Refactor Curve3D::_bake() method

The main change is to caculate tangent directly from bezier curve, without going
through discretized polyline, avoiding pitfalls of discretization.

Other changes are:
1. Add an bezier_derivative() method for Vector3, Vector2, and Math;
2. Add an tesselate_even_length() method to Curve3D, which tesselate bezier curve to even length segments adaptively;
3. Cache the tangent vectors in baked_tangent_vector_cache;

3 files changed, 46 insertions, 0 deletions

diff --git a/core/math/math_funcs.h b/core/math/math_funcs.h
index 0af529ad98..8dff8e6e7e 100644
--- a/core/math/math_funcs.h
+++ b/core/math/math_funcs.h
@@ -364,6 +364,26 @@ public:
 		return p_start * omt3 + p_control_1 * omt2 * p_t * 3.0f + p_control_2 * omt * t2 * 3.0f + p_end * t3;
 	}
 
+	static _ALWAYS_INLINE_ double bezier_derivative(double p_start, double p_control_1, double p_control_2, double p_end, double p_t) {
+		/* Formula from Wikipedia article on Bezier curves. */
+		double omt = (1.0 - p_t);
+		double omt2 = omt * omt;
+		double t2 = p_t * p_t;
+
+		double d = (p_control_1 - p_start) * 3.0 * omt2 + (p_control_2 - p_control_1) * 6.0 * omt * p_t + (p_end - p_control_2) * 3.0 * t2;
+		return d;
+	}
+
+	static _ALWAYS_INLINE_ float bezier_derivative(float p_start, float p_control_1, float p_control_2, float p_end, float p_t) {
+		/* Formula from Wikipedia article on Bezier curves. */
+		float omt = (1.0f - p_t);
+		float omt2 = omt * omt;
+		float t2 = p_t * p_t;
+
+		float d = (p_control_1 - p_start) * 3.0f * omt2 + (p_control_2 - p_control_1) * 6.0f * omt * p_t + (p_end - p_control_2) * 3.0f * t2;
+		return d;
+	}
+
 	static _ALWAYS_INLINE_ double lerp_angle(double p_from, double p_to, double p_weight) {
 		double difference = fmod(p_to - p_from, Math_TAU);
 		double distance = fmod(2.0 * difference, Math_TAU) - difference;
diff --git a/core/math/vector2.h b/core/math/vector2.h
index 5775d8e735..1266561a81 100644
--- a/core/math/vector2.h
+++ b/core/math/vector2.h
@@ -112,6 +112,7 @@ struct _NO_DISCARD_ Vector2 {
 	_FORCE_INLINE_ Vector2 cubic_interpolate(const Vector2 &p_b, const Vector2 &p_pre_a, const Vector2 &p_post_b, const real_t p_weight) const;
 	_FORCE_INLINE_ Vector2 cubic_interpolate_in_time(const Vector2 &p_b, const Vector2 &p_pre_a, const Vector2 &p_post_b, const real_t p_weight, const real_t &p_b_t, const real_t &p_pre_a_t, const real_t &p_post_b_t) const;
 	_FORCE_INLINE_ Vector2 bezier_interpolate(const Vector2 &p_control_1, const Vector2 &p_control_2, const Vector2 &p_end, const real_t p_t) const;
+	_FORCE_INLINE_ Vector2 bezier_derivative(const Vector2 &p_control_1, const Vector2 &p_control_2, const Vector2 &p_end, const real_t p_t) const;
 
 	Vector2 move_toward(const Vector2 &p_to, const real_t p_delta) const;
 
@@ -289,6 +290,18 @@ Vector2 Vector2::bezier_interpolate(const Vector2 &p_control_1, const Vector2 &p
 	return res * omt3 + p_control_1 * omt2 * p_t * 3.0 + p_control_2 * omt * t2 * 3.0 + p_end * t3;
 }
 
+Vector2 Vector2::bezier_derivative(const Vector2 &p_control_1, const Vector2 &p_control_2, const Vector2 &p_end, const real_t p_t) const {
+	Vector2 res = *this;
+
+	/* Formula from Wikipedia article on Bezier curves. */
+	real_t omt = (1.0 - p_t);
+	real_t omt2 = omt * omt;
+	real_t t2 = p_t * p_t;
+
+	Vector2 d = (p_control_1 - res) * 3.0 * omt2 + (p_control_2 - p_control_1) * 6.0 * omt * p_t + (p_end - p_control_2) * 3.0 * t2;
+	return d;
+}
+
 Vector2 Vector2::direction_to(const Vector2 &p_to) const {
 	Vector2 ret(p_to.x - x, p_to.y - y);
 	ret.normalize();
diff --git a/core/math/vector3.h b/core/math/vector3.h
index 19771eb312..f5fe76a92c 100644
--- a/core/math/vector3.h
+++ b/core/math/vector3.h
@@ -100,6 +100,7 @@ struct _NO_DISCARD_ Vector3 {
 	_FORCE_INLINE_ Vector3 cubic_interpolate(const Vector3 &p_b, const Vector3 &p_pre_a, const Vector3 &p_post_b, const real_t p_weight) const;
 	_FORCE_INLINE_ Vector3 cubic_interpolate_in_time(const Vector3 &p_b, const Vector3 &p_pre_a, const Vector3 &p_post_b, const real_t p_weight, const real_t &p_b_t, const real_t &p_pre_a_t, const real_t &p_post_b_t) const;
 	_FORCE_INLINE_ Vector3 bezier_interpolate(const Vector3 &p_control_1, const Vector3 &p_control_2, const Vector3 &p_end, const real_t p_t) const;
+	_FORCE_INLINE_ Vector3 bezier_derivative(const Vector3 &p_control_1, const Vector3 &p_control_2, const Vector3 &p_end, const real_t p_t) const;
 
 	Vector3 move_toward(const Vector3 &p_to, const real_t p_delta) const;
 
@@ -265,6 +266,18 @@ Vector3 Vector3::bezier_interpolate(const Vector3 &p_control_1, const Vector3 &p
 	return res * omt3 + p_control_1 * omt2 * p_t * 3.0 + p_control_2 * omt * t2 * 3.0 + p_end * t3;
 }
 
+Vector3 Vector3::bezier_derivative(const Vector3 &p_control_1, const Vector3 &p_control_2, const Vector3 &p_end, const real_t p_t) const {
+	Vector3 res = *this;
+
+	/* Formula from Wikipedia article on Bezier curves. */
+	real_t omt = (1.0 - p_t);
+	real_t omt2 = omt * omt;
+	real_t t2 = p_t * p_t;
+
+	Vector3 d = (p_control_1 - res) * 3.0 * omt2 + (p_control_2 - p_control_1) * 6.0 * omt * p_t + (p_end - p_control_2) * 3.0 * t2;
+	return d;
+}
+
 real_t Vector3::distance_to(const Vector3 &p_to) const {
 	return (p_to - *this).length();
 }