quaternion: update look, from_to, rotate

quaternion.cpp

@@ -34,7 +34,7 @@ template<> const quaternion<4, double> quaternion<4, double>::IDENTITY = { 1, 0,
 ///////////////////////////////////////////////////////////////////////////////
 template <size_t S, typename T>
 quaternion<S,T>
-quaternion<S,T>::rotation (const T radians, const vector<3,T> axis)
+quaternion<S,T>::angle_axis (const T radians, const vector<3,T> axis)
 {
     CHECK (is_normalised (axis));
 
@@ -50,19 +50,56 @@ quaternion<S,T>::rotation (const T radians, const vector<3,T> axis)
 //-----------------------------------------------------------------------------
 template <size_t S, typename T>
 quaternion<S,T>
-quaternion<S,T>::rotation (const vector<3,T> src, const vector<3,T> dst)
+quaternion<S,T>::from_euler (vector<3,T> angles)
 {
-    auto v = util::cross (src, dst);
+    auto half = angles / 2;
+
+    auto c = cos (half);
+    auto s = sin (half);
 
     return {
-        std::acos (dot (src, dst)),
+        c.x * c.y * c.z - s.x * s.y * s.z,
-        v.x,
+        s.x * c.y * c.z + c.x * s.y * s.z,
-        v.y,
+        c.x * s.y * c.z - s.x * c.y * s.z,
-        v.z
+        c.x * c.y * s.z + s.x * s.y * c.z,
     };
 }
 
 
+///////////////////////////////////////////////////////////////////////////////
+// vector-to-vector rotation algorithm from:
+//     http://lolengine.net/blog/2014/02/24/quaternion-from-two-vectors-final
+template <size_t S, typename T>
+quaternion<S,T>
+quaternion<S,T>::from_to (const vector<3,T> u, const vector<3,T> v)
+{
+    CHECK (is_normalised (u));
+    CHECK (is_normalised (v));
+
+    auto norm_u_norm_v = std::sqrt(dot(u, u) * dot(v, v));
+    auto real_part = norm_u_norm_v + dot(u, v);
+    util::vector<3,T> w;
+
+    if (real_part < 1.e-6f * norm_u_norm_v)
+    {
+        /* If u and v are exactly opposite, rotate 180 degrees
+         * around an arbitrary orthogonal axis. Axis normalisation
+         * can happen later, when we normalise the quaternion. */
+        real_part = 0.0f;
+        w = std::abs(u.x) > std::abs(u.z) ?
+            util::vector3<T> (-u.y, u.x, 0.f) :
+            util::vector3<T> (0.f, -u.z, u.y);
+    }
+    else
+    {
+        /* Otherwise, build quaternion the standard way. */
+        w = cross(u, v);
+    }
+
+    return normalised (util::quaternion<4,T> (real_part, w.x, w.y, w.z));
+}
+
+
 ///////////////////////////////////////////////////////////////////////////////
 template <typename T>
 quaternion<4,T>
@@ -72,6 +109,7 @@ util::conjugate (quaternion<4,T> q)
 }
 
 
+//-----------------------------------------------------------------------------
 template quaternion<4,float> util::conjugate (quaternion<4,float>);
 
 
@@ -130,6 +168,58 @@ quaternion<S, T>::as_matrix (void) const
 }
 
 
+///////////////////////////////////////////////////////////////////////////////
+// https://gamedev.stackexchange.com/questions/28395/rotating-vector3-by-a-quaternion
+template <typename T>
+util::vector3<T>
+util::rotate (vector3<T> v, quaternion<4,T> q)
+{
+    CHECK (is_normalised (v));
+
+#if 1
+    util::vector3<T> u { q.x, q.y, q.z };
+    return v + 2 * cross (u, cross (u, v) + q.w * v);
+#else
+    // Verbosely:
+    quaternionf p { 0, v.x, v.y, v.z };
+    auto p_ = q * p * conjugate (q);
+    return { p_.x, p_.y, p_.z };
+#endif
+}
+
+
+//-----------------------------------------------------------------------------
+template util::vector3f util::rotate (util::vector3f, util::quaternionf);
+template util::vector3d util::rotate (util::vector3d, util::quaterniond);
+
+
+///////////////////////////////////////////////////////////////////////////////
+template <size_t S, typename T>
+quaternion<S,T>
+quaternion<S,T>::look (vector<3,T> fwd, vector<3,T> up)
+{
+    CHECK (is_normalised (fwd));
+    CHECK (is_normalised (up));
+
+    constexpr util::vector3<T> RH_FWD { 0, 0, -1 };
+    constexpr util::vector3<T> RH_UP  { 0, 1,  0 };
+
+    // rotate the right-handed fwd to face the given fwd
+    auto q1 = from_to (RH_FWD, fwd);
+
+    // can't retain the up direction if fwd is the same as up
+    if (norm2 (cross (fwd, up)) < 1e-6)
+        return q1;
+
+    // find the right-handed up rotated to the given fwd
+    auto new_up = rotate (RH_UP, q1);
+
+    // rotate first to the new forward, then align the right handed up with
+    // the given up.
+    return from_to (new_up, up) * q1;
+}
+
+
 ///////////////////////////////////////////////////////////////////////////////
 template <size_t S, typename T>
 std::ostream&

quaternion.hpp

@@ -29,7 +29,7 @@ namespace util {
     // quaternions must be 4 elements, but we include a size parameter so it
     // fits with the generic coord infrastructure more easily.
     //
-    // specifically: 
+    // specifically:
     // large regions of base code require a template template parameter with
     // size and type arguments, which is annoying to work around for this one
     // case.
@@ -41,8 +41,11 @@ namespace util {
 
         using coord::base<S,T,::util::quaternion,::util::coord::wxyz,::util::coord::abcd>::base;
 
-        static quaternion rotation (T radians, vector<3,T> axis);
+        static quaternion angle_axis (T radians, vector<3,T> axis);
-        static quaternion rotation (vector<3,T> src, vector<3,T> dst);
+        static quaternion from_euler (vector<3,T>);
+
+        static quaternion from_to (vector<3,T>, vector<3,T>);
+        static quaternion look (vector<3,T> fwd, vector<3,T> up);
 
         matrix4<T> as_matrix (void) const;
 
@@ -61,7 +64,12 @@ namespace util {
     quaternion<S,T>
     operator/ (const quaternion<S,T>, const quaternion<S,T>);
 
-    typedef quaternion<4,float> quaternionf;
+    typedef quaternion<4,float>  quaternionf;
+    typedef quaternion<4,double> quaterniond;
+
+    template <typename T>
+    vector3<T>
+    rotate (vector3<T>, quaternion<4,T>);
 
     template <size_t S, typename T>
     std::ostream&

test/affine.cpp

@@ -54,7 +54,7 @@ test_mq_axis (util::TAP::logger &tap)
 
     for (auto t: TESTS) {
         auto m = util::matrix4f::rotation (1, t.euler);
-        auto q = util::quaternionf::rotation (1, t.euler);
+        auto q = util::quaternionf::angle_axis (1, t.euler);
 
         auto diff = sum (abs (m - q.as_matrix ()));
         tap.expect_le (diff, 1e-6f, "matrix/quaternion rotation identities, %s", t.msg);
@@ -83,9 +83,9 @@ test_mq_euler (util::TAP::logger &tap)
                  util::matrix4f::rotation (t.euler[1], { 0, 1, 0 }) *
                  util::matrix4f::rotation (t.euler[2], { 0, 0, 1 });
         auto q = (
-            util::quaternionf::rotation (t.euler[0], { 1, 0, 0 }) *
+            util::quaternionf::angle_axis (t.euler[0], { 1, 0, 0 }) *
-            util::quaternionf::rotation (t.euler[1], { 0, 1, 0 }) *
+            util::quaternionf::angle_axis (t.euler[1], { 0, 1, 0 }) *
-            util::quaternionf::rotation (t.euler[2], { 0, 0, 1 })
+            util::quaternionf::angle_axis (t.euler[2], { 0, 0, 1 })
         ).as_matrix ();
 
         auto diff = util::sum (abs (m - q));

test/matrix.cpp

@@ -192,9 +192,9 @@ main (void)
             constexpr auto PI2 = 2 * util::PI<float>;
 
             auto matrix = (
-                util::quaternionf::rotation (t.euler[2], { 0, 0, 1 }) *
+                util::quaternionf::angle_axis (t.euler[2], { 0, 0, 1 }) *
-                util::quaternionf::rotation (t.euler[1], { 0, 1, 0 }) *
+                util::quaternionf::angle_axis (t.euler[1], { 0, 1, 0 }) *
-                util::quaternionf::rotation (t.euler[0], { 1, 0, 0 })
+                util::quaternionf::angle_axis (t.euler[0], { 1, 0, 0 })
             ).as_matrix ();
 
             auto euler = to_euler (matrix);

test/quaternion.cpp

@@ -76,7 +76,7 @@ main (void)
         for (size_t i = 0; i < elems (ROTATIONS); ++i) {
             const auto &r = ROTATIONS[i];
 
-            auto q = quaternionf::rotation    (r.mag, r.axis).as_matrix ();
+            auto q = quaternionf::angle_axis  (r.mag, r.axis).as_matrix ();
             auto m = util::matrix4f::rotation (r.mag, r.axis);
             auto diff = util::abs (q - m);
 
@@ -87,7 +87,7 @@ main (void)
         auto m = util::matrix4f::IDENTITY;
 
         for (auto r: ROTATIONS) {
-            q = q.rotation (r.mag, r.axis) * q;
+            q = q.angle_axis (r.mag, r.axis) * q;
             m = m.rotation (r.mag, r.axis) * m;
         }