diff --git a/gtsam_unstable/geometry/Similarity3.cpp b/gtsam_unstable/geometry/Similarity3.cpp
index f0f673bca..dd4ca3a31 100644
--- a/gtsam_unstable/geometry/Similarity3.cpp
+++ b/gtsam_unstable/geometry/Similarity3.cpp
@@ -38,13 +38,13 @@ Similarity3::Similarity3(const Matrix3& R, const Vector3& t, double s) :
     R_(R), t_(t), s_(s) {
 }
 
-bool Similarity3::equals(const Similarity3& sim, double tol) const {
-  return R_.equals(sim.R_, tol) && t_.equals(sim.t_, tol) && s_ < (sim.s_ + tol)
-      && s_ > (sim.s_ - tol);
+bool Similarity3::equals(const Similarity3& other, double tol) const {
+  return R_.equals(other.R_, tol) && t_.equals(other.t_, tol)
+      && s_ < (other.s_ + tol) && s_ > (other.s_ - tol);
 }
 
 bool Similarity3::operator==(const Similarity3& other) const {
-  return equals(other, 1e-9);
+  return R_.matrix() == other.R_.matrix() && t_ == other.t_ && s_ == other.s_;
 }
 
 void Similarity3::print(const std::string& s) const {
@@ -70,40 +70,47 @@ Similarity3 Similarity3::inverse() const {
 
 Point3 Similarity3::transform_from(const Point3& p, //
     OptionalJacobian<3, 7> H1, OptionalJacobian<3, 3> H2) const {
+  Point3 q = R_ * p + t_;
   if (H1) {
     const Matrix3 R = R_.matrix();
     Matrix3 DR = s_ * R * skewSymmetric(-p.x(), -p.y(), -p.z());
-    *H1 << DR, R, R * p.vector();
-    print("From Derivative");
+    // TODO(frank): explain the derivative in lambda
+    *H1 << DR, s_ * R, s_ * p.vector();
   }
   if (H2)
     *H2 = s_ * R_.matrix(); // just 3*3 sub-block of matrix()
-  return R_ * (s_ * p) + t_;
-  // TODO: Effect of scale change is this, right?
-  // No, this is incorrect. Zhaoyang Lv
-  // sR t * (1+v)I 0 * p = s(1+v)R t * p = s(1+v)Rp + t = sRp + vRp + t
-  // 0001   000    1   1   000     1   1
+  return s_ * q;
 }
 
 Point3 Similarity3::operator*(const Point3& p) const {
   return transform_from(p);
 }
 
+Matrix4 Similarity3::wedge(const Vector7& xi) {
+  // http://www.ethaneade.org/latex2html/lie/node29.html
+  const auto w = xi.head<3>();
+  const auto u = xi.segment<3>(3);
+  double lambda = xi[6];
+  Matrix4 W;
+  W << skewSymmetric(w), u, 0, 0, 0, -lambda;
+  return W;
+}
+
 Matrix7 Similarity3::AdjointMap() const {
-//  ToDo:  This adjoint might not be correct, it is based on delta = [u, w, lambda]
-//  However, we use the convention delta = [w, u, lambda]
+  // http://www.ethaneade.org/latex2html/lie/node30.html
   const Matrix3 R = R_.matrix();
   const Vector3 t = t_.vector();
   Matrix3 A = s_ * skewSymmetric(t) * R;
   Matrix7 adj;
-  adj << s_ * R, A, -s_ * t, // 3*7
-  Z_3x3, R, Matrix31::Zero(), // 3*7
-  Matrix16::Zero(), 1; // 1*7
+  adj <<
+      R, Z_3x3, Matrix31::Zero(), // 3*7
+      A, s_ * R, -s_ * t, // 3*7
+      Matrix16::Zero(), 1; // 1*7
   return adj;
 }
 
-Matrix33 Similarity3::GetV(Vector3 w, double lambda){
-  Matrix33 wx = skewSymmetric(w[0], w[1], w[2]);
+Matrix3 Similarity3::GetV(Vector3 w, double lambda) {
+  Matrix3 wx = skewSymmetric(w[0], w[1], w[2]);
   double lambdasquared = lambda * lambda;
   double thetasquared = w.transpose() * w;
   double theta = sqrt(thetasquared);
@@ -122,13 +129,12 @@ Matrix33 Similarity3::GetV(Vector3 w, double lambda){
     A = (1 - exp(-lambda)) / lambda;
     B = alpha * (beta - gama) + gama;
     C = alpha * (mu - upsilon) + upsilon;
-  }
-  else if(thetasquared <= 1e-9 && lambdasquared > 1e-9) {
+  } else if (thetasquared <= 1e-9 && lambdasquared > 1e-9) {
     //Taylor series expansions
     X = 1;
-    Y = 0.5-thetasquared/24.0;
-    Z = 1.0/6.0 - thetasquared/120.0;
-    W = 1.0/24.0 - thetasquared/720.0;
+    Y = 0.5 - thetasquared / 24.0;
+    Z = 1.0 / 6.0 - thetasquared / 120.0;
+    W = 1.0 / 24.0 - thetasquared / 720.0;
     alpha = lambdasquared / (lambdasquared + thetasquared);
     beta = (exp(-lambda) - 1 + lambda) / lambdasquared;
     gama = Y - (lambda * Z);
@@ -138,8 +144,7 @@ Matrix33 Similarity3::GetV(Vector3 w, double lambda){
     A = (1 - exp(-lambda)) / lambda;
     B = alpha * (beta - gama) + gama;
     C = alpha * (mu - upsilon) + upsilon;
-  }
-  else if(thetasquared > 1e-9 && lambdasquared <= 1e-9) {
+  } else if (thetasquared > 1e-9 && lambdasquared <= 1e-9) {
     X = sin(theta) / theta;
     Y = (1 - cos(theta)) / thetasquared;
     Z = (1 - X) / thetasquared;
@@ -158,10 +163,9 @@ Matrix33 Similarity3::GetV(Vector3 w, double lambda){
     }
     B = alpha * (beta - gama) + gama;
     C = alpha * (mu - upsilon) + upsilon;
-  }
-  else {
+  } else {
     X = 1;
-    Y = 0.5-thetasquared/24.0;
+    Y = 0.5 - thetasquared / 24.0;
     Z = 1.0 / 6.0 - thetasquared / 120.0;
     W = 1.0 / 24.0 - thetasquared / 720.0;
     alpha = lambdasquared / (lambdasquared + thetasquared);
@@ -179,7 +183,7 @@ Matrix33 Similarity3::GetV(Vector3 w, double lambda){
     B = gama;
     C = upsilon;
   }
-  return A * Matrix33::Identity() + B * wx + C * wx * wx;
+  return A * I_3x3 + B * wx + C * wx * wx;
 }
 
 Vector7 Similarity3::Logmap(const Similarity3& s, OptionalJacobian<7, 7> Hm) {
@@ -196,26 +200,27 @@ Vector7 Similarity3::Logmap(const Similarity3& s, OptionalJacobian<7, 7> Hm) {
 }
 
 Similarity3 Similarity3::Expmap(const Vector7& v, OptionalJacobian<7, 7> Hm) {
-  Vector3 w(v.head<3>());
+  const auto w = v.head<3>();
+  const auto u = v.segment<3>(3);
   double lambda = v[6];
   if (Hm) {
-    Matrix6 J_pose = Pose3::ExpmapDerivative(v.head<6>());
+    // Matrix6 J_pose = Pose3::ExpmapDerivative(v.head<6>());
     // incomplete
   }
-  return Similarity3(Rot3::Expmap(w), Point3(GetV(w, lambda)*v.segment<3>(3)), 1.0/exp(-lambda));
+  const Matrix3 V = GetV(w, lambda);
+  return Similarity3(Rot3::Expmap(w), Point3(V * u), exp(lambda));
 }
 
-
 std::ostream &operator<<(std::ostream &os, const Similarity3& p) {
-  os << "[" << p.rotation().xyz().transpose() << " " << p.translation().vector().transpose() << " " <<
-      p.scale() << "]\';";
+  os << "[" << p.rotation().xyz().transpose() << " "
+      << p.translation().vector().transpose() << " " << p.scale() << "]\';";
   return os;
 }
 
 const Matrix4 Similarity3::matrix() const {
   Matrix4 T;
-  T.topRows<3>() << s_ * R_.matrix(), t_.vector();
-  T.bottomRows<1>() << 0, 0, 0, 1;
+  T.topRows<3>() << R_.matrix(), t_.vector();
+  T.bottomRows<1>() << 0, 0, 0, 1.0/s_;
   return T;
 }
 
diff --git a/gtsam_unstable/geometry/Similarity3.h b/gtsam_unstable/geometry/Similarity3.h
index c380c4aa7..92082ee02 100644
--- a/gtsam_unstable/geometry/Similarity3.h
+++ b/gtsam_unstable/geometry/Similarity3.h
@@ -67,7 +67,7 @@ public:
   /// Compare with tolerance
   bool equals(const Similarity3& sim, double tol) const;
 
-  /// Compare with standard tolerance
+  /// Exact equality
   bool operator==(const Similarity3& other) const;
 
   /// Print with optional string
@@ -92,6 +92,7 @@ public:
   /// @name Group action on Point3
   /// @{
 
+  /// Action on a point p is s*(R*p+t)
   Point3 transform_from(const Point3& p, //
       OptionalJacobian<3, 7> H1 = boost::none, //
       OptionalJacobian<3, 3> H2 = boost::none) const;
@@ -124,11 +125,19 @@ public:
     }
   };
 
+  using LieGroup<Similarity3, 7>::inverse;
+
+  /**
+   * wedge for Similarity3:
+   * @param xi 7-dim twist (w,u,lambda) where
+   * @return 4*4 element of Lie algebra that can be exponentiated
+   * TODO(frank): rename to Hat, make part of traits
+   */
+  static Matrix4 wedge(const Vector7& xi);
+
   /// Project from one tangent space to another
   Matrix7 AdjointMap() const;
 
-  using LieGroup<Similarity3, 7>::inverse;
-
   /// @}
   /// @name Standard interface
   /// @{
@@ -152,6 +161,7 @@ public:
   }
 
   /// Convert to a rigid body pose (R, s*t)
+  /// TODO(frank): why is this here? Red flag! Definitely don't have it as a cast.
   operator Pose3() const;
 
   /// Dimensionality of tangent space = 7 DOF - used to autodetect sizes
@@ -170,7 +180,7 @@ public:
 
   /// Calculate expmap and logmap coefficients.
 private:
-  static Matrix33 GetV(Vector3 w, double lambda);
+  static Matrix3 GetV(Vector3 w, double lambda);
 
   /// @}
 
diff --git a/gtsam_unstable/geometry/tests/testSimilarity3.cpp b/gtsam_unstable/geometry/tests/testSimilarity3.cpp
index 84aa14420..b8db48942 100644
--- a/gtsam_unstable/geometry/tests/testSimilarity3.cpp
+++ b/gtsam_unstable/geometry/tests/testSimilarity3.cpp
@@ -42,7 +42,7 @@ GTSAM_CONCEPT_TESTABLE_INST(Similarity3)
 static Point3 P(0.2, 0.7, -2);
 static Rot3 R = Rot3::Rodrigues(0.3, 0, 0);
 static double s = 4;
-static Similarity3 T_default(R, Point3(3.5, -8.2, 4.2), 1);
+static Similarity3 T1(R, Point3(3.5, -8.2, 4.2), 1);
 static Similarity3 T2(Rot3::Rodrigues(0.3, 0.2, 0.1), Point3(3.5, -8.2, 4.2), 1);
 static Similarity3 T3(Rot3::Rodrigues(-90, 0, 0), Point3(1, 2, 3), 1);
 static Similarity3 T4(R, P, s);
@@ -79,16 +79,13 @@ TEST(Similarity3, Getters) {
 
 //******************************************************************************
 TEST(Similarity3, AdjointMap) {
-  Similarity3 test(Rot3::Ypr(1, 2, 3).inverse(), Point3(4, 5, 6), 7);
-  Matrix7 result;
-  result << -1.5739, -2.4512, -6.3651, -50.7671, -11.2503, 16.8859, -28.0000,
-      6.3167, -2.9884, -0.4111, 0.8502, 8.6373, -49.7260, -35.0000,
-      -2.5734, -5.8362, 2.8839, 33.1363, 0.3024, 30.1811, -42.0000,
-      0, 0, 0, -0.2248, -0.3502, -0.9093, 0,
-      0, 0, 0, 0.9024, -0.4269, -0.0587, 0,
-      0, 0, 0, -0.3676, -0.8337, 0.4120, 0,
-      0, 0, 0, 0, 0, 0, 1.0000;
-  EXPECT(assert_equal(result, test.AdjointMap(), 1e-3));
+  const Matrix4 T = T2.matrix();
+  // Check Ad with actual definition
+  Vector7 delta;
+  delta << 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7;
+  Matrix4 W = Similarity3::wedge(delta);
+  Matrix4 TW = Similarity3::wedge(T2.AdjointMap() * delta);
+  EXPECT(assert_equal(TW, Matrix4(T * W * T.inverse()), 1e-9));
 }
 
 //******************************************************************************
@@ -169,13 +166,13 @@ TEST( Similarity3, retract_first_order) {
 TEST(Similarity3, localCoordinates_first_order) {
   Vector d12 = repeat(7, 0.1);
   d12(6) = 1.0;
-  Similarity3 t1 = T_default, t2 = t1.retract(d12);
+  Similarity3 t1 = T1, t2 = t1.retract(d12);
   EXPECT(assert_equal(d12, t1.localCoordinates(t2)));
 }
 
 //******************************************************************************
 TEST(Similarity3, manifold_first_order) {
-  Similarity3 t1 = T_default;
+  Similarity3 t1 = T1;
   Similarity3 t2 = T3;
   Similarity3 origin;
   Vector d12 = t1.localCoordinates(t2);
@@ -188,10 +185,11 @@ TEST(Similarity3, manifold_first_order) {
 // Return as a 4*4 Matrix
 TEST(Similarity3, Matrix) {
   Matrix4 expected;
-  expected << 2, 0, 0, 1,
-      0, 2, 0, 1,
-      0, 0, 2, 0,
-      0, 0, 0, 1;
+  expected <<
+      1, 0, 0, 1,
+      0, 1, 0, 1,
+      0, 0, 1, 0,
+      0, 0, 0, 0.5;
   Matrix4 actual = T6.matrix();
   EXPECT(assert_equal(expected, actual));
 }
@@ -226,55 +224,39 @@ TEST(Similarity3, ExpLogMap) {
 //******************************************************************************
 // Group action on Point3 (with simpler transform)
 TEST(Similarity3, GroupAction) {
-  EXPECT(assert_equal(Point3(1, 1, 0), T6 * Point3(0, 0, 0)));
+  EXPECT(assert_equal(Point3(2, 2, 0), T6 * Point3(0, 0, 0)));
+  EXPECT(assert_equal(Point3(4, 2, 0), T6 * Point3(1, 0, 0)));
 
-  // Test actual group action on R^4
+  // Test group action on R^4 via matrix representation
   Vector4 qh;
   qh << 1, 0, 0, 1;
   Vector4 ph;
-  ph << 3, 1, 0, 1;
+  ph << 2, 1, 0, 0.5; // equivalent to Point3(4, 2, 0)
   EXPECT(assert_equal((Vector )ph, T6.matrix() * qh));
 
+  // Test some more...
+  Point3 pa = Point3(1, 0, 0);
   Similarity3 Ta(Rot3(), Point3(1, 2, 3), 1.0);
   Similarity3 Tb(Rot3(), Point3(1, 2, 3), 2.0);
-
-  Point3 pa = Point3(1, 0, 0);
-  Point3 pTa = Point3(2, 2, 3);
-  Point3 pTb = Point3(3, 2, 3);
-
-  EXPECT(assert_equal(pTa, Ta.transform_from(pa)));
-  EXPECT(assert_equal(pTb, Tb.transform_from(pa)));
+  EXPECT(assert_equal(Point3(2, 2, 3), Ta.transform_from(pa)));
+  EXPECT(assert_equal(Point3(4, 4, 6), Tb.transform_from(pa)));
 
   Similarity3 Tc(Rot3::Rz(M_PI/2.0), Point3(1, 2, 3), 1.0);
   Similarity3 Td(Rot3::Rz(M_PI/2.0), Point3(1, 2, 3), 2.0);
-
-  Point3 pTc = Point3(1, 3, 3);
-  Point3 pTd = Point3(1, 4, 3);
-
-  EXPECT(assert_equal(pTc, Tc.transform_from(pa)));
-  EXPECT(assert_equal(pTd, Td.transform_from(pa)));
+  EXPECT(assert_equal(Point3(1, 3, 3), Tc.transform_from(pa)));
+  EXPECT(assert_equal(Point3(2, 6, 6), Td.transform_from(pa)));
 
   // Test derivative
   boost::function<Point3(Similarity3, Point3)> f = boost::bind(
       &Similarity3::transform_from, _1, _2, boost::none, boost::none);
 
-  { // T default
+  Point3 q(1, 2, 3);
+  for (const auto T : { T1, T2, T3, T4, T5, T6 }) {
     Point3 q(1, 0, 0);
-    Matrix H1 = numericalDerivative21<Point3, Similarity3, Point3>(f, T_default, q);
-    Matrix H2 = numericalDerivative22<Point3, Similarity3, Point3>(f, T_default, q);
+    Matrix H1 = numericalDerivative21<Point3, Similarity3, Point3>(f, T1, q);
+    Matrix H2 = numericalDerivative22<Point3, Similarity3, Point3>(f, T1, q);
     Matrix actualH1, actualH2;
-    T_default.transform_from(q, actualH1, actualH2);
-    EXPECT(assert_equal(H1, actualH1));
-    EXPECT(assert_equal(H2, actualH2));
-  }
-  { // T4
-    Point3 q(1, 0, 0);
-    Matrix H1 = numericalDerivative21<Point3, Similarity3, Point3>(f, T6, q);
-    Matrix H2 = numericalDerivative22<Point3, Similarity3, Point3>(f, T6, q);
-    Matrix actualH1, actualH2;
-    Point3 p = T6.transform_from(q, actualH1, actualH2);
-    EXPECT(assert_equal(Point3(3, 1, 0), p));
-    EXPECT(assert_equal(Point3(3, 1, 0), T6 * q));
+    T1.transform_from(q, actualH1, actualH2);
     EXPECT(assert_equal(H1, actualH1));
     EXPECT(assert_equal(H2, actualH2));
   }