diff --git a/gtsam/geometry/Unit3.cpp b/gtsam/geometry/Unit3.cpp
index e86121537..7729bd354 100644
--- a/gtsam/geometry/Unit3.cpp
+++ b/gtsam/geometry/Unit3.cpp
@@ -153,14 +153,14 @@ Unit3 Unit3::retract(const Vector2& v) const {
 /* ************************************************************************* */
 Vector2 Unit3::localCoordinates(const Unit3& other) const {
   const double x = p_.dot(other.p_);
-  // Crucial quantitity here is y = theta/sin(theta) with theta=acos(x)
+  // Crucial quantity here is y = theta/sin(theta) with theta=acos(x)
   // Now, y = acos(x) / sin(acos(x)) = acos(x)/sqrt(1-x^2)
-  // We treat the special caes 1 and -1 below
+  // We treat the special case 1 and -1 below
   const double x2 = x * x;
   const double z = 1 - x2;
   double y;
   if (z < std::numeric_limits<double>::epsilon()) {
-    if (x > 0)  // expand at x=1
+    if (x > 0)  // first order expansion at x=1
       y = 1.0 - (x - 1.0) / 3.0;
     else  // cop out
       return Vector2(M_PI, 0.0);
diff --git a/gtsam/geometry/tests/testUnit3.cpp b/gtsam/geometry/tests/testUnit3.cpp
index a4227acdf..e55caaa3c 100644
--- a/gtsam/geometry/tests/testUnit3.cpp
+++ b/gtsam/geometry/tests/testUnit3.cpp
@@ -192,6 +192,16 @@ TEST(Unit3, localCoordinates) {
     EXPECT(assert_equal(expected, actual, 1e-8));
     EXPECT(assert_equal(q, p.retract(expected), 1e-8));
   }
+  {
+    Unit3 p(0,1,0), q(0,-1,0);
+    Vector2 actual = p.localCoordinates(q);
+    EXPECT(assert_equal(q, p.retract(actual), 1e-8));
+  }
+  {
+    Unit3 p(0,0,1), q(0,0,-1);
+    Vector2 actual = p.localCoordinates(q);
+    EXPECT(assert_equal(q, p.retract(actual), 1e-8));
+  }
 
   double twist = 1e-4;
   {