Add tests file for Quaternion unit tests, with initial UTs

- Test constructors and quaternion product.
- Add test case for Axis-Angle construction about Y-axis.
- Add test case for xform of i-, j-, & k-unit vectors.
- Add test case for construction from Basis.
- Add test case for xform of arbitrary vector.
- Add stress test case: many Quaternions xform many vectors.
- Make comments consistent with style guide.

2 files changed, 390 insertions, 0 deletions

diff --git a/tests/core/math/test_quaternion.h b/tests/core/math/test_quaternion.h
new file mode 100644
index 0000000000..94eef6c463
--- /dev/null
+++ b/tests/core/math/test_quaternion.h
@@ -0,0 +1,389 @@
+/*************************************************************************/
+/*  test_quaternion.h                                                    */
+/*************************************************************************/
+/*                       This file is part of:                           */
+/*                           GODOT ENGINE                                */
+/*                      https://godotengine.org                          */
+/*************************************************************************/
+/* Copyright (c) 2007-2022 Juan Linietsky, Ariel Manzur.                 */
+/* Copyright (c) 2014-2022 Godot Engine contributors (cf. AUTHORS.md).   */
+/*                                                                       */
+/* Permission is hereby granted, free of charge, to any person obtaining */
+/* a copy of this software and associated documentation files (the       */
+/* "Software"), to deal in the Software without restriction, including   */
+/* without limitation the rights to use, copy, modify, merge, publish,   */
+/* distribute, sublicense, and/or sell copies of the Software, and to    */
+/* permit persons to whom the Software is furnished to do so, subject to */
+/* the following conditions:                                             */
+/*                                                                       */
+/* The above copyright notice and this permission notice shall be        */
+/* included in all copies or substantial portions of the Software.       */
+/*                                                                       */
+/* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,       */
+/* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF    */
+/* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.*/
+/* IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY  */
+/* CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,  */
+/* TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE     */
+/* SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.                */
+/*************************************************************************/
+
+#ifndef TEST_QUATERNION_H
+#define TEST_QUATERNION_H
+
+#include "core/math/math_defs.h"
+#include "core/math/math_funcs.h"
+#include "core/math/quaternion.h"
+#include "core/math/vector3.h"
+
+#include "tests/test_macros.h"
+
+namespace TestQuaternion {
+
+Quaternion quat_euler_yxz_deg(Vector3 angle) {
+	double yaw = Math::deg2rad(angle[1]);
+	double pitch = Math::deg2rad(angle[0]);
+	double roll = Math::deg2rad(angle[2]);
+
+	// Generate YXZ (Z-then-X-then-Y) Quaternion using single-axis Euler
+	// constructor and quaternion product, both tested separately.
+	Quaternion q_y(Vector3(0.0, yaw, 0.0));
+	Quaternion q_p(Vector3(pitch, 0.0, 0.0));
+	Quaternion q_r(Vector3(0.0, 0.0, roll));
+	// Roll-Z is followed by Pitch-X, then Yaw-Y.
+	Quaternion q_yxz = q_y * q_p * q_r;
+
+	return q_yxz;
+}
+
+TEST_CASE("[Quaternion] Default Construct") {
+	Quaternion q;
+
+	CHECK(q[0] == 0.0);
+	CHECK(q[1] == 0.0);
+	CHECK(q[2] == 0.0);
+	CHECK(q[3] == 1.0);
+}
+
+TEST_CASE("[Quaternion] Construct x,y,z,w") {
+	// Values are taken from actual use in another project & are valid (except roundoff error).
+	Quaternion q(0.2391, 0.099, 0.3696, 0.8924);
+
+	CHECK(q[0] == doctest::Approx(0.2391));
+	CHECK(q[1] == doctest::Approx(0.099));
+	CHECK(q[2] == doctest::Approx(0.3696));
+	CHECK(q[3] == doctest::Approx(0.8924));
+}
+
+TEST_CASE("[Quaternion] Construct AxisAngle 1") {
+	// Easy to visualize: 120 deg about X-axis.
+	Quaternion q(Vector3(1.0, 0.0, 0.0), Math::deg2rad(120.0));
+
+	// 0.866 isn't close enough; doctest::Approx doesn't cut much slack!
+	CHECK(q[0] == doctest::Approx(0.866025)); // Sine of half the angle.
+	CHECK(q[1] == doctest::Approx(0.0));
+	CHECK(q[2] == doctest::Approx(0.0));
+	CHECK(q[3] == doctest::Approx(0.5)); // Cosine of half the angle.
+}
+
+TEST_CASE("[Quaternion] Construct AxisAngle 2") {
+	// Easy to visualize: 30 deg about Y-axis.
+	Quaternion q(Vector3(0.0, 1.0, 0.0), Math::deg2rad(30.0));
+
+	CHECK(q[0] == doctest::Approx(0.0));
+	CHECK(q[1] == doctest::Approx(0.258819)); // Sine of half the angle.
+	CHECK(q[2] == doctest::Approx(0.0));
+	CHECK(q[3] == doctest::Approx(0.965926)); // Cosine of half the angle.
+}
+
+TEST_CASE("[Quaternion] Construct AxisAngle 3") {
+	// Easy to visualize: 60 deg about Z-axis.
+	Quaternion q(Vector3(0.0, 0.0, 1.0), Math::deg2rad(60.0));
+
+	CHECK(q[0] == doctest::Approx(0.0));
+	CHECK(q[1] == doctest::Approx(0.0));
+	CHECK(q[2] == doctest::Approx(0.5)); // Sine of half the angle.
+	CHECK(q[3] == doctest::Approx(0.866025)); // Cosine of half the angle.
+}
+
+TEST_CASE("[Quaternion] Construct AxisAngle 4") {
+	// More complex & hard to visualize, so test w/ data from online calculator.
+	Vector3 axis(1.0, 2.0, 0.5);
+	Quaternion q(axis.normalized(), Math::deg2rad(35.0));
+
+	CHECK(q[0] == doctest::Approx(0.131239));
+	CHECK(q[1] == doctest::Approx(0.262478));
+	CHECK(q[2] == doctest::Approx(0.0656194));
+	CHECK(q[3] == doctest::Approx(0.953717));
+}
+
+TEST_CASE("[Quaternion] Construct from Quaternion") {
+	Vector3 axis(1.0, 2.0, 0.5);
+	Quaternion q_src(axis.normalized(), Math::deg2rad(35.0));
+	Quaternion q(q_src);
+
+	CHECK(q[0] == doctest::Approx(0.131239));
+	CHECK(q[1] == doctest::Approx(0.262478));
+	CHECK(q[2] == doctest::Approx(0.0656194));
+	CHECK(q[3] == doctest::Approx(0.953717));
+}
+
+TEST_CASE("[Quaternion] Construct Euler SingleAxis") {
+	double yaw = Math::deg2rad(45.0);
+	double pitch = Math::deg2rad(30.0);
+	double roll = Math::deg2rad(10.0);
+
+	Vector3 euler_y(0.0, yaw, 0.0);
+	Quaternion q_y(euler_y);
+	CHECK(q_y[0] == doctest::Approx(0.0));
+	CHECK(q_y[1] == doctest::Approx(0.382684));
+	CHECK(q_y[2] == doctest::Approx(0.0));
+	CHECK(q_y[3] == doctest::Approx(0.923879));
+
+	Vector3 euler_p(pitch, 0.0, 0.0);
+	Quaternion q_p(euler_p);
+	CHECK(q_p[0] == doctest::Approx(0.258819));
+	CHECK(q_p[1] == doctest::Approx(0.0));
+	CHECK(q_p[2] == doctest::Approx(0.0));
+	CHECK(q_p[3] == doctest::Approx(0.965926));
+
+	Vector3 euler_r(0.0, 0.0, roll);
+	Quaternion q_r(euler_r);
+	CHECK(q_r[0] == doctest::Approx(0.0));
+	CHECK(q_r[1] == doctest::Approx(0.0));
+	CHECK(q_r[2] == doctest::Approx(0.0871558));
+	CHECK(q_r[3] == doctest::Approx(0.996195));
+}
+
+TEST_CASE("[Quaternion] Construct Euler YXZ dynamic axes") {
+	double yaw = Math::deg2rad(45.0);
+	double pitch = Math::deg2rad(30.0);
+	double roll = Math::deg2rad(10.0);
+
+	// Generate YXZ comparision data (Z-then-X-then-Y) using single-axis Euler
+	// constructor and quaternion product, both tested separately.
+	Vector3 euler_y(0.0, yaw, 0.0);
+	Quaternion q_y(euler_y);
+	Vector3 euler_p(pitch, 0.0, 0.0);
+	Quaternion q_p(euler_p);
+	Vector3 euler_r(0.0, 0.0, roll);
+	Quaternion q_r(euler_r);
+
+	// Roll-Z is followed by Pitch-X.
+	Quaternion check_xz = q_p * q_r;
+	// Then Yaw-Y follows both.
+	Quaternion check_yxz = q_y * check_xz;
+
+	// Test construction from YXZ Euler angles.
+	Vector3 euler_yxz(pitch, yaw, roll);
+	Quaternion q(euler_yxz);
+	CHECK(q[0] == doctest::Approx(check_yxz[0]));
+	CHECK(q[1] == doctest::Approx(check_yxz[1]));
+	CHECK(q[2] == doctest::Approx(check_yxz[2]));
+	CHECK(q[3] == doctest::Approx(check_yxz[3]));
+
+	// Sneak in a test of is_equal_approx.
+	CHECK(q.is_equal_approx(check_yxz));
+}
+
+TEST_CASE("[Quaternion] Construct Basis Euler") {
+	double yaw = Math::deg2rad(45.0);
+	double pitch = Math::deg2rad(30.0);
+	double roll = Math::deg2rad(10.0);
+	Vector3 euler_yxz(pitch, yaw, roll);
+	Quaternion q_yxz(euler_yxz);
+	Basis basis_axes(euler_yxz);
+	Quaternion q(basis_axes);
+	CHECK(q.is_equal_approx(q_yxz));
+}
+
+TEST_CASE("[Quaternion] Construct Basis Axes") {
+	// Arbitrary Euler angles.
+	Vector3 euler_yxz(Math::deg2rad(31.41), Math::deg2rad(-49.16), Math::deg2rad(12.34));
+	// Basis vectors from online calculation of rotation matrix.
+	Vector3 i_unit(0.5545787, 0.1823950, 0.8118957);
+	Vector3 j_unit(-0.5249245, 0.8337420, 0.1712555);
+	Vector3 k_unit(-0.6456754, -0.5211586, 0.5581192);
+	// Quaternion from online calculation.
+	Quaternion q_calc(0.2016913, -0.4245716, 0.206033, 0.8582598);
+	// Quaternion from local calculation.
+	Quaternion q_local = quat_euler_yxz_deg(Vector3(31.41, -49.16, 12.34));
+	// Quaternion from Euler angles constructor.
+	Quaternion q_euler(euler_yxz);
+	CHECK(q_calc.is_equal_approx(q_local));
+	CHECK(q_local.is_equal_approx(q_euler));
+
+	// Calculate Basis and construct Quaternion.
+	// When this is written, C++ Basis class does not construct from basis vectors.
+	// This is by design, but may be subject to change.
+	// Workaround by constructing Basis from Euler angles.
+	// basis_axes = Basis(i_unit, j_unit, k_unit);
+	Basis basis_axes(euler_yxz);
+	Quaternion q(basis_axes);
+
+	CHECK(basis_axes.get_column(0).is_equal_approx(i_unit));
+	CHECK(basis_axes.get_column(1).is_equal_approx(j_unit));
+	CHECK(basis_axes.get_column(2).is_equal_approx(k_unit));
+
+	CHECK(q.is_equal_approx(q_calc));
+	CHECK_FALSE(q.inverse().is_equal_approx(q_calc));
+	CHECK(q.is_equal_approx(q_local));
+	CHECK(q.is_equal_approx(q_euler));
+	CHECK(q[0] == doctest::Approx(0.2016913));
+	CHECK(q[1] == doctest::Approx(-0.4245716));
+	CHECK(q[2] == doctest::Approx(0.206033));
+	CHECK(q[3] == doctest::Approx(0.8582598));
+}
+
+TEST_CASE("[Quaternion] Product (book)") {
+	// Example from "Quaternions and Rotation Sequences" by Jack Kuipers, p. 108.
+	Quaternion p(1.0, -2.0, 1.0, 3.0);
+	Quaternion q(-1.0, 2.0, 3.0, 2.0);
+
+	Quaternion pq = p * q;
+	CHECK(pq[0] == doctest::Approx(-9.0));
+	CHECK(pq[1] == doctest::Approx(-2.0));
+	CHECK(pq[2] == doctest::Approx(11.0));
+	CHECK(pq[3] == doctest::Approx(8.0));
+}
+
+TEST_CASE("[Quaternion] Product") {
+	double yaw = Math::deg2rad(45.0);
+	double pitch = Math::deg2rad(30.0);
+	double roll = Math::deg2rad(10.0);
+
+	Vector3 euler_y(0.0, yaw, 0.0);
+	Quaternion q_y(euler_y);
+	CHECK(q_y[0] == doctest::Approx(0.0));
+	CHECK(q_y[1] == doctest::Approx(0.382684));
+	CHECK(q_y[2] == doctest::Approx(0.0));
+	CHECK(q_y[3] == doctest::Approx(0.923879));
+
+	Vector3 euler_p(pitch, 0.0, 0.0);
+	Quaternion q_p(euler_p);
+	CHECK(q_p[0] == doctest::Approx(0.258819));
+	CHECK(q_p[1] == doctest::Approx(0.0));
+	CHECK(q_p[2] == doctest::Approx(0.0));
+	CHECK(q_p[3] == doctest::Approx(0.965926));
+
+	Vector3 euler_r(0.0, 0.0, roll);
+	Quaternion q_r(euler_r);
+	CHECK(q_r[0] == doctest::Approx(0.0));
+	CHECK(q_r[1] == doctest::Approx(0.0));
+	CHECK(q_r[2] == doctest::Approx(0.0871558));
+	CHECK(q_r[3] == doctest::Approx(0.996195));
+
+	// Test ZYX dynamic-axes since test data is available online.
+	// Rotate first about X axis, then new Y axis, then new Z axis.
+	// (Godot uses YXZ Yaw-Pitch-Roll order).
+	Quaternion q_yp = q_y * q_p;
+	CHECK(q_yp[0] == doctest::Approx(0.239118));
+	CHECK(q_yp[1] == doctest::Approx(0.369644));
+	CHECK(q_yp[2] == doctest::Approx(-0.099046));
+	CHECK(q_yp[3] == doctest::Approx(0.892399));
+
+	Quaternion q_ryp = q_r * q_yp;
+	CHECK(q_ryp[0] == doctest::Approx(0.205991));
+	CHECK(q_ryp[1] == doctest::Approx(0.389078));
+	CHECK(q_ryp[2] == doctest::Approx(-0.0208912));
+	CHECK(q_ryp[3] == doctest::Approx(0.897636));
+}
+
+TEST_CASE("[Quaternion] xform unit vectors") {
+	// Easy to visualize: 120 deg about X-axis.
+	// Transform the i, j, & k unit vectors.
+	Quaternion q(Vector3(1.0, 0.0, 0.0), Math::deg2rad(120.0));
+	Vector3 i_t = q.xform(Vector3(1.0, 0.0, 0.0));
+	Vector3 j_t = q.xform(Vector3(0.0, 1.0, 0.0));
+	Vector3 k_t = q.xform(Vector3(0.0, 0.0, 1.0));
+	//
+	CHECK(i_t.is_equal_approx(Vector3(1.0, 0.0, 0.0)));
+	CHECK(j_t.is_equal_approx(Vector3(0.0, -0.5, 0.866025)));
+	CHECK(k_t.is_equal_approx(Vector3(0.0, -0.866025, -0.5)));
+	CHECK(i_t.length_squared() == doctest::Approx(1.0));
+	CHECK(j_t.length_squared() == doctest::Approx(1.0));
+	CHECK(k_t.length_squared() == doctest::Approx(1.0));
+
+	// Easy to visualize: 30 deg about Y-axis.
+	q = Quaternion(Vector3(0.0, 1.0, 0.0), Math::deg2rad(30.0));
+	i_t = q.xform(Vector3(1.0, 0.0, 0.0));
+	j_t = q.xform(Vector3(0.0, 1.0, 0.0));
+	k_t = q.xform(Vector3(0.0, 0.0, 1.0));
+	//
+	CHECK(i_t.is_equal_approx(Vector3(0.866025, 0.0, -0.5)));
+	CHECK(j_t.is_equal_approx(Vector3(0.0, 1.0, 0.0)));
+	CHECK(k_t.is_equal_approx(Vector3(0.5, 0.0, 0.866025)));
+	CHECK(i_t.length_squared() == doctest::Approx(1.0));
+	CHECK(j_t.length_squared() == doctest::Approx(1.0));
+	CHECK(k_t.length_squared() == doctest::Approx(1.0));
+
+	// Easy to visualize: 60 deg about Z-axis.
+	q = Quaternion(Vector3(0.0, 0.0, 1.0), Math::deg2rad(60.0));
+	i_t = q.xform(Vector3(1.0, 0.0, 0.0));
+	j_t = q.xform(Vector3(0.0, 1.0, 0.0));
+	k_t = q.xform(Vector3(0.0, 0.0, 1.0));
+	//
+	CHECK(i_t.is_equal_approx(Vector3(0.5, 0.866025, 0.0)));
+	CHECK(j_t.is_equal_approx(Vector3(-0.866025, 0.5, 0.0)));
+	CHECK(k_t.is_equal_approx(Vector3(0.0, 0.0, 1.0)));
+	CHECK(i_t.length_squared() == doctest::Approx(1.0));
+	CHECK(j_t.length_squared() == doctest::Approx(1.0));
+	CHECK(k_t.length_squared() == doctest::Approx(1.0));
+}
+
+TEST_CASE("[Quaternion] xform vector") {
+	// Arbitrary quaternion rotates an arbitrary vector.
+	Vector3 euler_yzx(Math::deg2rad(31.41), Math::deg2rad(-49.16), Math::deg2rad(12.34));
+	Basis basis_axes(euler_yzx);
+	Quaternion q(basis_axes);
+
+	Vector3 v_arb(3.0, 4.0, 5.0);
+	Vector3 v_rot = q.xform(v_arb);
+	Vector3 v_compare = basis_axes.xform(v_arb);
+
+	CHECK(v_rot.length_squared() == doctest::Approx(v_arb.length_squared()));
+	CHECK(v_rot.is_equal_approx(v_compare));
+}
+
+// Test vector xform for a single combination of Quaternion and Vector.
+void test_quat_vec_rotate(Vector3 euler_yzx, Vector3 v_in) {
+	Basis basis_axes(euler_yzx);
+	Quaternion q(basis_axes);
+
+	Vector3 v_rot = q.xform(v_in);
+	Vector3 v_compare = basis_axes.xform(v_in);
+
+	CHECK(v_rot.length_squared() == doctest::Approx(v_in.length_squared()));
+	CHECK(v_rot.is_equal_approx(v_compare));
+}
+
+TEST_CASE("[Stress][Quaternion] Many vector xforms") {
+	// Many arbitrary quaternions rotate many arbitrary vectors.
+	// For each trial, check that rotation by Quaternion yields same result as
+	// rotation by Basis.
+	const int STEPS = 100; // Number of test steps in each dimension
+	const double delta = 2.0 * Math_PI / STEPS; // Angle increment per step
+	const double delta_vec = 20.0 / STEPS; // Vector increment per step
+	Vector3 vec_arb(1.0, 1.0, 1.0);
+	double x_angle = -Math_PI;
+	double y_angle = -Math_PI;
+	double z_angle = -Math_PI;
+	for (double i = 0; i < STEPS; ++i) {
+		vec_arb[0] = -10.0 + i * delta_vec;
+		x_angle = i * delta - Math_PI;
+		for (double j = 0; j < STEPS; ++j) {
+			vec_arb[1] = -10.0 + j * delta_vec;
+			y_angle = j * delta - Math_PI;
+			for (double k = 0; k < STEPS; ++k) {
+				vec_arb[2] = -10.0 + k * delta_vec;
+				z_angle = k * delta - Math_PI;
+				Vector3 euler_yzx(x_angle, y_angle, z_angle);
+				test_quat_vec_rotate(euler_yzx, vec_arb);
+			}
+		}
+	}
+}
+
+} // namespace TestQuaternion
+
+#endif // TEST_QUATERNION_H
diff --git a/tests/test_main.cpp b/tests/test_main.cpp
index fca484a6b3..8a9abd0f33 100644
--- a/tests/test_main.cpp
+++ b/tests/test_main.cpp
@@ -48,6 +48,7 @@
 #include "tests/core/math/test_geometry_2d.h"
 #include "tests/core/math/test_geometry_3d.h"
 #include "tests/core/math/test_plane.h"
+#include "tests/core/math/test_quaternion.h"
 #include "tests/core/math/test_random_number_generator.h"
 #include "tests/core/math/test_rect2.h"
 #include "tests/core/math/test_rect2i.h"