0) + (quaternion->x3 * tx1 - quaternion->x1 * tx3);
+    const double x3 = (original_vector->x3 + tx3 * quaternion->s0) + (quaternion->x1 * tx2 - quaternion->x2 * tx1);
+
+    turned_vector->x1 = x1;
+    turned_vector->x2 = x2;
+    turned_vector->x3 = x3;
+}
+
+// ================ Turn Vector3 ================ //
+
+inline int bgc_fp32_quaternion_turn_vector(BGC_FP32_Vector3* turned_vector, const BGC_FP32_Quaternion* quaternion, const BGC_FP32_Vector3* original_vector)
+{
+    const float square_modulus = bgc_fp32_quaternion_get_square_modulus(quaternion);
+
+    if (square_modulus < BGC_FP32_SQUARE_EPSILON || isnan(square_modulus)) {
+        return BGC_FAILED;
+    }
+
+    const float multiplier = 2.0f / square_modulus;
+
+    const float tx1 = multiplier * (quaternion->x2 * original_vector->x3 - quaternion->x3 * original_vector->x2);
+    const float tx2 = multiplier * (quaternion->x3 * original_vector->x1 - quaternion->x1 * original_vector->x3);
+    const float tx3 = multiplier * (quaternion->x1 * original_vector->x2 - quaternion->x2 * original_vector->x1);
+
+    const float x1 = (original_vector->x1 + tx1 * quaternion->s0) + (quaternion->x2 * tx3 - quaternion->x3 * tx2);
+    const float x2 = (original_vector->x2 + tx2 * quaternion->s0) + (quaternion->x3 * tx1 - quaternion->x1 * tx3);
+    const float x3 = (original_vector->x3 + tx3 * quaternion->s0) + (quaternion->x1 * tx2 - quaternion->x2 * tx1);
+
+    turned_vector->x1 = x1;
+    turned_vector->x2 = x2;
+    turned_vector->x3 = x3;
+
+
+    return BGC_SUCCESS;
+}
+
+inline int bgc_fp64_quaternion_turn_vector(BGC_FP64_Vector3* turned_vector, const BGC_FP64_Quaternion* quaternion, const BGC_FP64_Vector3* original_vector)
+{
+    const double square_modulus = bgc_fp64_quaternion_get_square_modulus(quaternion);
+
+    if (square_modulus < BGC_FP64_SQUARE_EPSILON || isnan(square_modulus)) {
+        return BGC_FAILED;
+    }
+
+    const double multiplier = 2.0 / square_modulus;
+
+    const double tx1 = multiplier * (quaternion->x2 * original_vector->x3 - quaternion->x3 * original_vector->x2);
+    const double tx2 = multiplier * (quaternion->x3 * original_vector->x1 - quaternion->x1 * original_vector->x3);
+    const double tx3 = multiplier * (quaternion->x1 * original_vector->x2 - quaternion->x2 * original_vector->x1);
+
+    const double x1 = (original_vector->x1 + tx1 * quaternion->s0) + (quaternion->x2 * tx3 - quaternion->x3 * tx2);
+    const double x2 = (original_vector->x2 + tx2 * quaternion->s0) + (quaternion->x3 * tx1 - quaternion->x1 * tx3);
+    const double x3 = (original_vector->x3 + tx3 * quaternion->s0) + (quaternion->x1 * tx2 - quaternion->x2 * tx1);
+
+    turned_vector->x1 = x1;
+    turned_vector->x2 = x2;
+    turned_vector->x3 = x3;
+
+
+    return BGC_SUCCESS;
+}
+
+// =========== Turn Vector3 Backwards =========== //
+
+inline int bgc_fp32_quaternion_turn_vector_back(BGC_FP32_Vector3* turned_vector, const BGC_FP32_Quaternion* quaternion, const BGC_FP32_Vector3* original_vector)
+{
+    const float square_modulus = bgc_fp32_quaternion_get_square_modulus(quaternion);
+
+    if (square_modulus < BGC_FP32_SQUARE_EPSILON || isnan(square_modulus)) {
+        return BGC_FAILED;
+    }
+
+    const float multiplier = 2.0f / square_modulus;
+
+    const float tx1 = multiplier * (quaternion->x2 * original_vector->x3 - quaternion->x3 * original_vector->x2);
+    const float tx2 = multiplier * (quaternion->x3 * original_vector->x1 - quaternion->x1 * original_vector->x3);
+    const float tx3 = multiplier * (quaternion->x1 * original_vector->x2 - quaternion->x2 * original_vector->x1);
+
+    const float x1 = (original_vector->x1 - tx1 * quaternion->s0) + (quaternion->x2 * tx3 - quaternion->x3 * tx2);
+    const float x2 = (original_vector->x2 - tx2 * quaternion->s0) + (quaternion->x3 * tx1 - quaternion->x1 * tx3);
+    const float x3 = (original_vector->x3 - tx3 * quaternion->s0) + (quaternion->x1 * tx2 - quaternion->x2 * tx1);
+
+    turned_vector->x1 = x1;
+    turned_vector->x2 = x2;
+    turned_vector->x3 = x3;
+
+
+    return BGC_SUCCESS;
+}
+
+inline int bgc_fp64_quaternion_turn_vector_back(BGC_FP64_Vector3* turned_vector, const BGC_FP64_Quaternion* quaternion, const BGC_FP64_Vector3* original_vector)
+{
+    const double square_modulus = bgc_fp64_quaternion_get_square_modulus(quaternion);
+
+    if (square_modulus < BGC_FP64_SQUARE_EPSILON || isnan(square_modulus)) {
+        return BGC_FAILED;
+    }
+
+    const double multiplier = 2.0 / square_modulus;
+
+    const double tx1 = multiplier * (quaternion->x2 * original_vector->x3 - quaternion->x3 * original_vector->x2);
+    const double tx2 = multiplier * (quaternion->x3 * original_vector->x1 - quaternion->x1 * original_vector->x3);
+    const double tx3 = multiplier * (quaternion->x1 * original_vector->x2 - quaternion->x2 * original_vector->x1);
+
+    const double x1 = (original_vector->x1 - tx1 * quaternion->s0) + (quaternion->x2 * tx3 - quaternion->x3 * tx2);
+    const double x2 = (original_vector->x2 - tx2 * quaternion->s0) + (quaternion->x3 * tx1 - quaternion->x1 * tx3);
+    const double x3 = (original_vector->x3 - tx3 * quaternion->s0) + (quaternion->x1 * tx2 - quaternion->x2 * tx1);
+
+    turned_vector->x1 = x1;
+    turned_vector->x2 = x2;
+    turned_vector->x3 = x3;
+
+
+    return BGC_SUCCESS;
+}
+
 // ============ Get Rotation Matrix ============= //
 
 inline int bgc_fp32_quaternion_get_rotation_matrix(BGC_FP32_Matrix3x3* rotation, const BGC_FP32_Quaternion* quaternion)
@@ -647,7 +824,7 @@ inline int bgc_fp32_quaternion_get_rotation_matrix(BGC_FP32_Matrix3x3* rotation,
     if (square_modulus <= BGC_FP32_SQUARE_EPSILON || isnan(square_modulus))
     {
         bgc_fp32_matrix3x3_make_identity(rotation);
-        return 0;
+        return BGC_FAILED;
     }
 
     const float corrector1 = 1.0f / square_modulus;
@@ -673,7 +850,7 @@ inline int bgc_fp32_quaternion_get_rotation_matrix(BGC_FP32_Matrix3x3* rotation,
     rotation->r3c2 = corrector2 * (x2x3 + s0x1);
     rotation->r1c3 = corrector2 * (x1x3 + s0x2);
 
-    return 1;
+    return BGC_SUCCESS;
 }
 
 inline int bgc_fp64_quaternion_get_rotation_matrix(BGC_FP64_Matrix3x3* rotation, const BGC_FP64_Quaternion* quaternion)
@@ -688,7 +865,7 @@ inline int bgc_fp64_quaternion_get_rotation_matrix(BGC_FP64_Matrix3x3* rotation,
     if (square_modulus <= BGC_FP64_SQUARE_EPSILON || isnan(square_modulus))
     {
         bgc_fp64_matrix3x3_make_identity(rotation);
-        return 0;
+        return BGC_FAILED;
     }
 
     const double corrector1 = 1.0f / square_modulus;
@@ -714,7 +891,7 @@ inline int bgc_fp64_quaternion_get_rotation_matrix(BGC_FP64_Matrix3x3* rotation,
     rotation->r3c2 = corrector2 * (x2x3 + s0x1);
     rotation->r1c3 = corrector2 * (x1x3 + s0x2);
 
-    return 1;
+    return BGC_SUCCESS;
 }
 
 // ============= Get Reverse Matrix ============= //
@@ -731,7 +908,7 @@ inline int bgc_fp32_quaternion_get_reverse_matrix(BGC_FP32_Matrix3x3* reverse, c
     if (square_modulus <= BGC_FP32_SQUARE_EPSILON || isnan(square_modulus))
     {
         bgc_fp32_matrix3x3_make_identity(reverse);
-        return 0;
+        return BGC_FAILED;
     }
 
     const float corrector1 = 1.0f / square_modulus;
@@ -757,7 +934,7 @@ inline int bgc_fp32_quaternion_get_reverse_matrix(BGC_FP32_Matrix3x3* reverse, c
     reverse->r3c2 = corrector2 * (x2x3 - s0x1);
     reverse->r1c3 = corrector2 * (x1x3 - s0x2);
 
-    return 1;
+    return BGC_SUCCESS;
 }
 
 inline int bgc_fp64_quaternion_get_reverse_matrix(BGC_FP64_Matrix3x3* reverse, const BGC_FP64_Quaternion* quaternion)
@@ -772,7 +949,7 @@ inline int bgc_fp64_quaternion_get_reverse_matrix(BGC_FP64_Matrix3x3* reverse, c
     if (square_modulus <= BGC_FP64_SQUARE_EPSILON || isnan(square_modulus))
     {
         bgc_fp64_matrix3x3_make_identity(reverse);
-        return 0;
+        return BGC_FAILED;
     }
 
     const double corrector1 = 1.0f / square_modulus;
@@ -805,22 +982,22 @@ inline int bgc_fp64_quaternion_get_reverse_matrix(BGC_FP64_Matrix3x3* reverse, c
 
 inline int bgc_fp32_quaternion_get_both_matrices(BGC_FP32_Matrix3x3* rotation, BGC_FP32_Matrix3x3* reverse, const BGC_FP32_Quaternion* quaternion)
 {
-    if (bgc_fp32_quaternion_get_reverse_matrix(reverse, quaternion)) {
+    if (bgc_fp32_quaternion_get_reverse_matrix(reverse, quaternion) == BGC_SUCCESS) {
         bgc_fp32_matrix3x3_get_transposed(rotation, reverse);
-        return 1;
+        return BGC_SUCCESS;
     }
 
-    return 0;
+    return BGC_FAILED;
 }
 
 inline int bgc_fp64_quaternion_get_both_matrices(BGC_FP64_Matrix3x3* rotation, BGC_FP64_Matrix3x3* reverse, const BGC_FP64_Quaternion* quaternion)
 {
-    if (bgc_fp64_quaternion_get_reverse_matrix(reverse, quaternion)) {
+    if (bgc_fp64_quaternion_get_reverse_matrix(reverse, quaternion) == BGC_SUCCESS) {
         bgc_fp64_matrix3x3_get_transposed(rotation, reverse);
-        return 1;
+        return BGC_SUCCESS;
     }
 
-    return 0;
+    return BGC_FAILED;
 }
 
 // ================== Are Close ================= //

basic-geometry/turn3.c

@@ -85,7 +85,7 @@ void _bgc_fp32_turn3_normalize(BGC_FP32_Turn3* turn, const float square_modulus)
         return;
     }
 
-    bgc_fp32_quaternion_multiply(&turn->_versor, &turn->_versor, sqrtf(1.0f / square_modulus));
+    bgc_fp32_quaternion_multiply_by_number(&turn->_versor, &turn->_versor, sqrtf(1.0f / square_modulus));
 }
 
 void _bgc_fp64_turn3_normalize(BGC_FP64_Turn3* turn, const double square_modulus)
@@ -95,7 +95,7 @@ void _bgc_fp64_turn3_normalize(BGC_FP64_Turn3* turn, const double square_modulus
         return;
     }
 
-    bgc_fp64_quaternion_multiply(&turn->_versor, &turn->_versor, sqrt(1.0 / square_modulus));
+    bgc_fp64_quaternion_multiply_by_number(&turn->_versor, &turn->_versor, sqrt(1.0 / square_modulus));
 }
 
 
@@ -289,190 +289,246 @@ int bgc_fp64_turn3_find_direction_difference(BGC_FP64_Turn3* difference, const B
     return _bgc_fp64_turn3_make_direction_turn(difference, start, end, start_square_modulus * end_square_modulus);
 }
 
-// =============== Set Directions =============== //
+// ============ Make Orthogonal Pair ============ //
 
-static int _bgc_fp32_turn3_validate_basis(const float primary_square_modulus, const float auxiliary_square_modulus, const float orthogonal_square_modulus)
+static inline int _bgc_fp32_turn3_get_orthogonal_pair(BGC_FP32_Vector3* unit_main, BGC_FP32_Vector3* unit_branch, const BGC_FP32_Vector3* main, const BGC_FP32_Vector3* branch)
 {
-    if (primary_square_modulus <= BGC_FP32_SQUARE_EPSILON) {
+    const float main_square_modulus = bgc_fp32_vector3_get_square_modulus(main);
-        //TODO: add error code for: primary_vector is zero
+
-        return BGC_FAILED;
+    if (main_square_modulus <= BGC_FP32_SQUARE_EPSILON) {
+        return _BGC_ERROR_TURN3_EMPTY_MAIN;
     }
 
-    if (auxiliary_square_modulus <= BGC_FP32_SQUARE_EPSILON) {
+    const float branch_square_modulus = bgc_fp32_vector3_get_square_modulus(branch);
-        //TODO: add error code for: auxiliary_vector is zero
+
-        return BGC_FAILED;
+    if (branch_square_modulus <= BGC_FP32_SQUARE_EPSILON) {
+        return _BGC_ERROR_TURN3_EMPTY_BRANCH;
     }
 
-    if (orthogonal_square_modulus <= BGC_FP32_SQUARE_EPSILON * primary_square_modulus * auxiliary_square_modulus) {
+    bgc_fp32_vector3_multiply(unit_main, main, sqrtf(1.0f / main_square_modulus));
-        //TODO: add error code for: primary_vector and auxiliary_vector are parallel
+
-        return BGC_FAILED;
+    bgc_fp32_vector3_add_scaled(unit_branch, branch, unit_main, -bgc_fp32_vector3_get_dot_product(branch, unit_main));
+
+    const float orthogonal_square_modulus = bgc_fp32_vector3_get_square_modulus(unit_branch);
+
+    if (orthogonal_square_modulus <= BGC_FP32_SQUARE_EPSILON) {
+        return _BGC_ERROR_TURN3_PAIR_PARALLEL;
+    }
+
+    bgc_fp32_vector3_multiply(unit_branch, unit_branch, sqrtf(1.0f / orthogonal_square_modulus));
+
+    return BGC_SUCCESS;
+}
+
+static inline int _bgc_fp64_turn3_get_orthogonal_pair(BGC_FP64_Vector3* unit_main, BGC_FP64_Vector3* unit_branch, const BGC_FP64_Vector3* main, const BGC_FP64_Vector3* branch)
+{
+    const double main_square_modulus = bgc_fp64_vector3_get_square_modulus(main);
+
+    if (main_square_modulus <= BGC_FP64_SQUARE_EPSILON) {
+        return _BGC_ERROR_TURN3_EMPTY_MAIN;
+    }
+
+    const double branch_square_modulus = bgc_fp64_vector3_get_square_modulus(branch);
+
+    if (branch_square_modulus <= BGC_FP64_SQUARE_EPSILON) {
+        return _BGC_ERROR_TURN3_EMPTY_BRANCH;
+    }
+
+    bgc_fp64_vector3_multiply(unit_main, main, sqrtf(1.0 / main_square_modulus));
+
+    bgc_fp64_vector3_add_scaled(unit_branch, branch, unit_main, -bgc_fp64_vector3_get_dot_product(branch, unit_main));
+
+    const double orthogonal_square_modulus = bgc_fp64_vector3_get_square_modulus(unit_branch);
+
+    if (orthogonal_square_modulus <= BGC_FP64_SQUARE_EPSILON) {
+        return _BGC_ERROR_TURN3_PAIR_PARALLEL;
+    }
+
+    bgc_fp64_vector3_multiply(unit_branch, unit_branch, sqrt(1.0 / orthogonal_square_modulus));
+
+    return BGC_SUCCESS;
+}
+
+// ========= Make Direction Difference ========== //
+
+static inline void _bgc_fp32_turn3_get_turning_quaternion(BGC_FP32_Quaternion* quaternion, const BGC_FP32_Vector3* unit_start, const BGC_FP32_Vector3* unit_end, const BGC_FP32_Vector3* unit_orthogonal)
+{
+    BGC_FP32_Vector3 axis;
+
+    bgc_fp32_vector3_get_cross_product(&axis, unit_start, unit_end);
+
+    const float dot_product = bgc_fp32_vector3_get_dot_product(unit_start, unit_end);
+
+    const float axis_square_modulus = bgc_fp32_vector3_get_square_modulus(&axis);
+
+    // unit_start and unit_end are parallel
+    if (axis_square_modulus <= BGC_FP32_SQUARE_EPSILON) {
+        // unit_start and unit_end are co-directional, angle = 180 degrees
+        if (dot_product >= 0.0f) {
+            quaternion->s0 = 1.0f;
+            quaternion->x1 = 0.0f;
+            quaternion->x2 = 0.0f;
+            quaternion->x3 = 0.0f;
+            return;
+        }
+
+        // unit_start and unit_end are opposite, angle = 180 degrees
+        quaternion->s0 = 0.0f;
+        quaternion->x1 = unit_orthogonal->x1;
+        quaternion->x2 = unit_orthogonal->x2;
+        quaternion->x3 = unit_orthogonal->x3;
+        return;
+    }
+
+    const float axis_modulus = sqrtf(axis_square_modulus);
+
+    const float angle = 0.5f * atan2f(axis_modulus, dot_product);
+
+    const float multiplier = sinf(angle) / axis_modulus;
+
+    quaternion->s0 = cosf(angle);
+    quaternion->x1 = axis.x1 * multiplier;
+    quaternion->x2 = axis.x2 * multiplier;
+    quaternion->x3 = axis.x3 * multiplier;
+}
+
+static inline void _bgc_fp64_turn3_get_turning_quaternion(BGC_FP64_Quaternion* quaternion, const BGC_FP64_Vector3* unit_start, const BGC_FP64_Vector3* unit_end, const BGC_FP64_Vector3* unit_orthogonal)
+{
+    BGC_FP64_Vector3 axis;
+
+    bgc_fp64_vector3_get_cross_product(&axis, unit_start, unit_end);
+
+    const double dot_product = bgc_fp64_vector3_get_dot_product(unit_start, unit_end);
+
+    const double axis_square_modulus = bgc_fp64_vector3_get_square_modulus(&axis);
+
+    // unit_start and unit_end are parallel
+    if (axis_square_modulus <= BGC_FP64_SQUARE_EPSILON) {
+        // unit_start and unit_end are co-directional, angle = 180 degrees
+        if (dot_product >= 0.0) {
+            quaternion->s0 = 1.0;
+            quaternion->x1 = 0.0;
+            quaternion->x2 = 0.0;
+            quaternion->x3 = 0.0;
+            return;
+        }
+
+        // unit_start and unit_end are opposite, angle = 180 degrees
+        quaternion->s0 = 0.0;
+        quaternion->x1 = unit_orthogonal->x1;
+        quaternion->x2 = unit_orthogonal->x2;
+        quaternion->x3 = unit_orthogonal->x3;
+        return;
+    }
+
+    const double axis_modulus = sqrt(axis_square_modulus);
+
+    const double angle = 0.5 * atan2(axis_modulus, dot_product);
+
+    const double multiplier = sin(angle) / axis_modulus;
+
+    quaternion->s0 = cos(angle);
+    quaternion->x1 = axis.x1 * multiplier;
+    quaternion->x2 = axis.x2 * multiplier;
+    quaternion->x3 = axis.x3 * multiplier;
+}
+
+// ============ Make 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
+) {
+    BGC_FP32_Vector3 first_fixed_main, first_fixed_branch, first_turned_branch, second_fixed_main, second_fixed_branch;
+
+    int status = _bgc_fp32_turn3_get_orthogonal_pair(&first_fixed_main, &first_fixed_branch, first_pair_main, first_pair_branch);
+
+    if (status != BGC_SUCCESS) {
+        bgc_fp32_turn3_reset(turn);
+        return status + _BGC_ERROR_TURN3_FIRST_PAIR;
+    }
+
+    status = _bgc_fp32_turn3_get_orthogonal_pair(&second_fixed_main, &second_fixed_branch, second_pair_main, second_pair_branch);
+
+    if (status != BGC_SUCCESS) {
+        bgc_fp32_turn3_reset(turn);
+        return status + _BGC_ERROR_TURN3_SECOND_PAIR;
+    }
+
+    BGC_FP32_Quaternion q1, q2;
+
+    // Calculation of a turn (q1) which turns first_fixed_main into second_fixed_main
+    _bgc_fp32_turn3_get_turning_quaternion(&q1, &first_fixed_main, &second_fixed_main, &first_fixed_branch);
+
+    // Roughly turn first_fixed_branch with q1 turn
+    _bgc_fp32_quaternion_turn_vector_roughly(&first_turned_branch, &q1, &first_fixed_branch);
+
+    // Calculation of a turn (q2) which turns first_turned_branch into second_fixed_branch
+    _bgc_fp32_turn3_get_turning_quaternion(&q2, &first_turned_branch, &second_fixed_branch, &second_fixed_main);
+
+    // Composing two turns with multiplication of quaterntions (q2 * q1)
+    bgc_fp32_quaternion_multiply_by_quaternion(&turn->_versor, &q2, &q1);
+
+    // Making a final versor (a normalized quaternion)
+    const float square_modulus = bgc_fp32_quaternion_get_square_modulus(&turn->_versor);
+
+    if (!bgc_fp32_is_square_unit(square_modulus)) {
+        _bgc_fp32_turn3_normalize(turn, square_modulus);
     }
 
     return BGC_SUCCESS;
 }
 
-static int _bgc_fp64_turn3_validate_basis(const double primary_square_modulus, const double auxiliary_square_modulus, const double orthogonal_square_modulus)
+int bgc_fp64_turn3_find_pair_difference(
-{
+    BGC_FP64_Turn3* turn,
-    if (primary_square_modulus <= BGC_FP64_SQUARE_EPSILON) {
+    const BGC_FP64_Vector3* first_pair_main,
-        //TODO: add error code for: primary_vector is zero
+    const BGC_FP64_Vector3* first_pair_branch,
-        return BGC_FAILED;
+    const BGC_FP64_Vector3* second_pair_main,
+    const BGC_FP64_Vector3* second_pair_branch
+) {
+    BGC_FP64_Vector3 first_fixed_main, first_fixed_branch, first_turned_branch, second_fixed_main, second_fixed_branch;
+
+    int status = _bgc_fp64_turn3_get_orthogonal_pair(&first_fixed_main, &first_fixed_branch, first_pair_main, first_pair_branch);
+
+    if (status != BGC_SUCCESS) {
+        bgc_fp64_turn3_reset(turn);
+        return status + _BGC_ERROR_TURN3_FIRST_PAIR;
     }
 
-    if (auxiliary_square_modulus <= BGC_FP64_SQUARE_EPSILON) {
+    status = _bgc_fp64_turn3_get_orthogonal_pair(&second_fixed_main, &second_fixed_branch, second_pair_main, second_pair_branch);
-        //TODO: add error code for: auxiliary_vector is zero
+
-        return BGC_FAILED;
+    if (status != BGC_SUCCESS) {
+        bgc_fp64_turn3_reset(turn);
+        return status + _BGC_ERROR_TURN3_SECOND_PAIR;
     }
 
-    if (orthogonal_square_modulus <= BGC_FP64_SQUARE_EPSILON * primary_square_modulus * auxiliary_square_modulus) {
+    BGC_FP64_Quaternion q1, q2;
-        //TODO: add error code for: primary_vector and auxiliary_vector are parallel
+
-        return BGC_FAILED;
+    // Calculation of a turn (q1) which turns first_fixed_main into second_fixed_main
+    _bgc_fp64_turn3_get_turning_quaternion(&q1, &first_fixed_main, &second_fixed_main, &first_fixed_branch);
+
+    // Roughly turn first_fixed_branch with q1 turn
+    _bgc_fp64_quaternion_turn_vector_roughly(&first_turned_branch, &q1, &first_fixed_branch);
+
+    // Calculation of a turn (q2) which turns first_turned_branch into second_fixed_branch
+    _bgc_fp64_turn3_get_turning_quaternion(&q2, &first_turned_branch, &second_fixed_branch, &second_fixed_main);
+
+    // Composing two turns with multiplication of quaterntions (q2 * q1)
+    bgc_fp64_quaternion_multiply_by_quaternion(&turn->_versor, &q2, &q1);
+
+    // Making a final versor (a normalized quaternion)
+    const double square_modulus = bgc_fp64_quaternion_get_square_modulus(&turn->_versor);
+
+    if (!bgc_fp64_is_square_unit(square_modulus)) {
+        _bgc_fp64_turn3_normalize(turn, square_modulus);
     }
 
     return BGC_SUCCESS;
 }
 
-int bgc_fp32_turn3_make_basis_difference(
-    BGC_FP32_Turn3* versor,
-    const BGC_FP32_Vector3* initial_primary_direction,
-    const BGC_FP32_Vector3* initial_auxiliary_direction,
-    const BGC_FP32_Vector3* final_primary_direction,
-    const BGC_FP32_Vector3* final_auxiliary_direction
-)
-{
-    BGC_FP32_Vector3 initial_orthogonal_direction, turned_orthogonal_direction, final_orthogonal_direction;
-
-    // Step 1: Validate initial basis:
-    bgc_fp32_vector3_get_cross_product(&initial_orthogonal_direction, initial_primary_direction, initial_auxiliary_direction);
-
-    const float initial_primary_square_modulus = bgc_fp32_vector3_get_square_modulus(initial_primary_direction);
-    const float initial_auxiliary_square_modulus = bgc_fp32_vector3_get_square_modulus(initial_auxiliary_direction);
-    const float initial_orthogonal_square_modulus = bgc_fp32_vector3_get_square_modulus(&initial_orthogonal_direction);
-
-    const int initial_basis_valudation = _bgc_fp32_turn3_validate_basis(initial_primary_square_modulus, initial_auxiliary_square_modulus, initial_orthogonal_square_modulus);
-
-    if (initial_basis_valudation != BGC_SUCCESS) {
-        return initial_basis_valudation;
-    }
-
-    // Step 1: Validate final basis:
-    bgc_fp32_vector3_get_cross_product(&final_orthogonal_direction, final_primary_direction, final_auxiliary_direction);
-
-    const float final_primary_square_modulus = bgc_fp32_vector3_get_square_modulus(final_primary_direction);
-    const float final_auxiliary_square_modulus = bgc_fp32_vector3_get_square_modulus(final_auxiliary_direction);
-    const float final_orthogonal_square_modulus = bgc_fp32_vector3_get_square_modulus(&final_orthogonal_direction);
-
-    const int final_basis_valudation = _bgc_fp32_turn3_validate_basis(final_primary_square_modulus, final_auxiliary_square_modulus, final_orthogonal_square_modulus);
-
-    if (final_basis_valudation != BGC_SUCCESS) {
-        return final_basis_valudation;
-    }
-
-    // Step 3: Validate normalize orthogonal vectors:
-    bgc_fp32_vector3_divide(&initial_orthogonal_direction, &initial_orthogonal_direction, sqrtf(initial_orthogonal_square_modulus));
-    bgc_fp32_vector3_divide(&final_orthogonal_direction, &final_orthogonal_direction, sqrtf(final_orthogonal_square_modulus));
-
-    BGC_FP32_Turn3 turn1, turn2;
-
-    // Step 4: Find turn1
-    int turn1_code = _bgc_fp32_turn3_make_direction_turn(&turn1, initial_primary_direction, final_primary_direction, initial_primary_square_modulus * final_primary_square_modulus);
-
-    if (turn1_code == BGC_OPPOSITE) {
-        bgc_fp32_turn3_set_raw_values(&turn1, 0.0f, initial_orthogonal_direction.x1, initial_orthogonal_direction.x2, initial_orthogonal_direction.x3);
-    }
-
-    bgc_fp32_turn3_vector(&turned_orthogonal_direction, &turn1, &initial_orthogonal_direction);
-
-    // Step 5: Find turn2:
-    int turn2_code = _bgc_fp32_turn3_make_direction_turn(&turn2, &turned_orthogonal_direction, &final_orthogonal_direction, 1.0f);
-
-    if (turn2_code == BGC_OPPOSITE) {
-        const float turn2_multiplier = sqrtf(1.0f / final_primary_square_modulus);
-
-        bgc_fp32_turn3_set_raw_values(&turn2,
-            0.0f,
-            final_primary_direction->x1 * turn2_multiplier,
-            final_primary_direction->x2 * turn2_multiplier,
-            final_primary_direction->x3 * turn2_multiplier
-        );
-    }
-
-    // Step 6: Combine turn1 and turn2:
-    bgc_fp32_turn3_combine(versor, &turn1, &turn2);
-
-    return BGC_SUCCESS;
-}
-
-int bgc_fp64_turn3_make_basis_difference(
-    BGC_FP64_Turn3* versor,
-    const BGC_FP64_Vector3* initial_primary_direction,
-    const BGC_FP64_Vector3* initial_auxiliary_direction,
-    const BGC_FP64_Vector3* final_primary_direction,
-    const BGC_FP64_Vector3* final_auxiliary_direction
-)
-{
-    BGC_FP64_Vector3 initial_orthogonal_direction, turned_orthogonal_direction, final_orthogonal_direction;
-
-    // Step 1: Validate initial basis:
-    bgc_fp64_vector3_get_cross_product(&initial_orthogonal_direction, initial_primary_direction, initial_auxiliary_direction);
-
-    const double initial_primary_square_modulus = bgc_fp64_vector3_get_square_modulus(initial_primary_direction);
-    const double initial_auxiliary_square_modulus = bgc_fp64_vector3_get_square_modulus(initial_auxiliary_direction);
-    const double initial_orthogonal_square_modulus = bgc_fp64_vector3_get_square_modulus(&initial_orthogonal_direction);
-
-    const int initial_basis_valudation = _bgc_fp64_turn3_validate_basis(initial_primary_square_modulus, initial_auxiliary_square_modulus, initial_orthogonal_square_modulus);
-
-    if (initial_basis_valudation != BGC_SUCCESS) {
-        return initial_basis_valudation;
-    }
-
-    // Step 1: Validate final basis:
-    bgc_fp64_vector3_get_cross_product(&final_orthogonal_direction, final_primary_direction, final_auxiliary_direction);
-
-    const double final_primary_square_modulus = bgc_fp64_vector3_get_square_modulus(final_primary_direction);
-    const double final_auxiliary_square_modulus = bgc_fp64_vector3_get_square_modulus(final_auxiliary_direction);
-    const double final_orthogonal_square_modulus = bgc_fp64_vector3_get_square_modulus(&final_orthogonal_direction);
-
-    const int final_basis_valudation = _bgc_fp64_turn3_validate_basis(final_primary_square_modulus, final_auxiliary_square_modulus, final_orthogonal_square_modulus);
-
-    if (final_basis_valudation != BGC_SUCCESS) {
-        return final_basis_valudation;
-    }
-
-    // Step 3: Validate normalize orthogonal vectors:
-    bgc_fp64_vector3_divide(&initial_orthogonal_direction, &initial_orthogonal_direction, sqrt(initial_orthogonal_square_modulus));
-    bgc_fp64_vector3_divide(&final_orthogonal_direction, &final_orthogonal_direction, sqrt(final_orthogonal_square_modulus));
-
-    BGC_FP64_Turn3 turn1, turn2;
-
-    // Step 4: Find turn1
-    int turn1_code = _bgc_fp64_turn3_make_direction_turn(&turn1, initial_primary_direction, final_primary_direction, initial_primary_square_modulus * final_primary_square_modulus);
-
-    if (turn1_code == BGC_OPPOSITE) {
-        bgc_fp64_turn3_set_raw_values(&turn1, 0.0, initial_orthogonal_direction.x1, initial_orthogonal_direction.x2, initial_orthogonal_direction.x3);
-    }
-
-    bgc_fp64_turn3_vector(&turned_orthogonal_direction, &turn1, &initial_orthogonal_direction);
-
-    // Step 5: Find turn2:
-    int turn2_code = _bgc_fp64_turn3_make_direction_turn(&turn2, &turned_orthogonal_direction, &final_orthogonal_direction, 1.0f);
-
-    if (turn2_code == BGC_OPPOSITE) {
-        const double turn2_multiplier = sqrt(1.0 / final_primary_square_modulus);
-
-        bgc_fp64_turn3_set_raw_values(&turn2,
-            0.0,
-            final_primary_direction->x1 * turn2_multiplier,
-            final_primary_direction->x2 * turn2_multiplier,
-            final_primary_direction->x3 * turn2_multiplier
-        );
-    }
-
-    // Step 6: Combine turn1 and turn2:
-    bgc_fp64_turn3_combine(versor, &turn1, &turn2);
-
-    return BGC_SUCCESS;
-}
-
 // =============== Get Exponation =============== //
 
 void bgc_fp32_turn3_get_exponation(BGC_FP32_Turn3* power, const BGC_FP32_Turn3* base, const float exponent)

basic-geometry/turn3.h

@@ -13,6 +13,20 @@
 #define BGC_ZERO_TURN 0
 #define BGC_OPPOSITE -1
 
+#define _BGC_ERROR_TURN3_FIRST_PAIR 3000
+#define _BGC_ERROR_TURN3_SECOND_PAIR 3010
+#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_EMPTY_MAIN 3001
+#define BGC_ERROR_TURN3_FIRST_PAIR_EMPTY_BRANCH 3002
+#define BGC_ERROR_TURN3_FIRST_PAIR_PARALLEL 3003
+
+#define BGC_ERROR_TURN3_SECOND_PAIR_EMPTY_MAIN 3011
+#define BGC_ERROR_TURN3_SECOND_PAIR_EMPTY_BRANCH 3012
+#define BGC_ERROR_TURN3_SECOND_PAIR_PARALLEL 3013
+
 #define BGC_ERROR_PRIMARY_DIRECTION_UNKNOWN -3001
 #define BGC_ERROR_PRIMARY_VECTOR_IS_ZERO -3002
 
@@ -133,22 +147,22 @@ int bgc_fp32_turn3_find_direction_difference(BGC_FP32_Turn3* difference, const B
 
 int bgc_fp64_turn3_find_direction_difference(BGC_FP64_Turn3* difference, const BGC_FP64_Vector3* start, const BGC_FP64_Vector3* end);
 
-// =============== Set Directions =============== //
+// ======= Find Direction Pair Difference ======= //
 
-int bgc_fp32_turn3_make_basis_difference(
+int bgc_fp32_turn3_find_pair_difference(
     BGC_FP32_Turn3* turn,
-    const BGC_FP32_Vector3* initial_primary_direction,
+    const BGC_FP32_Vector3* first_pair_main,
-    const BGC_FP32_Vector3* initial_auxiliary_direction,
+    const BGC_FP32_Vector3* first_pair_branch,
-    const BGC_FP32_Vector3* final_primary_direction,
+    const BGC_FP32_Vector3* second_pair_main,
-    const BGC_FP32_Vector3* final_auxiliary_direction
+    const BGC_FP32_Vector3* second_pair_branch
 );
 
-int bgc_fp64_turn3_make_basis_difference(
+int bgc_fp64_turn3_find_pair_difference(
     BGC_FP64_Turn3* turn,
-    const BGC_FP64_Vector3* initial_primary_direction,
+    const BGC_FP64_Vector3* first_pair_main,
-    const BGC_FP64_Vector3* initial_auxiliary_direction,
+    const BGC_FP64_Vector3* first_pair_branch,
-    const BGC_FP64_Vector3* final_primary_direction,
+    const BGC_FP64_Vector3* second_pair_main,
-    const BGC_FP64_Vector3* final_auxiliary_direction
+    const BGC_FP64_Vector3* second_pair_branch
 );
 
 // ==================== Copy ==================== //
@@ -301,7 +315,7 @@ void bgc_fp64_turn3_get_exponation(BGC_FP64_Turn3* power, const BGC_FP64_Turn3*
 
 inline void bgc_fp32_turn3_combine(BGC_FP32_Turn3* combination, const BGC_FP32_Turn3* first, const BGC_FP32_Turn3* second)
 {
-    bgc_fp32_quaternion_get_product(&combination->_versor, &second->_versor, &first->_versor);
+    bgc_fp32_quaternion_multiply_by_quaternion(&combination->_versor, &second->_versor, &first->_versor);
 
     const float square_modulus = bgc_fp32_quaternion_get_square_modulus(&combination->_versor);
 
@@ -312,7 +326,7 @@ inline void bgc_fp32_turn3_combine(BGC_FP32_Turn3* combination, const BGC_FP32_T
 
 inline void bgc_fp64_turn3_combine(BGC_FP64_Turn3* combination, const BGC_FP64_Turn3* first, const BGC_FP64_Turn3* second)
 {
-    bgc_fp64_quaternion_get_product(&combination->_versor, &second->_versor, &first->_versor);
+    bgc_fp64_quaternion_multiply_by_quaternion(&combination->_versor, &second->_versor, &first->_versor);
 
     const double square_modulus = bgc_fp64_quaternion_get_square_modulus(&combination->_versor);
 
@@ -327,9 +341,9 @@ inline void bgc_fp32_turn3_combine3(BGC_FP32_Turn3* combination, const BGC_FP32_
 {
     BGC_FP32_Quaternion product;
 
-    bgc_fp32_quaternion_get_product(&product, &second->_versor, &first->_versor);
+    bgc_fp32_quaternion_multiply_by_quaternion(&product, &second->_versor, &first->_versor);
 
-    bgc_fp32_quaternion_get_product(&combination->_versor, &third->_versor, &product);
+    bgc_fp32_quaternion_multiply_by_quaternion(&combination->_versor, &third->_versor, &product);
 
     const float square_modulus = bgc_fp32_quaternion_get_square_modulus(&combination->_versor);
 
@@ -342,9 +356,9 @@ inline void bgc_fp64_turn3_combine3(BGC_FP64_Turn3* combination, const BGC_FP64_
 {
     BGC_FP64_Quaternion product;
 
-    bgc_fp64_quaternion_get_product(&product, &second->_versor, &first->_versor);
+    bgc_fp64_quaternion_multiply_by_quaternion(&product, &second->_versor, &first->_versor);
 
-    bgc_fp64_quaternion_get_product(&combination->_versor, &third->_versor, &product);
+    bgc_fp64_quaternion_multiply_by_quaternion(&combination->_versor, &third->_versor, &product);
 
     const double square_modulus = bgc_fp64_quaternion_get_square_modulus(&combination->_versor);
 
@@ -357,7 +371,7 @@ inline void bgc_fp64_turn3_combine3(BGC_FP64_Turn3* combination, const BGC_FP64_
 
 inline void bgc_fp32_turn3_exclude(BGC_FP32_Turn3* difference, const BGC_FP32_Turn3* base, const BGC_FP32_Turn3* excludant)
 {
-    bgc_fp32_quaternion_get_product_by_conjugate(&difference->_versor, &base->_versor, &excludant->_versor);
+    bgc_fp32_quaternion_multiply_by_conjugate(&difference->_versor, &base->_versor, &excludant->_versor);
 
     const float square_modulus = bgc_fp32_quaternion_get_square_modulus(&difference->_versor);
 
@@ -368,7 +382,7 @@ inline void bgc_fp32_turn3_exclude(BGC_FP32_Turn3* difference, const BGC_FP32_Tu
 
 inline void bgc_fp64_turn3_exclude(BGC_FP64_Turn3* difference, const BGC_FP64_Turn3* base, const BGC_FP64_Turn3* excludant)
 {
-    bgc_fp64_quaternion_get_product_by_conjugate(&difference->_versor, &base->_versor, &excludant->_versor);
+    bgc_fp64_quaternion_multiply_by_conjugate(&difference->_versor, &base->_versor, &excludant->_versor);
 
     const double square_modulus = bgc_fp64_quaternion_get_square_modulus(&difference->_versor);
 
@@ -421,66 +435,26 @@ inline void bgc_fp64_turn3_get_both_matrices(BGC_FP64_Matrix3x3* rotation, BGC_F
 
 // ================ Turn Vector ================= //
 
-inline void bgc_fp32_turn3_vector(BGC_FP32_Vector3* turned_vector, const BGC_FP32_Turn3* versor, const BGC_FP32_Vector3* vector)
+inline void bgc_fp32_turn3_vector(BGC_FP32_Vector3* turned_vector, const BGC_FP32_Turn3* turn, const BGC_FP32_Vector3* original_vector)
 {
-    const float tx1 = 2.0f * (versor->_versor.x2 * vector->x3 - versor->_versor.x3 * vector->x2);
+    _bgc_fp32_quaternion_turn_vector_roughly(turned_vector, &turn->_versor, original_vector);
-    const float tx2 = 2.0f * (versor->_versor.x3 * vector->x1 - versor->_versor.x1 * vector->x3);
-    const float tx3 = 2.0f * (versor->_versor.x1 * vector->x2 - versor->_versor.x2 * vector->x1);
-
-    const float x1 = (vector->x1 + tx1 * versor->_versor.s0) + (versor->_versor.x2 * tx3 - versor->_versor.x3 * tx2);
-    const float x2 = (vector->x2 + tx2 * versor->_versor.s0) + (versor->_versor.x3 * tx1 - versor->_versor.x1 * tx3);
-    const float x3 = (vector->x3 + tx3 * versor->_versor.s0) + (versor->_versor.x1 * tx2 - versor->_versor.x2 * tx1);
-
-    turned_vector->x1 = x1;
-    turned_vector->x2 = x2;
-    turned_vector->x3 = x3;
 }
 
-inline void bgc_fp64_turn3_vector(BGC_FP64_Vector3* turned_vector, const BGC_FP64_Turn3* turn, const BGC_FP64_Vector3* vector)
+inline void bgc_fp64_turn3_vector(BGC_FP64_Vector3* turned_vector, const BGC_FP64_Turn3* turn, const BGC_FP64_Vector3* original_vector)
 {
-    const double tx1 = 2.0 * (turn->_versor.x2 * vector->x3 - turn->_versor.x3 * vector->x2);
+    _bgc_fp64_quaternion_turn_vector_roughly(turned_vector, &turn->_versor, original_vector);
-    const double tx2 = 2.0 * (turn->_versor.x3 * vector->x1 - turn->_versor.x1 * vector->x3);
-    const double tx3 = 2.0 * (turn->_versor.x1 * vector->x2 - turn->_versor.x2 * vector->x1);
-
-    const double x1 = (vector->x1 + tx1 * turn->_versor.s0) + (turn->_versor.x2 * tx3 - turn->_versor.x3 * tx2);
-    const double x2 = (vector->x2 + tx2 * turn->_versor.s0) + (turn->_versor.x3 * tx1 - turn->_versor.x1 * tx3);
-    const double x3 = (vector->x3 + tx3 * turn->_versor.s0) + (turn->_versor.x1 * tx2 - turn->_versor.x2 * tx1);
-
-    turned_vector->x1 = x1;
-    turned_vector->x2 = x2;
-    turned_vector->x3 = x3;
 }
 
 // ============== Turn Vector Back ============== //
 
-inline void bgc_fp32_turn3_vector_back(BGC_FP32_Vector3* turned_vector, const BGC_FP32_Turn3* turn, const BGC_FP32_Vector3* vector)
+inline void bgc_fp32_turn3_vector_back(BGC_FP32_Vector3* turned_vector, const BGC_FP32_Turn3* turn, const BGC_FP32_Vector3* original_vector)
 {
-    const float tx1 = 2.0f * (turn->_versor.x2 * vector->x3 - turn->_versor.x3 * vector->x2);
+    _bgc_fp32_quaternion_turn_vector_back_roughly(turned_vector, &turn->_versor, original_vector);
-    const float tx2 = 2.0f * (turn->_versor.x3 * vector->x1 - turn->_versor.x1 * vector->x3);
-    const float tx3 = 2.0f * (turn->_versor.x1 * vector->x2 - turn->_versor.x2 * vector->x1);
-
-    const float x1 = (vector->x1 - tx1 * turn->_versor.s0) + (turn->_versor.x2 * tx3 - turn->_versor.x3 * tx2);
-    const float x2 = (vector->x2 - tx2 * turn->_versor.s0) + (turn->_versor.x3 * tx1 - turn->_versor.x1 * tx3);
-    const float x3 = (vector->x3 - tx3 * turn->_versor.s0) + (turn->_versor.x1 * tx2 - turn->_versor.x2 * tx1);
-
-    turned_vector->x1 = x1;
-    turned_vector->x2 = x2;
-    turned_vector->x3 = x3;
 }
 
-inline void bgc_fp64_turn3_vector_back(BGC_FP64_Vector3* turned_vector, const BGC_FP64_Turn3* turn, const BGC_FP64_Vector3* vector)
+inline void bgc_fp64_turn3_vector_back(BGC_FP64_Vector3* turned_vector, const BGC_FP64_Turn3* turn, const BGC_FP64_Vector3* original_vector)
 {
-    const double tx1 = 2.0 * (turn->_versor.x2 * vector->x3 - turn->_versor.x3 * vector->x2);
+    _bgc_fp64_quaternion_turn_vector_back_roughly(turned_vector, &turn->_versor, original_vector);
-    const double tx2 = 2.0 * (turn->_versor.x3 * vector->x1 - turn->_versor.x1 * vector->x3);
-    const double tx3 = 2.0 * (turn->_versor.x1 * vector->x2 - turn->_versor.x2 * vector->x1);
-
-    const double x1 = (vector->x1 - tx1 * turn->_versor.s0) + (turn->_versor.x2 * tx3 - turn->_versor.x3 * tx2);
-    const double x2 = (vector->x2 - tx2 * turn->_versor.s0) + (turn->_versor.x3 * tx1 - turn->_versor.x1 * tx3);
-    const double x3 = (vector->x3 - tx3 * turn->_versor.s0) + (turn->_versor.x1 * tx2 - turn->_versor.x2 * tx1);
-
-    turned_vector->x1 = x1;
-    turned_vector->x2 = x2;
-    turned_vector->x3 = x3;
 }
 
 // ================== Are Close ================= //