Quaternion slerp overload which interpolates with extra spins

glm/ext/quaternion_common.hpp

@@ -76,6 +76,21 @@ namespace glm
 	template<typename T, qualifier Q>
 	GLM_FUNC_DECL qua<T, Q> slerp(qua<T, Q> const& x, qua<T, Q> const& y, T a);
 
+    /// Spherical linear interpolation of two quaternions with multiple spins over rotation axis.
+    /// The interpolation always take the short path when the spin count is positive and long path
+    /// when count is negative. Rotation is performed at constant speed.
+    ///
+    /// @param x A quaternion
+    /// @param y A quaternion
+    /// @param a Interpolation factor. The interpolation is defined beyond the range [0, 1].
+    /// @param k Additional spin count. If Value is negative interpolation will be on "long" path.
+    ///
+    /// @tparam T A floating-point scalar type
+    /// @tparam S An integer scalar type
+    /// @tparam Q A value from qualifier enum
+    template<typename T, typename S, qualifier Q>
+    GLM_FUNC_DECL qua<T, Q> slerp(qua<T, Q> const& x, qua<T, Q> const& y, T a, S k);
+
 	/// Returns the q conjugate.
 	///
 	/// @tparam T A floating-point scalar type

glm/ext/quaternion_common.inl

@@ -72,6 +72,43 @@ namespace glm
 		}
 	}
 
+    template<typename T, typename S, qualifier Q>
+    GLM_FUNC_QUALIFIER qua<T, Q> slerp(qua<T, Q> const& x, qua<T, Q> const& y, T a, S k)
+    {
+        GLM_STATIC_ASSERT(std::numeric_limits<T>::is_iec559, "'slerp' only accept floating-point inputs");
+        GLM_STATIC_ASSERT(std::numeric_limits<S>::is_integer, "'slerp' only accept integer for spin count");
+
+        qua<T, Q> z = y;
+
+        T cosTheta = dot(x, y);
+
+        // If cosTheta < 0, the interpolation will take the long way around the sphere.
+        // To fix this, one quat must be negated.
+        if (cosTheta < static_cast<T>(0))
+        {
+            z = -y;
+            cosTheta = -cosTheta;
+        }
+
+        // Perform a linear interpolation when cosTheta is close to 1 to avoid side effect of sin(angle) becoming a zero denominator
+        if (cosTheta > static_cast<T>(1) - epsilon<T>())
+        {
+            // Linear interpolation
+            return qua<T, Q>(
+                mix(x.w, z.w, a),
+                mix(x.x, z.x, a),
+                mix(x.y, z.y, a),
+                mix(x.z, z.z, a));
+        }
+        else
+        {
+            // Graphics Gems III, page 96
+            T angle = acos(cosTheta);
+            T phi = angle + k * glm::pi<T>();
+            return (sin(angle - a * phi)* x + sin(a * phi) * z) / sin(angle);
+        }
+    }
+
 	template<typename T, qualifier Q>
 	GLM_FUNC_QUALIFIER qua<T, Q> conjugate(qua<T, Q> const& q)
 	{

test/gtc/gtc_quaternion.cpp

@@ -194,6 +194,84 @@ int test_quat_slerp()
 	return Error;
 }
 
+int test_quat_slerp_spins()
+{
+    int Error = 0;
+
+    float const Epsilon = 0.0001f;//glm::epsilon<float>();
+
+    float sqrt2 = std::sqrt(2.0f) / 2.0f;
+    glm::quat id(static_cast<float>(1), static_cast<float>(0), static_cast<float>(0), static_cast<float>(0));
+    glm::quat Y90rot(sqrt2, 0.0f, sqrt2, 0.0f);
+    glm::quat Y180rot(0.0f, 0.0f, 1.0f, 0.0f);
+
+    // Testing a == 0, k == 1
+    // Must be id
+    glm::quat id2 = glm::slerp(id, id, 1.0f, 1);
+    Error += glm::all(glm::equal(id, id2, Epsilon)) ? 0 : 1;
+
+    // Testing a == 1, k == 2
+    // Must be id
+    glm::quat id3 = glm::slerp(id, id, 1.0f, 2);
+    Error += glm::all(glm::equal(id, id3, Epsilon)) ? 0 : 1;
+
+    // Testing a == 1, k == 1
+    // Must be 90� rotation on Y : 0 0.7 0 0.7
+    // Negative quaternion is representing same orientation
+    glm::quat Y90rot2 = glm::slerp(id, Y90rot, 1.0f, 1);
+    Error += glm::all(glm::equal(Y90rot, -Y90rot2, Epsilon)) ? 0 : 1;
+
+    // Testing a == 1, k == 2
+    // Must be id
+    glm::quat Y90rot3 = glm::slerp(id, Y90rot, 8.0f / 9.0f, 2);
+    Error += glm::all(glm::equal(id, Y90rot3, Epsilon)) ? 0 : 1;
+
+    // Testing a == 1, k == 1
+    // Must be 90� rotation on Y : 0 0.7 0 0.7
+    glm::quat Y90rot4 = glm::slerp(id, Y90rot, 0.2f, 1);
+    Error += glm::all(glm::equal(Y90rot, Y90rot4, Epsilon)) ? 0 : 1;
+
+    // Testing reverse case
+    // Must be 45� rotation on Y : 0 0.38 0 0.92
+    // Negative quaternion is representing same orientation
+    glm::quat Ym45rot2 = glm::slerp(Y90rot, id, 0.9f, 1);
+    glm::quat Ym45rot3 = glm::slerp(Y90rot, id, 0.5f);
+    Error += glm::all(glm::equal(-Ym45rot2, Ym45rot3, Epsilon)) ? 0 : 1;
+
+    // Testing against full circle around the sphere instead of shortest path
+    // Must be 45� rotation on Y
+    // certainly not a 135� rotation
+    glm::quat Y45rot3 = glm::slerp(id, -Y90rot, 0.5f, 0);
+    float Y45angle3 = glm::angle(Y45rot3);
+    Error += glm::equal(Y45angle3, glm::pi<float>() * 0.25f, Epsilon) ? 0 : 1;
+    Error += glm::all(glm::equal(Ym45rot3, Y45rot3, Epsilon)) ? 0 : 1;
+
+    // Same, but inverted
+    // Must also be 45� rotation on Y :  0 0.38 0 0.92
+    // -0 -0.38 -0 -0.92 is ok too
+    glm::quat Y45rot4 = glm::slerp(-Y90rot, id, 0.5f, 0);
+    Error += glm::all(glm::equal(Ym45rot2, Y45rot4, Epsilon)) ? 0 : 1;
+
+    // Testing q1 = q2 k == 2
+    // Must be 90� rotation on Y : 0 0.7 0 0.7
+    glm::quat Y90rot5 = glm::slerp(Y90rot, Y90rot, 0.5f, 2);
+    Error += glm::all(glm::equal(Y90rot, Y90rot5, Epsilon)) ? 0 : 1;
+
+    // Testing 180� rotation
+    // Must be 90� rotation on almost any axis that is on the XZ plane
+    glm::quat XZ90rot = glm::slerp(id, -Y90rot, 0.5f, 1);
+    float XZ90angle = glm::angle(XZ90rot); // Must be PI/4 = 0.78;
+    Error += glm::equal(XZ90angle, glm::pi<float>() * 1.25f, Epsilon) ? 0 : 1;
+
+    // Testing rotation over long arc
+    // Distance from id to 90� is 270�, so 2/3 of it should be 180�
+    // Negative quaternion is representing same orientation
+    glm::quat Neg90rot = glm::slerp(id, Y90rot, 2.0f / 3.0f, -1);
+    Error += glm::all(glm::equal(Y180rot, -Neg90rot, Epsilon)) ? 0 : 1;
+
+    return Error;
+}
+
 static int test_quat_mul_vec()
 {
 	int Error(0);
@@ -260,6 +338,7 @@ int main()
 	Error += test_quat_normalize();
 	Error += test_quat_euler();
 	Error += test_quat_slerp();
+    Error += test_quat_slerp_spins();
 	Error += test_identity();
 
 	return Error;