Eric Lengyel • May 20, 2022
The concept of duality can be understood geometrically in an n-dimensional projective setting by considering both the subspace that an object occupies and the complementary subspace thatthe object concurrently does not occupy. The dimensionalities of these two components always sum to n, and they represent the space and antispace associated with the object.(Antispace is also known as negative space or counterspace.) The example shown in Figure 1 demonstrates the duality between hom*ogeneous points and lines in a three-dimensional projective space.The triplet of coordinates \((p_x, p_y, p_z)\) can be interpreted as a vector pointing from the origin toward a specific location on the projectionplane \(z = 1.\) This vector corresponds to the one-dimensional space of the point that it represents. The dual of a point materializes when we consider all of the directions of space that areorthogonal to the single direction \((p_x, p_y, p_z).\) As illustrated by the figure, these directions span an (\(n - 1\))-dimensionalsubspace that intersects the projection plane at a line when \(n = 3.\) In this way, the coordinates \((p_x, p_y, p_z)\) can be interpreted asboth a point and a line, and they are geometric duals of each other.
Figure 1. The coordinates \((p_x, p_y, p_z)\) can be interpreted as the one-dimensional span of a single vector representinga hom*ogeneous point or as the (\(n - 1\))-dimensional span of all orthogonal vectors representing a hom*ogeneous hyperplane, which is a line when \(n = 3.\) Geometrically, these twointerpretations are dual to each other, and their distances to the origin are reciprocals of each other.
When we express the coordinates \((p_x, p_y, p_z)\) on the vector basis as \(p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3,\)it explicitly states that we are working with a single spatial dimension representing a point, and the ambiguity is removed. Similarly, if we express thecoordinates on the bivector basis as \(p_x \mathbf e_{23} + p_y \mathbf e_{31} + p_z \mathbf e_{12},\) then we are working with thetwo orthogonal spatial dimensions representing a line. In each case, the subscripts of the basis elements tell us which basis vectors are present in the representation, and this defines the spaceof the object. The subscripts also tell us which basis vectors are absent in the representation, and this defines the antispace of the object. Acknowledging the existence of both the space andthe antispace of any object and assigning equal meaningfulness to them allows us to explore the nature of duality to its fullest. A vector \(p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3\)is never only a point, but both a point and a line simultaneously, where the point exists in space, and the lineexists in antispace. Likewise, a bivector \(p_x \mathbf e_{23} + p_y \mathbf e_{31} + p_z \mathbf e_{12}\) is never only a line,but both a line and a point simultaneously, where the line exists in space, and the point exists in antispace. If we study only the spatial facet of these objects and their higher-dimensionalcounterparts, then we are missing half of a bigger picture.
It is particularly interesting to consider the Euclidean isometries that map n-dimensional space onto itself while preserving distances and angles. We know how each isometry transformsthe space of a point, line, plane, etc., but for a complete understanding of the geometry, we must ask ourselves what happens to the antispace of those objects at the same time. Equivalently,when an object is transformed by an isometry, we would like to know how its dual is transformed. The answer requires that we first look at the invariants associated with each Euclidean isometry.
In the two-dimensional plane, the Euclidean isometries consist of a rotation about point, a translation in a specific direction, and a transflection with respect to a specific mirroring line.A reflection is a special case of transflection in which there is no motion parallel to the line, and translation is a special case of rotation in which the center lies in the horizon. Naturally,the invariant of a rotation is its center point, and the invariant of a transflection is its mirroring line. These objects are mapped onto themselves by their associated transforms, and thisnecessitates that their duals also be mapped onto themselves by whatever corresponding transforms occur in antispace. Because a rotation fixes its center point, the corresponding transform inantispace must fix the line that is dual to that point. And because a transflection fixes its mirroring line, the corresponding transform in antispace must fix the point that is dual to that line.These transforms occurring in antispace are reciprocal analogs of the transforms occurring in space. There is a direct correspondence between the two transforms, and they are inextricably linked.Whenever one transform is applied in space, the other is applied in antispace, and vice-versa.
Figure 2 shows a two-dimensional rotation transform and its reciprocal analog. The green point represents the center of rotation, and the green line is the dual of that point. Under the regularrotation, the center point is fixed, and under the reciprocal rotation, the line dual to the center point is fixed. These are not, however, the only fixed geometries. A regular rotation also fixesthe horizon line, and thus its dual, the origin, must be fixed in the reciprocal rotation. This is illustrated by the red point in the figure, which is the focus of the various conic-section orbits.Here, the green line is the directrix.
Figure 2. (Left) A regular rotation fixes the green center point at \((1,0)\) and the horizon. (Right) The corresponding complement rotation fixes the green line at \(x = -1\)dual to the center point and the origin.
Figure 3 shows a two-dimensional reflection transform and its reciprocal analog. The green line represents the mirroring plane of the reflection, and the green point is the dual of that line.Under the regular reflection, the mirroring line is fixed, and under the reciprocal reflection, the point dual to the mirroring line is fixed. As with rotation, there are additional fixed geometriesunder these transforms. A regular reflection fixes the point in the horizon in the direction perpendicular to the mirroring plane. The dual of this point is a line parallel to the mirroring line andcontaining the origin that remains fixed by the reciprocal reflection. This is illustrated by the red line in the figure, which is clearly a reflection boundary in a sense.
Figure 3. (Left) A regular reflection fixes the green mirroring line at \(x = -1/2\) and the point in the horizon in the perpendicular direction. (Right) Thecorresponding complement reflection fixes the green point \((2,0)\) dual to the mirroring line and the line through the origin parallel to the mirroring line.
Finally, Figure 4 shows a two-dimensional translation transform and its reciprocal analog. As mentioned above, a translation is a special case of rotation in which the center lies in the horizon.As such, there is no finite fixed geometry that can be shown in the figure for a regular translation. However, the dual of the center in the horizon must be a line containing the origin that is fixedby the reciprocal translation, and that is illustrated by the red line in the figure. Reciprocal translation is especially important because it is the one to which we can most easily assign somepractical meaning. It is a perspective projection onto the line through the origin perpendicular to the direction of translation.
Figure 4. (Left) A regular translation fixes the point in the horizon perpendicular to the direction of translation and every line parallel to the direction of translation.(Right) The corresponding complement translation (a perspective projection) fixes the line parallel to the direction of translation through the origin and every point in the line through the originperpendicular to the direction of translation.
In the three-dimensional projective geometric algebra \(\mathbb R_{2,0,1},\) a hom*ogeneous representation of two-dimensional space, a regular rotation about a center point c is given by
\(\mathbf Q = c_x \mathbf e_1 + c_y \mathbf e_2 + c_z \mathbf e_3 + r \unicode{x1D7D9},\)
and this becomes a translation when \(c_z = 0.\) A regular transflection across the line h is given by
\(\mathbf F = s + h_x \mathbf e_{23} + h_x \mathbf e_{31} + h_x \mathbf e_{12},\)
and this becomes a pure reflection when \(s = 0.\) Together, these operators include all possible Euclidean isometries in the two-dimensional plane. Under the geometric antiproduct ⟇,arbitrary products of these operators form the group \(\mathrm E(2)\) with 𝟙 as the identity, and they covariantly transform any object a in the algebra through the sandwich products
\(\mathbf{a'} = \mathbf Q \mathbin{\unicode{x27C7}} \mathbf a \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}\)and\(\mathbf{a'} = \mathbf F \mathbin{\unicode{x27C7}} \mathbf a \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}}.\)
Symmetrically, a complement rotation about the line c is given by
\(\mathcal Q = c_x \mathbf e_{23} + c_y \mathbf e_{31} + c_z \mathbf e_{12} - r,\)
and a complement transflection across the point h is given by
\(\mathcal F = h_x \mathbf e_1 + h_y \mathbf e_2 + h_z \mathbf e_3 - s \unicode{x1D7D9}.\)
These two operators generate a different group of transformations under the geometric product ⟑. Arbitrary products of these operators form the complement Euclidean group \(\overline{\mathrm E}(2)\)with 1 as the identity, and they covariantly transform any object a in the algebra through the sandwich products
\(\mathbf{a'} = \mathcal Q \mathbin{\unicode{x27D1}} \mathbf a \mathbin{\unicode{x27D1}} \tilde{\mathcal Q}\)and\(\mathbf{a'} = \mathcal F \mathbin{\unicode{x27D1}} \mathbf a \mathbin{\unicode{x27D1}} \tilde{\mathcal F}.\)
The groups \(\mathrm E(2)\) and \(\overline{\mathrm E}(2)\) are isomorphic, and they each contain the orthogonal group \(\mathrm O(2)\) as a common subgroup. The complement operation provides a two-way mapping between transformsassociated with members of \(\mathrm E(2)\) and \(\overline{\mathrm E}(2).\)
The invariant geometries of the four types of transforms described above are summarized in Table 1. The Euclidean isometries always fix a coinvariant contained in the horizon, and thecorresponding complement transforms always fix a coinvariant containing the origin. In general, if x is the primary invariant of a Euclidean isometry, then the weight dual of xgives the coinvariant. Symmetrically, if x is the primary invariant of a complement transform, then the bulk dual of x gives the coinvariant. When the primary invariant of aEuclidean isometry contains the origin, there is a corresponding complement transform that performs the same operation. Symmetrically, when the primary invariant of a complement transform iscontained in the horizon, there is a corresponding Euclidean isometry that performs the same operation. These are where \(\mathrm E(2)\) and \(\overline{\mathrm E}(2)\) intersect at \(\mathrm O(2).\)
| Transform | Primary Invariant | Coinvariant |
|---|---|---|
| Regular rotation \(\mathbf Q = c_x \mathbf e_1 + c_y \mathbf e_2 + c_z \mathbf e_3 + r \unicode{x1D7D9}\) | Point \(c_x \mathbf e_1 + c_y \mathbf e_2 + c_z \mathbf e_3\) | Horizon line \(\mathbf e_{23}\) |
| Regular transflection \(\mathbf F = s + h_x \mathbf e_{23} + h_x \mathbf e_{31} + h_x \mathbf e_{12}\) | Line \(h_x \mathbf e_{23} + h_y \mathbf e_{31} + h_z \mathbf e_{12}\) | Point in horizon \(h_x \mathbf e_1 + h_y \mathbf e_2\) |
| Complement rotation \(\mathcal Q = c_x \mathbf e_{23} + c_y \mathbf e_{31} + c_z \mathbf e_{12} - r,\) | Line \(c_x \mathbf e_{23} + c_y \mathbf e_{31} + c_z \mathbf e_{12}\) | Origin point \(\mathbf e_3\) |
| Complement transflection \(\mathcal F = h_x \mathbf e_1 + h_y \mathbf e_2 + h_z \mathbf e_3 - s \unicode{x1D7D9}\) | Point \(h_x \mathbf e_1 + h_y \mathbf e_2 + h_z \mathbf e_3\) | Line through origin \(h_x \mathbf e_{23} + h_y \mathbf e_{31}\) |
Table 1. These are the invariants of transforms occurring in the 3D projective geometric algebra representing the 2D plane. The primary invariant of any regulartransform (a Euclidean isometry) or complement transform is given by the vector or bivector components of the operator itself. The coinvariant is given by the weight dual of the primary invariantin the case of regular transforms and by the bulk dual of the primary invariant in the case of complement transforms.
In the four-dimensional projective geometric algebra \(\mathbb R_{3,0,1},\) representing three-dimensional space, every Euclidean isometry is either a screw transform Qor a rotoreflection F. (The only geometrical difference between these is that the displacement along the rotation axis in a screw transform is replaced by a reflection in a plane perpendicularto the rotation axis in a rotoreflection.) The primary invariant of a screw transform is its bivector components, which corresponds to the line about which a rotation is taking place.A rotoreflection can have two primary invariants, one associated with its vector components and a second associated with its trivector components. These invariants, the invariants of thecorresponding complement transforms, and the coinvariants for each are summarized in Table 2.
| Transform | Primary Invariants | Coinvariants |
|---|---|---|
| Regular screw transform \(\mathbf Q = r_x \mathbf e_{41} + r_y \mathbf e_{42} + r_z \mathbf e_{43} + r_w \unicode{x1D7D9}\) \(+\,u_x \mathbf e_{23} + u_y \mathbf e_{31} + u_z \mathbf e_{12} + u_w\) | Line \(r_x \mathbf e_{41} + r_y \mathbf e_{42} + r_z \mathbf e_{43} + u_x \mathbf e_{23} + u_y \mathbf e_{31} + u_z \mathbf e_{12}\) | Line in horizon \(r_x \mathbf e_{23} + r_y \mathbf e_{31} + r_z \mathbf e_{12}\) |
| Regular rotoreflection \(\mathbf F = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 + s_w \mathbf e_4\) \(+\,h_x \mathbf e_{423} + h_y \mathbf e_{431} + h_z \mathbf e_{412} + h_w \mathbf e_{321}\) | Plane \(h_x \mathbf e_{423} + h_y \mathbf e_{431} + h_z \mathbf e_{412} + h_w \mathbf e_{321},\) Point \(s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 + s_w \mathbf e_4\) | Point in horizon \(h_x \mathbf e_1 + h_y \mathbf e_2 + h_z \mathbf e_3,\) Horizon \(\mathbf e_{321}\) |
| Complement screw transform \(\mathcal Q = u_x \mathbf e_{41} + u_y \mathbf e_{42} + u_z \mathbf e_{43} - u_w \unicode{x1D7D9}\) \(+\,r_x \mathbf e_{23} + r_y \mathbf e_{31} + r_z \mathbf e_{12} - r_w\) | Line \(u_x \mathbf e_{41} + u_y \mathbf e_{42} + u_z \mathbf e_{43} + r_x \mathbf e_{23} + r_y \mathbf e_{31} + r_z \mathbf e_{12}\) | Line through origin \(r_x \mathbf e_{41} + r_y \mathbf e_{42} + r_z \mathbf e_{43}\) |
| Complement rotoreflection \(\mathcal F = h_x \mathbf e_1 + h_y \mathbf e_2 + h_z \mathbf e_3 + h_w \mathbf e_4\) \(+\,s_x \mathbf e_{423} + s_y \mathbf e_{431} + s_z \mathbf e_{412} + s_w \mathbf e_{321}\) | Point \(h_x \mathbf e_1 + h_y \mathbf e_2 + h_z \mathbf e_3 + h_w \mathbf e_4,\) Plane \(s_x \mathbf e_{423} + s_y \mathbf e_{431} + s_z \mathbf e_{412} + s_w \mathbf e_{321}\) | Plane through origin \(h_x \mathbf e_{423} + h_y \mathbf e_{431} + h_z \mathbf e_{412},\) Origin \(\mathbf e_4\) |
Table 2. These are the invariants of transforms occurring in the 4D projective geometric algebra representing 3D space. The primary invariant of any regular transform(a Euclidean isometry) or complement transform is given by the vector, bivector, or trivector components of the operator itself. The coinvariants are given by the weight dual of each primary invariantin the case of regular transforms and by the bulk dual of each primary invariant in the case of complement transforms.
The unit-weight invertible elements of the \((n + 1)\)-dimensional projective geometric algebra \(\mathbb R_{n,0,1}\) constitute a double cover of both the groups \(\mathrm E(n)\) and \(\overline{\mathrm E}(n).\)The geometric product corresponds to transform composition in the group \(\overline{\mathrm E}(n),\) and the geometric antiproduct corresponds to transform composition in the group \(\mathrm E(n).\) Regular reflections acrossplanes are represented by antivectors (having antigrade one), and they meet at lower-dimensional invariants under the geometric antiproduct. Symmetrically, complement reflections across points arerepresented by vectors (having grade one), and they join at higher-dimensional invariants under the geometric product. A sandwich product
\(\mathcal Q \mathbin{\unicode{x27D1}} \mathbf a \mathbin{\unicode{x27D1}} \tilde{\mathcal Q}\)
transforms the space of a with an element of \(\overline{\mathrm E}(n),\) and it transforms the antispace of a with the complementary element of \(\mathrm E(n).\) Symmetrically, a sandwich product
\(\mathbf Q \mathbin{\unicode{x27C7}} \mathbf a \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}\)
transforms the space of a with an element of \(\mathrm E(n),\) and it transforms the antispace of a with the complementary element of \(\overline{\mathrm E}(n).\)
The groups \(\mathrm E(n)\) and \(\overline{\mathrm E}(n)\) have a number of subgroups, and the hierarchical relationships among them are shown in Figure 5. In particular, the Euclidean group \(\mathrm E(n)\) contains the specialEuclidean subgroup \(\mathrm{SE}(n)\) consisting of all combinations of regular rotations, which are covered by the antigrade 2 elements of \(\mathbb R_{n,0,1}.\) Correspondingly, the complementEuclidean group \(\overline{\mathrm E}(n)\) contains the special complement Euclidean subgroup \(\mathrm{S\overline{E}}(n)\) consisting of all combinations of reciprocal rotations, which are covered by the grade 2 elements of\(\mathbb R_{n,0,1}.\) The subgroups \(\mathrm{SE}(n)\) and \(\mathrm{S\overline{E}}(n)\) further contain translation subgroups \(\mathrm T(n)\) and \(\overline{\mathrm T}(n),\) respectively.
Figure 5. The unit-weight invertible elements of the projective geometric algebra \(\mathbb R_{n,0,1}\) are a double cover of both the Euclidean group \(\mathrm E(n)\)and the complement Euclidean group \(\overline{\mathrm E}(n).\) Elements corresponding to transforms in \(\mathrm E(n)\) are composed with the geometric antiproduct, and elements corresponding to transforms in \(\overline{\mathrm E}(n)\)are composed with the geometric product. Transforms belonging to the common subgroup \(\mathrm O(n)\) have two representations in \(\mathbb R_{n,0,1},\) one associated with the geometric product,and one associated with the geometric antiproduct.
Transforms about invariants containing the origin are the same in both \(\mathrm E(n)\) and \(\overline{\mathrm E}(n),\) and they constitute the common subgroup \(\mathrm O(n).\) Every member of \(\mathrm O(n)\) has a representation in\(\mathbb R_{n,0,1}\) that transforms elements with the geometric product and a complementary representation that transforms elements with the geometric antiproduct. For example, in\(\mathbb R_{3,0,1},\) a conventional quaternion q can be expressed as
\(\mathbf q = x \mathbf e_{23} + x \mathbf e_{31} + x \mathbf e_{12} - w \mathbf 1,\)
which covariantly transforms any object a with the sandwich product \(\mathbf q \mathbin{\unicode{x27D1}} \mathbf a \mathbin{\unicode{x27D1}} \mathbf{\tilde q},\) and it can be expressed as
\(\mathbf q = x \mathbf e_{41} + y \mathbf e_{42} + z \mathbf e_{43} + w \unicode{x1D7D9}\)
which covariantly transforms any object a with the sandwich product \(\mathbf q \mathbin{\unicode{x27C7}} \mathbf a \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{q}}}.\)
In terms of matrix multiplication, a general element of the group \(\mathrm E(n)\) transforms a point by multiplying on the left by an \((n + 1) \times (n + 1)\) matrix of the form
\(\begin{bmatrix}\mathbf M_{n \times n} & \boldsymbol \tau_{n \times 1} \\ \mathbf 0_{1 \times n} & 1\end{bmatrix}\)
where the \(n \times n\) submatrix M is orthogonal. A general element of the corresponding group \(\overline{\mathrm E}(n)\) transforms points with matrices of the form
\(\begin{bmatrix}\mathbf M_{n \times n} & \mathbf 0_{n \times 1} \\ \boldsymbol \tau_{1 \times n} & 1\end{bmatrix}\)
In the special subgroups of \(\mathrm E(n)\) and \(\overline{\mathrm E}(n),\) the submatrix M has a determinant of \(+1\). In the translation subgroups \(\mathrm T(n)\) and \(\overline{\mathrm T}(n),\) M is the identity matrix.Finally, when \(\boldsymbol \tau = \mathbf 0,\) both matrices have the same form and belong to \(\mathrm O(n).\)
The isomorphic mapping between \(\mathrm E(n)\) and \(\overline{\mathrm E}(n)\) is given by the inverse transpose operation on the matrix representatives. That is, if H is an \((n + 1) \times (n + 1)\)matrix representing an element of \(\mathrm E(n),\) then the corresponding element of \(\overline{\mathrm E}(n)\) is given by \((\mathbf H^{-1})^{\mathrm T}.\) Of course, this operation is an involution,and the mapping works both ways.
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