fix(ga): change default dual mode to Iinv+ for correct B*dual(B)=I

doc/changelog.rst

@@ -8,6 +8,11 @@ Changelog
   \newcommand {\es}[1] {\mathbf{e}_{#1}}
   \newcommand {\til}[1] {\widetilde{#1}}
 
+- :bug:`514` The default dual mode has changed from ``'I+'`` to ``'Iinv+'`` so that
+  ``B * B.dual() == I`` holds for unit blades, matching the standard definition
+  (Doran & Lasenby). To preserve the old behavior, call ``Ga.dual_mode('I+')``
+  at the start of your program.
+
 - :bug:`518` :class:`~galgebra.mv.Mv` now correctly returns ``Mv`` instance when raise to power of zero. But in general, if one needs to call ``Mv`` methods on a result returned by some GA operations, it would be more prudent to initialize it as an ``Mv`` instance first, as sometimes the result becomes a ``sympy`` object.
 
 - :bug:`516` :attr:`~galgebra.mv.Mv.grades` no longer incorrectly returns ``None`` under some circumstances, as now all initialization branch will correctly call :meth:`~galgebra.mv.Mv.characterise_Mv`.

doc/module-components.rst

@@ -339,7 +339,7 @@ If we can instantiate multivectors we can use all the multivector class function
 .. method:: galgebra.mv.Mv.dual()
    :noindex:
 
-   The mode of the ``dual()`` function is set by the ``Ga`` class static member function, ``GA.dual_mode(mode='I+')`` of the ``GA`` geometric galgebra which sets the following return values (:math:`I` is the pseudo-scalar for the geometric algebra ``GA``)
+   The mode of the ``dual()`` function is set by the ``Ga`` class static member function, ``GA.dual_mode(mode='Iinv+')`` of the ``GA`` geometric galgebra which sets the following return values (:math:`I` is the pseudo-scalar for the geometric algebra ``GA``)
 
    =========== ================
    ``mode``    Return Value
@@ -356,7 +356,7 @@ If we can instantiate multivectors we can use all the multivector class function
 
    For example if the geometric algebra is ``o3d``, ``A`` is a multivector in ``o3d``, and we wish to use ``mode='I-'``. We set the mode with the function ``o3d.dual('I-')`` and get the dual of ``A`` with the function ``A.dual()`` which returns :math:`-AI`.
 
-   If ``o3d.dual(mode)`` is not called the default for the dual mode is ``mode='I+'`` and ``A*I`` is returned.
+   If ``o3d.dual(mode)`` is not called the default for the dual mode is ``mode='Iinv+'`` and ``A*I^{-1}`` is returned.
 
    Note that ``Ga.dual(mode)`` used the function ``Ga.I()`` to calculate the normalized pseudoscalar. Thus if the metric tensor is not numerical and orthogonal the correct hint for then ``sig`` input of the *Ga* constructor is required.
 

examples/ipython/Smith Sphere.ipynb

@@ -19,7 +19,14 @@
   {
    "cell_type": "code",
    "execution_count": 1,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-03-30T07:43:51.081888Z",
+     "iopub.status.busy": "2026-03-30T07:43:51.081668Z",
+     "iopub.status.idle": "2026-03-30T07:43:52.052483Z",
+     "shell.execute_reply": "2026-03-30T07:43:52.051405Z"
+    }
+   },
    "outputs": [],
    "source": [
     "#from IPython.display import SVG\n",
@@ -30,7 +37,14 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-03-30T07:43:52.056481Z",
+     "iopub.status.busy": "2026-03-30T07:43:52.056286Z",
+     "iopub.status.idle": "2026-03-30T07:43:52.110490Z",
+     "shell.execute_reply": "2026-03-30T07:43:52.109778Z"
+    }
+   },
    "outputs": [],
    "source": [
     "from galgebra import ga\n",
@@ -50,7 +64,7 @@
     "    '''\n",
     "    stereographically project a vector in G3 downto the bivector N\n",
     "    '''\n",
-    "    n= -1*N.dual()\n",
+    "    n= N.dual()\n",
     "    return -(n^p)*(n-n*(n|p)).inv() \n",
     "\n",
     "def up(p):\n",
@@ -62,7 +76,7 @@
     "    elif  (p^Bs).obj == 0:\n",
     "        N = Bs\n",
     "        \n",
-    "    n = -N.dual()\n",
+    "    n = N.dual()\n",
     "    \n",
     "    return   n + 2*(p*p + 1).inv()*(p-n)\n",
     "    \n",
@@ -81,15 +95,22 @@
   {
    "cell_type": "code",
    "execution_count": 3,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-03-30T07:43:52.114557Z",
+     "iopub.status.busy": "2026-03-30T07:43:52.114284Z",
+     "iopub.status.idle": "2026-03-30T07:43:52.121067Z",
+     "shell.execute_reply": "2026-03-30T07:43:52.119835Z"
+    }
+   },
    "outputs": [
     {
      "data": {
       "text/latex": [
-       "\\begin{equation*}- \\boldsymbol{e}_{s}\\end{equation*}"
+       "\\begin{equation*} \\boldsymbol{e}_{s}\\end{equation*}"
       ],
       "text/plain": [
-       "-e_s"
+       "e_s"
       ]
      },
      "execution_count": 3,
@@ -104,7 +125,14 @@
   {
    "cell_type": "code",
    "execution_count": 4,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-03-30T07:43:52.181509Z",
+     "iopub.status.busy": "2026-03-30T07:43:52.181279Z",
+     "iopub.status.idle": "2026-03-30T07:43:52.185078Z",
+     "shell.execute_reply": "2026-03-30T07:43:52.184386Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -124,7 +152,14 @@
   {
    "cell_type": "code",
    "execution_count": 5,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-03-30T07:43:52.190028Z",
+     "iopub.status.busy": "2026-03-30T07:43:52.189840Z",
+     "iopub.status.idle": "2026-03-30T07:43:52.194750Z",
+     "shell.execute_reply": "2026-03-30T07:43:52.193673Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -155,7 +190,14 @@
   {
    "cell_type": "code",
    "execution_count": 6,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-03-30T07:43:52.197764Z",
+     "iopub.status.busy": "2026-03-30T07:43:52.197554Z",
+     "iopub.status.idle": "2026-03-30T07:43:52.378352Z",
+     "shell.execute_reply": "2026-03-30T07:43:52.377614Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -179,7 +221,14 @@
   {
    "cell_type": "code",
    "execution_count": 7,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-03-30T07:43:52.383180Z",
+     "iopub.status.busy": "2026-03-30T07:43:52.382998Z",
+     "iopub.status.idle": "2026-03-30T07:43:52.431466Z",
+     "shell.execute_reply": "2026-03-30T07:43:52.430376Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -209,7 +258,14 @@
   {
    "cell_type": "code",
    "execution_count": 8,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-03-30T07:43:52.435047Z",
+     "iopub.status.busy": "2026-03-30T07:43:52.434874Z",
+     "iopub.status.idle": "2026-03-30T07:43:52.529889Z",
+     "shell.execute_reply": "2026-03-30T07:43:52.529102Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -232,7 +288,14 @@
   {
    "cell_type": "code",
    "execution_count": 9,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-03-30T07:43:52.534863Z",
+     "iopub.status.busy": "2026-03-30T07:43:52.534694Z",
+     "iopub.status.idle": "2026-03-30T07:43:52.958048Z",
+     "shell.execute_reply": "2026-03-30T07:43:52.957068Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -255,7 +318,14 @@
   {
    "cell_type": "code",
    "execution_count": 10,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-03-30T07:43:52.961133Z",
+     "iopub.status.busy": "2026-03-30T07:43:52.960973Z",
+     "iopub.status.idle": "2026-03-30T07:43:53.012279Z",
+     "shell.execute_reply": "2026-03-30T07:43:53.011641Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -278,7 +348,14 @@
   {
    "cell_type": "code",
    "execution_count": 11,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-03-30T07:43:53.016181Z",
+     "iopub.status.busy": "2026-03-30T07:43:53.015975Z",
+     "iopub.status.idle": "2026-03-30T07:43:53.067710Z",
+     "shell.execute_reply": "2026-03-30T07:43:53.065599Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -301,7 +378,14 @@
   {
    "cell_type": "code",
    "execution_count": 12,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-03-30T07:43:53.072173Z",
+     "iopub.status.busy": "2026-03-30T07:43:53.072020Z",
+     "iopub.status.idle": "2026-03-30T07:43:53.097579Z",
+     "shell.execute_reply": "2026-03-30T07:43:53.096840Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -325,7 +409,14 @@
   {
    "cell_type": "code",
    "execution_count": 13,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-03-30T07:43:53.101304Z",
+     "iopub.status.busy": "2026-03-30T07:43:53.100943Z",
+     "iopub.status.idle": "2026-03-30T07:43:53.171192Z",
+     "shell.execute_reply": "2026-03-30T07:43:53.169464Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -348,7 +439,14 @@
   {
    "cell_type": "code",
    "execution_count": 14,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-03-30T07:43:53.174960Z",
+     "iopub.status.busy": "2026-03-30T07:43:53.174821Z",
+     "iopub.status.idle": "2026-03-30T07:43:54.002785Z",
+     "shell.execute_reply": "2026-03-30T07:43:54.001491Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -385,7 +483,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.8.0"
+   "version": "3.11.14"
   }
  },
  "nbformat": 4,

galgebra/ga.py

@@ -565,7 +565,7 @@ class Ga(metric.Metric):
         =======  ======================
     """
 
-    dual_mode_value = 'I+'
+    dual_mode_value = 'Iinv+'
     dual_mode_lst = ['+I', 'I+', '-I', 'I-', '+Iinv', 'Iinv+', '-Iinv', 'Iinv-']
 
     presets = {'o3d': 'x,y,z:[1,1,1]:[1,1,0]',
@@ -574,12 +574,12 @@ class Ga(metric.Metric):
                'para3d': 'u,v,z:[u**2+v**2,u**2+v**2,1]:[1,1,0]:norm=True'}
 
     @staticmethod
-    def dual_mode(mode='I+'):
+    def dual_mode(mode='Iinv+'):
         """
         Sets mode of multivector dual function for all geometric algebras
         in users program.
 
-        If Ga.dual_mode(mode) not called the default mode is ``'I+'``.
+        If Ga.dual_mode(mode) not called the default mode is ``'Iinv+'``.
 
         =====  ============
         mode   return value

galgebra/primer.py

@@ -32,9 +32,8 @@
 # Default `Dmode=True` causes partial differentiation
 #   operators to be displayed in shortened form.
 Ga.dual_mode('Iinv+')
-# Sets multivector dualization to be right multiplication by the
-# by the inverse unit pseudoscalar (the convention used in the
-# textbooks LAGA, VAGC, and GACS).
+# Confirms the default dual mode: right multiplication by the inverse
+# pseudoscalar (convention of LAGA, VAGC, and GACS).
 initializations_list = r"""\textsf{The following initialization commands were executed:}\\
 \quad\texttt{from sys import version}\\
 \quad\texttt{import sympy}\\

test/test_test.py

@@ -530,21 +530,33 @@ def test_dual_mode(self):
         ga, e1, e2 = Ga.build('e*1|2', g=[1, 1])
 
         default = Ga.dual_mode_value
-        assert default == 'I+'
+        assert default == 'Iinv+'
         with pytest.raises(ValueError):
             Ga.dual_mode('illegal')
 
         d_default = e1.dual()
 
         # note: this is a global setting, so we have to make sure we put it back
         try:
-            Ga.dual_mode('I-')
+            Ga.dual_mode('Iinv-')
             d_negated = e1.dual()
         finally:
             Ga.dual_mode(default)
 
         assert d_negated == -d_default
 
+    def test_dual_correctness(self):
+        """Test Iinv+ dual mode on a unit bivector in 3D Euclidean space (issue 514)."""
+        from sympy import symbols as S_symbols
+        ga = Ga('e', g=[1, 1, 1], coords=S_symbols('x y z', real=True), wedge=False)
+        ex, ey, ez = ga.mv()
+        I = ga.I()
+
+        # For a unit bivector B with B*B = -1, B * dual(B) == I under Iinv+ mode
+        bivector = ex * I  # = e_yz, a unit bivector
+        assert bivector * bivector.dual() == I
+        assert bivector.dual() * bivector == I
+
     def test_basis_dict(self):
         ga = Ga('e*1|2', g=[1, 1])
         b = ga.bases_dict()