diff --git a/basic-geometry/turn3.c b/basic-geometry/turn3.c
index 0050c64..a1f986f 100644
--- a/basic-geometry/turn3.c
+++ b/basic-geometry/turn3.c
@@ -201,92 +201,95 @@ void bgc_fp64_turn3_set_rotation(BGC_FP64_Turn3* turn, const double x1, const do
     }
 }
 
-// ========= Make Direction Difference ========== //
-
-static int _bgc_fp32_turn3_make_direction_turn(BGC_FP32_Turn3* turn, const BGC_FP32_Vector3* start, const BGC_FP32_Vector3* end, const float square_modulus_product)
-{
-    BGC_FP32_Vector3 orthogonal_axis;
-
-    bgc_fp32_vector3_get_cross_product(&orthogonal_axis, start, end);
-
-    const float scalar_product = bgc_fp32_vector3_get_dot_product(start, end);
-    const float square_modulus = bgc_fp32_vector3_get_square_modulus(&orthogonal_axis);
-    const float square_sine = square_modulus / square_modulus_product;
-
-    if (square_sine > BGC_FP32_SQUARE_EPSILON) {
-        const float cosine = scalar_product / sqrtf(square_modulus_product);
-        const float angle = 0.5f * atan2f(sqrtf(square_sine), cosine);
-
-        const float multiplier = sinf(angle) * sqrtf(1.0f / square_modulus);
-
-        bgc_fp32_turn3_set_raw_values(turn, cosf(angle), orthogonal_axis.x1 * multiplier, orthogonal_axis.x2 * multiplier, orthogonal_axis.x3 * multiplier);
-        return BGC_SOME_TURN;
-    }
-
-    if (scalar_product < 0.0f) {
-        return BGC_OPPOSITE;
-    }
-
-    bgc_fp32_turn3_reset(turn);
-
-    return BGC_ZERO_TURN;
-}
-
-static int _bgc_fp64_turn3_make_direction_turn(BGC_FP64_Turn3* versor, const BGC_FP64_Vector3* start, const BGC_FP64_Vector3* end, const double square_modulus_product)
-{
-    BGC_FP64_Vector3 orthogonal_axis;
-
-    bgc_fp64_vector3_get_cross_product(&orthogonal_axis, start, end);
-
-    const double scalar_product = bgc_fp64_vector3_get_dot_product(start, end);
-    const double square_modulus = bgc_fp64_vector3_get_square_modulus(&orthogonal_axis);
-    const double square_sine = square_modulus / square_modulus_product;
-
-    if (square_sine > BGC_FP64_SQUARE_EPSILON) {
-        const double cosine = scalar_product / sqrt(square_modulus_product);
-        const double angle = 0.5 * atan2(sqrt(square_sine), cosine);
-
-        const double multiplier = sin(angle) * sqrt(1.0f / square_modulus);
-
-        bgc_fp64_turn3_set_raw_values(versor, cos(angle), orthogonal_axis.x1 * multiplier, orthogonal_axis.x2 * multiplier, orthogonal_axis.x3 * multiplier);
-        return BGC_SOME_TURN;
-    }
-
-    if (scalar_product < 0.0) {
-        return BGC_OPPOSITE;
-    }
-
-    bgc_fp64_turn3_reset(versor);
-
-    return BGC_ZERO_TURN;
-}
-
 // ========= Find Direction Difference ========== //
 
-int bgc_fp32_turn3_find_direction_difference(BGC_FP32_Turn3* difference, const BGC_FP32_Vector3* start, const BGC_FP32_Vector3* end)
+int bgc_fp32_turn3_find_direction_difference(BGC_FP32_Turn3* turn, const BGC_FP32_Vector3* first, const BGC_FP32_Vector3* second)
 {
-    const float start_square_modulus = bgc_fp32_vector3_get_square_modulus(start);
-    const float end_square_modulus = bgc_fp32_vector3_get_square_modulus(end);
+    const float first_square_modulus = bgc_fp32_vector3_get_square_modulus(first);
+    
+    bgc_fp32_turn3_reset(turn);
 
-    if (start_square_modulus <= BGC_FP32_SQUARE_EPSILON || end_square_modulus <= BGC_FP32_SQUARE_EPSILON) {
-        bgc_fp32_turn3_reset(difference);
-        return BGC_ZERO_TURN;
+    if (first_square_modulus <= BGC_FP32_SQUARE_EPSILON) {
+        return BGC_ERROR_TURN3_FIRST_VECTOR_ZERO;
     }
 
-    return _bgc_fp32_turn3_make_direction_turn(difference, start, end, start_square_modulus * end_square_modulus);
+    const float second_square_modulus = bgc_fp32_vector3_get_square_modulus(second);
+
+    if (second_square_modulus <= BGC_FP32_SQUARE_EPSILON) {
+        return BGC_ERROR_TURN3_SECOND_VECTOR_ZERO;
+    }
+
+    BGC_FP32_Vector3 axis;
+
+    bgc_fp32_vector3_get_cross_product(&axis, first, second);
+
+    const float square_product = first_square_modulus * second_square_modulus;
+    const float dot_product = bgc_fp32_vector3_get_dot_product(first, second);
+    const float axis_square_modulus = bgc_fp32_vector3_get_square_modulus(&axis);
+
+    if (axis_square_modulus <= BGC_FP32_SQUARE_EPSILON * square_product) {
+        if (dot_product < 0.0f) {
+            return BGC_ERROR_TURN3_VECTORS_OPPOSITE;
+        }
+
+        return BGC_SUCCESS;
+    }
+
+    const float axis_modulus = sqrtf(axis_square_modulus);
+    const float trigonometry_fix = sqrtf(1.0f / square_product);
+
+    const float angle = 0.5f * atan2f(axis_modulus * trigonometry_fix, dot_product * trigonometry_fix);
+
+    const float vector_multiplier = sinf(angle) / axis_modulus;
+
+    bgc_fp32_turn3_set_raw_values(turn, cosf(angle), axis.x1 * vector_multiplier, axis.x2 * vector_multiplier, axis.x3 * vector_multiplier);
+
+    return BGC_SUCCESS;
 }
 
-int bgc_fp64_turn3_find_direction_difference(BGC_FP64_Turn3* difference, const BGC_FP64_Vector3* start, const BGC_FP64_Vector3* end)
+int bgc_fp64_turn3_find_direction_difference(BGC_FP64_Turn3* turn, const BGC_FP64_Vector3* first, const BGC_FP64_Vector3* second)
 {
-    const double start_square_modulus = bgc_fp64_vector3_get_square_modulus(start);
-    const double end_square_modulus = bgc_fp64_vector3_get_square_modulus(end);
+    const double first_square_modulus = bgc_fp64_vector3_get_square_modulus(first);
 
-    if (start_square_modulus <= BGC_FP64_SQUARE_EPSILON || end_square_modulus <= BGC_FP64_SQUARE_EPSILON) {
-        bgc_fp64_turn3_reset(difference);
-        return BGC_ZERO_TURN;
+    bgc_fp64_turn3_reset(turn);
+
+    if (first_square_modulus <= BGC_FP64_SQUARE_EPSILON) {
+        return BGC_ERROR_TURN3_FIRST_VECTOR_ZERO;
     }
 
-    return _bgc_fp64_turn3_make_direction_turn(difference, start, end, start_square_modulus * end_square_modulus);
+    const double second_square_modulus = bgc_fp64_vector3_get_square_modulus(second);
+
+    if (second_square_modulus <= BGC_FP64_SQUARE_EPSILON) {
+        return BGC_ERROR_TURN3_SECOND_VECTOR_ZERO;
+    }
+
+    BGC_FP64_Vector3 axis;
+
+    bgc_fp64_vector3_get_cross_product(&axis, first, second);
+
+    const double square_product = first_square_modulus * second_square_modulus;
+    const double dot_product = bgc_fp64_vector3_get_dot_product(first, second);
+    const double axis_square_modulus = bgc_fp64_vector3_get_square_modulus(&axis);
+
+    if (axis_square_modulus <= BGC_FP64_SQUARE_EPSILON * square_product) {
+        bgc_fp64_turn3_reset(turn);
+        if (dot_product < 0.0) {
+            return BGC_ERROR_TURN3_VECTORS_OPPOSITE;
+        }
+
+        return BGC_SUCCESS;
+    }
+
+    const double axis_modulus = sqrt(axis_square_modulus);
+    const double trigonometry_fix = sqrt(1.0 / square_product);
+
+    const double angle = 0.5 * atan2(axis_modulus * trigonometry_fix, dot_product * trigonometry_fix);
+
+    const double vector_multiplier = sin(angle) / axis_modulus;
+
+    bgc_fp64_turn3_set_raw_values(turn, cos(angle), axis.x1 * vector_multiplier, axis.x2 * vector_multiplier, axis.x3 * vector_multiplier);
+
+    return BGC_SUCCESS;
 }
 
 // ============ Make Orthogonal Pair ============ //
@@ -334,7 +337,7 @@ static inline int _bgc_fp64_turn3_get_orthogonal_pair(BGC_FP64_Vector3* unit_mai
         return _BGC_ERROR_TURN3_EMPTY_BRANCH;
     }
 
-    bgc_fp64_vector3_multiply(unit_main, main, sqrtf(1.0 / main_square_modulus));
+    bgc_fp64_vector3_multiply(unit_main, main, sqrt(1.0 / main_square_modulus));
 
     bgc_fp64_vector3_add_scaled(unit_branch, branch, unit_main, -bgc_fp64_vector3_get_dot_product(branch, unit_main));
 
diff --git a/basic-geometry/turn3.h b/basic-geometry/turn3.h
index f4ea1ce..8a48605 100644
--- a/basic-geometry/turn3.h
+++ b/basic-geometry/turn3.h
@@ -9,32 +9,23 @@
 #include "matrix3x3.h"
 #include "quaternion.h"
 
-#define BGC_SOME_TURN 1
-#define BGC_ZERO_TURN 0
-#define BGC_OPPOSITE -1
+#define BGC_ERROR_TURN3_FIRST_VECTOR_ZERO -3010
+#define BGC_ERROR_TURN3_SECOND_VECTOR_ZERO -3011
+#define BGC_ERROR_TURN3_VECTORS_OPPOSITE -3012
 
-#define _BGC_ERROR_TURN3_FIRST_PAIR 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 -3020
+#define _BGC_ERROR_TURN3_SECOND_PAIR -3030
+#define _BGC_ERROR_TURN3_EMPTY_MAIN -1
+#define _BGC_ERROR_TURN3_EMPTY_BRANCH -2
+#define _BGC_ERROR_TURN3_PAIR_PARALLEL -3
 
-#define BGC_ERROR_TURN3_FIRST_PAIR_EMPTY_MAIN 3001
-#define BGC_ERROR_TURN3_FIRST_PAIR_EMPTY_BRANCH 3002
-#define BGC_ERROR_TURN3_FIRST_PAIR_PARALLEL 3003
+#define BGC_ERROR_TURN3_FIRST_PAIR_ZERO_MAIN -3021
+#define BGC_ERROR_TURN3_FIRST_PAIR_ZERO_BRANCH -3022
+#define BGC_ERROR_TURN3_FIRST_PAIR_PARALLEL -3023
 
-#define BGC_ERROR_TURN3_SECOND_PAIR_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
-
-#define BGC_ERROR_AUXILIARY_DIRECTION_UNKNOWN -3011
-#define BGC_ERROR_AUXILIARY_VECTOR_IS_ZERO -3012
-
-#define BGC_ERROR_DIRECTIONS_PARALLEL -3021
-#define BGC_ERROR_VECTORS_PARALLEL -3022
+#define BGC_ERROR_TURN3_SECOND_PAIR_ZERO_MAIN -3031
+#define BGC_ERROR_TURN3_SECOND_PAIR_ZERO_BRANCH -3032
+#define BGC_ERROR_TURN3_SECOND_PAIR_PARALLEL -3033
 
 // =================== Types ==================== //
 
@@ -143,9 +134,9 @@ void bgc_fp64_turn3_set_rotation(BGC_FP64_Turn3* turn, const double x1, const do
 
 // ========= Find Direction Difference ========== //
 
-int bgc_fp32_turn3_find_direction_difference(BGC_FP32_Turn3* difference, const BGC_FP32_Vector3* start, const BGC_FP32_Vector3* end);
+int bgc_fp32_turn3_find_direction_difference(BGC_FP32_Turn3* turn, const BGC_FP32_Vector3* start, const BGC_FP32_Vector3* end);
 
-int bgc_fp64_turn3_find_direction_difference(BGC_FP64_Turn3* difference, const BGC_FP64_Vector3* start, const BGC_FP64_Vector3* end);
+int bgc_fp64_turn3_find_direction_difference(BGC_FP64_Turn3* turn, const BGC_FP64_Vector3* start, const BGC_FP64_Vector3* end);
 
 // ======= Find Direction Pair Difference ======= //
 
@@ -401,36 +392,134 @@ void bgc_fp64_turn3_spherically_interpolate(BGC_FP64_Turn3* interpolation, const
 
 inline void bgc_fp32_turn3_get_rotation_matrix(BGC_FP32_Matrix3x3* matrix, const BGC_FP32_Turn3* turn)
 {
-    bgc_fp32_quaternion_get_rotation_matrix(matrix, &turn->_versor);
+    const float s0s0 = turn->_versor.s0 * turn->_versor.s0;
+    const float x1x1 = turn->_versor.x1 * turn->_versor.x1;
+    const float x2x2 = turn->_versor.x2 * turn->_versor.x2;
+    const float x3x3 = turn->_versor.x3 * turn->_versor.x3;
+
+    const float s0x1 = turn->_versor.s0 * turn->_versor.x1;
+    const float s0x2 = turn->_versor.s0 * turn->_versor.x2;
+    const float s0x3 = turn->_versor.s0 * turn->_versor.x3;
+
+    const float x1x2 = turn->_versor.x1 * turn->_versor.x2;
+    const float x1x3 = turn->_versor.x1 * turn->_versor.x3;
+
+    const float x2x3 = turn->_versor.x2 * turn->_versor.x3;
+
+    matrix->r1c1 = ((s0s0 + x1x1) - (x2x2 + x3x3));
+    matrix->r2c2 = ((s0s0 + x2x2) - (x1x1 + x3x3));
+    matrix->r3c3 = ((s0s0 + x3x3) - (x1x1 + x2x2));
+
+    matrix->r1c2 = 2.0f * (x1x2 - s0x3);
+    matrix->r2c3 = 2.0f * (x2x3 - s0x1);
+    matrix->r3c1 = 2.0f * (x1x3 - s0x2);
+
+    matrix->r2c1 = 2.0f * (x1x2 + s0x3);
+    matrix->r3c2 = 2.0f * (x2x3 + s0x1);
+    matrix->r1c3 = 2.0f * (x1x3 + s0x2);
 }
 
 inline void bgc_fp64_turn3_get_rotation_matrix(BGC_FP64_Matrix3x3* matrix, const BGC_FP64_Turn3* turn)
 {
-    bgc_fp64_quaternion_get_rotation_matrix(matrix, &turn->_versor);
+    const double s0s0 = turn->_versor.s0 * turn->_versor.s0;
+    const double x1x1 = turn->_versor.x1 * turn->_versor.x1;
+    const double x2x2 = turn->_versor.x2 * turn->_versor.x2;
+    const double x3x3 = turn->_versor.x3 * turn->_versor.x3;
+
+    const double s0x1 = turn->_versor.s0 * turn->_versor.x1;
+    const double s0x2 = turn->_versor.s0 * turn->_versor.x2;
+    const double s0x3 = turn->_versor.s0 * turn->_versor.x3;
+
+    const double x1x2 = turn->_versor.x1 * turn->_versor.x2;
+    const double x1x3 = turn->_versor.x1 * turn->_versor.x3;
+
+    const double x2x3 = turn->_versor.x2 * turn->_versor.x3;
+
+    matrix->r1c1 = ((s0s0 + x1x1) - (x2x2 + x3x3));
+    matrix->r2c2 = ((s0s0 + x2x2) - (x1x1 + x3x3));
+    matrix->r3c3 = ((s0s0 + x3x3) - (x1x1 + x2x2));
+
+    matrix->r1c2 = 2.0 * (x1x2 - s0x3);
+    matrix->r2c3 = 2.0 * (x2x3 - s0x1);
+    matrix->r3c1 = 2.0 * (x1x3 - s0x2);
+
+    matrix->r2c1 = 2.0 * (x1x2 + s0x3);
+    matrix->r3c2 = 2.0 * (x2x3 + s0x1);
+    matrix->r1c3 = 2.0 * (x1x3 + s0x2);
 }
 
 // ============= Get Reverse Matrix ============= //
 
 inline void bgc_fp32_turn3_get_reverse_matrix(BGC_FP32_Matrix3x3* matrix, const BGC_FP32_Turn3* turn)
 {
-    bgc_fp32_quaternion_get_reverse_matrix(matrix, &turn->_versor);
+    const float s0s0 = turn->_versor.s0 * turn->_versor.s0;
+    const float x1x1 = turn->_versor.x1 * turn->_versor.x1;
+    const float x2x2 = turn->_versor.x2 * turn->_versor.x2;
+    const float x3x3 = turn->_versor.x3 * turn->_versor.x3;
+
+    const float s0x1 = turn->_versor.s0 * turn->_versor.x1;
+    const float s0x2 = turn->_versor.s0 * turn->_versor.x2;
+    const float s0x3 = turn->_versor.s0 * turn->_versor.x3;
+
+    const float x1x2 = turn->_versor.x1 * turn->_versor.x2;
+    const float x1x3 = turn->_versor.x1 * turn->_versor.x3;
+
+    const float x2x3 = turn->_versor.x2 * turn->_versor.x3;
+
+    matrix->r1c1 = ((s0s0 + x1x1) - (x2x2 + x3x3));
+    matrix->r2c2 = ((s0s0 + x2x2) - (x1x1 + x3x3));
+    matrix->r3c3 = ((s0s0 + x3x3) - (x1x1 + x2x2));
+
+    matrix->r1c2 = 2.0f * (x1x2 + s0x3);
+    matrix->r2c3 = 2.0f * (x2x3 + s0x1);
+    matrix->r3c1 = 2.0f * (x1x3 + s0x2);
+
+    matrix->r2c1 = 2.0f * (x1x2 - s0x3);
+    matrix->r3c2 = 2.0f * (x2x3 - s0x1);
+    matrix->r1c3 = 2.0f * (x1x3 - s0x2);
 }
 
 inline void bgc_fp64_turn3_get_reverse_matrix(BGC_FP64_Matrix3x3* matrix, const BGC_FP64_Turn3* turn)
 {
-    bgc_fp64_quaternion_get_reverse_matrix(matrix, &turn->_versor);
+    const double s0s0 = turn->_versor.s0 * turn->_versor.s0;
+    const double x1x1 = turn->_versor.x1 * turn->_versor.x1;
+    const double x2x2 = turn->_versor.x2 * turn->_versor.x2;
+    const double x3x3 = turn->_versor.x3 * turn->_versor.x3;
+
+    const double s0x1 = turn->_versor.s0 * turn->_versor.x1;
+    const double s0x2 = turn->_versor.s0 * turn->_versor.x2;
+    const double s0x3 = turn->_versor.s0 * turn->_versor.x3;
+
+    const double x1x2 = turn->_versor.x1 * turn->_versor.x2;
+    const double x1x3 = turn->_versor.x1 * turn->_versor.x3;
+
+    const double x2x3 = turn->_versor.x2 * turn->_versor.x3;
+
+    matrix->r1c1 = ((s0s0 + x1x1) - (x2x2 + x3x3));
+    matrix->r2c2 = ((s0s0 + x2x2) - (x1x1 + x3x3));
+    matrix->r3c3 = ((s0s0 + x3x3) - (x1x1 + x2x2));
+
+    matrix->r1c2 = 2.0 * (x1x2 + s0x3);
+    matrix->r2c3 = 2.0 * (x2x3 + s0x1);
+    matrix->r3c1 = 2.0 * (x1x3 + s0x2);
+
+    matrix->r2c1 = 2.0 * (x1x2 - s0x3);
+    matrix->r3c2 = 2.0 * (x2x3 - s0x1);
+    matrix->r1c3 = 2.0 * (x1x3 - s0x2);
 }
 
 // ============= Get Both Matrixes ============== //
 
 inline void bgc_fp32_turn3_get_both_matrices(BGC_FP32_Matrix3x3* rotation, BGC_FP32_Matrix3x3* reverse, const BGC_FP32_Turn3* turn)
 {
-    bgc_fp32_quaternion_get_both_matrices(rotation, reverse, &turn->_versor);
+    bgc_fp32_turn3_get_reverse_matrix(reverse, turn);
+    bgc_fp32_matrix3x3_get_transposed(rotation, reverse);
 }
 
 inline void bgc_fp64_turn3_get_both_matrices(BGC_FP64_Matrix3x3* rotation, BGC_FP64_Matrix3x3* reverse, const BGC_FP64_Turn3* turn)
 {
-    bgc_fp64_quaternion_get_both_matrices(rotation, reverse, &turn->_versor);
+    bgc_fp64_turn3_get_reverse_matrix(reverse, turn);
+    bgc_fp64_matrix3x3_get_transposed(rotation, reverse);
 }
 
 // ================ Turn Vector ================= //