Inversions and Space-Antispace Transform Correspondence in Projective Geometric Algebra: Difference between pages

From Rigid Geometric Algebra
(Difference between pages)
Jump to navigation Jump to search
(Redirected page to Inversion)
Tag: New redirect
 
No edit summary
 
Line 1: Line 1:
#REDIRECT [[Inversion]]
'''By Eric Lengyel'''<br />
May 20, 2022
 
[[Image:Antispace1.svg|480px|thumb|right|'''Figure 1.''' The coordinates $$(p_x, p_y, p_z)$$ can be interpreted as the one-dimensional span of a single vector representing a homogeneous point or as the $$(n - 1)$$-dimensional span of all orthogonal vectors representing a homogeneous hyperplane, which is a line when $$n = 3$$. Geometrically, these two interpretations are dual to each other, and their distances to the origin are reciprocals of each other.]]
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 that the 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 homogeneous 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 projection plane $$z = 1$$. 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 are orthogonal to the single direction $$(p_x, p_y, p_z)$$. As illustrated by the figure, these directions span an $$(n - 1)$$-dimensional subspace 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 as both a point and a line, and they are ''duals'' 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 the coordinates 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 the two 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 ''space'' of 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 and the 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 line exists 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-dimensional counterparts, 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 transforms the 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 this necessitates 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 in antispace 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 dual 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.
 
[[Image:Antispace2.svg|640px|thumb|right|'''Figure 2.''' (Left) A regular rotation fixes the green center point at $$(1, 0)$$ and the horizon. (Right) The corresponding dual rotation fixes the green line at $$x = -1$$ dual to the center point and the origin.]]
Figure 2 shows a two-dimensional rotation transform and its dual analog. The green point represents the center of rotation, and the green line is the dual of that point. Under the regular rotation, the center point is fixed, and under the dual rotation, the line dual to the center point is fixed. These are not, however, the only fixed geometries. A regular rotation also fixes the horizon line, and thus its dual, the origin, must be fixed in the dual 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.
 
[[Image:Antispace3.svg|640px|thumb|right|'''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) The corresponding dual reflection fixes the green point $$(2, 0)$$ dual to the mirroring line and the line through the origin parallel to the mirroring line.]]
Figure 3 shows a two-dimensional reflection transform and its dual 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 dual reflection, the point dual to the mirroring line is fixed. As with rotation, there are additional fixed geometries under 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 and containing the origin that remains fixed by the dual reflection. This is illustrated by the red line in the figure, which is clearly a reflection boundary in a sense.
 
[[Image:Antispace4.svg|640px|thumb|right|'''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 dual translation (a perspective projection) fixes the line parallel to the direction of translation through the origin and every point in the line through the origin perpendicular to the direction of translation.]]
Finally, Figure 4 shows a two-dimensional translation transform and its dual 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 fixed by the dual translation, and that is illustrated by the red line in the figure. Dual translation is especially important because it is the one to which we can most easily assign some practical meaning. It is a ''perspective projection'' onto the line through the origin perpendicular to the direction of translation.
 
In the three-dimensional projective geometric algebra $$\mathbb R_{2,0,1}$$, a homogeneous representation of two-dimensional space, a regular rotation about a center point $$\mathbf c$$ is given by
 
:$$\mathbf Q = c_x \mathbf e_1 + c_y \mathbf e_2 + c_z \mathbf e_3 + r {\large\unicode{x1D7D9}}$$,
 
and this becomes a translation when $$c_z = 0$$. A regular transflection across the line $$\mathbf h$$ is given by
 
:$$\mathbf F = s + h_x \mathbf e_{23} + h_y \mathbf e_{31} + h_z \mathbf e_{12}$$,
 
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 $$\unicode{x27C7}$$, arbitrary products of these operators form the group $$\mathrm E(2)$$ with $${\large\unicode{x1D7D9}}$$ as the identity, and they covariantly transform any object $$a$$ in the algebra through the sandwich products
 
:$$a' = \mathbf Q \mathbin{\unicode{x27C7}} a \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}$$
 
and
 
:$$a' = \mathbf F \mathbin{\unicode{x27C7}} a \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}}$$.
 
Symmetrically, a dual rotation about the line $$\mathbf c$$ is given by
 
:$$\mathcal Q = c_x \mathbf e_{23} + c_y \mathbf e_{31} + c_z \mathbf e_{12} - r$$,
 
and a dual transflection across the point $$\mathbf h$$ is given by
 
:$$\mathcal F = h_x \mathbf e_1 + h_y \mathbf e_2 + h_z \mathbf e_3 - s {\large\unicode{x1D7D9}}$$.
 
These two operators generate a different group of transformations under the geometric product $$\unicode{x27D1}$$. Arbitrary products of these operators form the dual Euclidean group $$\mathrm{RE}(2)$$ with $$\mathbf 1$$ as the identity, and they covariantly transform any object $$a$$ in the algebra through the sandwich products
 
:$$a' = \mathcal Q \mathbin{\unicode{x27D1}} a \mathbin{\unicode{x27D1}} \mathcal{\tilde Q}$$
 
and
 
:$$a' = \mathcal F \mathbin{\unicode{x27D1}} a \mathbin{\unicode{x27D1}} \mathcal{\tilde F}$$
 
The groups $$\mathrm E(2)$$ and $$\mathrm{RE}(2)$$ 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 transforms associated with members of $$\mathrm E(2)$$ and $$\mathrm{RE}(2)$$.
 
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 the corresponding dual transforms always fix a coinvariant containing the origin. In general, if $$x$$ is the primary invariant of a Euclidean isometry, then the complement of the weight of $$x$$ gives the coinvariant. Symmetrically, if $$x$$ is the primary invariant of a dual transform, then the complement of the bulk of $$x$$ gives the coinvariant. When the primary invariant of a Euclidean isometry contains the origin, there is a corresponding dual transform that performs the same operation. Symmetrically, when the primary invariant of a projective transform is contained in the horizon, there is a corresponding Euclidean isometry that performs the same operation. These are where $$\mathrm E(2)$$ and $$\mathrm{RE}(2)$$ intersect at $$\mathrm O(2)$$.
 
{| class="wikitable"
|+ style="caption-side: bottom; padding-top: 0.5em; text-align: left; font-size: 92%; font-weight: normal;" | '''Table 1.''' These are the invariants of transforms occurring in the 3D projective geometric algebra representing the 2D plane. The primary invariant of any regular transform (a Euclidean isometry) or dual transform is given by the vector or bivector components of the operator itself. The coinvariant is given by the weight complement of the primary invariant in the case of regular transforms and by the bulk complement of the primary invariant in the case of dual transforms.
|-
! Transform !! Primary Invariant !! Coinvariant
|-
| style="padding-left: 1em; padding-right: 1em;" | Regular rotation $$\mathbf Q = c_x \mathbf e_1 + c_y \mathbf e_2 + c_z \mathbf e_3 + r {\large\unicode{x1D7D9}}$$
| style="padding-left: 1em; padding-right: 1em;" | Point $$c_x \mathbf e_1 + c_y \mathbf e_2 + c_z \mathbf e_3$$
| style="padding-left: 1em; padding-right: 1em;" | Horizon line $$\mathbf e_{12}$$
|-
| style="padding-left: 1em; padding-right: 1em;" | Regular transflection $$\mathbf F = s + h_x \mathbf e_{23} + h_y \mathbf e_{31} + h_z \mathbf e_{12}$$
| style="padding-left: 1em; padding-right: 1em;" | Line $$h_x \mathbf e_{23} + h_y \mathbf e_{31} + h_z \mathbf e_{12}$$
| style="padding-left: 1em; padding-right: 1em;" | Point in horizon $$h_x \mathbf e_1 + h_y \mathbf e_2$$
|-
| style="padding-left: 1em; padding-right: 1em;" | Dual rotation $$\mathcal Q = c_x \mathbf e_{23} + c_y \mathbf e_{31} + c_z \mathbf e_{12} - r$$
| style="padding-left: 1em; padding-right: 1em;" | Line $$c_x \mathbf e_{23} + c_y \mathbf e_{31} + c_z \mathbf e_{12}$$
| style="padding-left: 1em; padding-right: 1em;" | Origin point $$\mathbf e_3$$
|-
| style="padding-left: 1em; padding-right: 1em;" | Dual transflection $$\mathcal F = h_x \mathbf e_1 + h_y \mathbf e_2 + h_z \mathbf e_3 - s {\large\unicode{x1D7D9}}$$
| style="padding-left: 1em; padding-right: 1em;" | Point $$h_x \mathbf e_1 + h_y \mathbf e_2 + h_z \mathbf e_3$$
| style="padding-left: 1em; padding-right: 1em;" | Line through origin $$h_x \mathbf e_{23} + h_y \mathbf e_{31}$$
|}

Revision as of 04:38, 5 August 2023

By Eric Lengyel
May 20, 2022

Figure 1. The coordinates $$(p_x, p_y, p_z)$$ can be interpreted as the one-dimensional span of a single vector representing a homogeneous point or as the $$(n - 1)$$-dimensional span of all orthogonal vectors representing a homogeneous hyperplane, which is a line when $$n = 3$$. Geometrically, these two interpretations are dual to each other, and their distances to the origin are reciprocals of each other.

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 that the 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 homogeneous 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 projection plane $$z = 1$$. 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 are orthogonal to the single direction $$(p_x, p_y, p_z)$$. As illustrated by the figure, these directions span an $$(n - 1)$$-dimensional subspace 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 as both a point and a line, and they are duals 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 the coordinates 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 the two 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 space of 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 and the 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 line exists 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-dimensional counterparts, 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 transforms the 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 this necessitates 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 in antispace 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 dual 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. (Left) A regular rotation fixes the green center point at $$(1, 0)$$ and the horizon. (Right) The corresponding dual rotation fixes the green line at $$x = -1$$ dual to the center point and the origin.

Figure 2 shows a two-dimensional rotation transform and its dual analog. The green point represents the center of rotation, and the green line is the dual of that point. Under the regular rotation, the center point is fixed, and under the dual rotation, the line dual to the center point is fixed. These are not, however, the only fixed geometries. A regular rotation also fixes the horizon line, and thus its dual, the origin, must be fixed in the dual 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 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) The corresponding dual reflection fixes the green point $$(2, 0)$$ dual to the mirroring line and the line through the origin parallel to the mirroring line.

Figure 3 shows a two-dimensional reflection transform and its dual 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 dual reflection, the point dual to the mirroring line is fixed. As with rotation, there are additional fixed geometries under 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 and containing the origin that remains fixed by the dual reflection. This is illustrated by the red line in the figure, which is clearly a reflection boundary in a sense.

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 dual translation (a perspective projection) fixes the line parallel to the direction of translation through the origin and every point in the line through the origin perpendicular to the direction of translation.

Finally, Figure 4 shows a two-dimensional translation transform and its dual 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 fixed by the dual translation, and that is illustrated by the red line in the figure. Dual translation is especially important because it is the one to which we can most easily assign some practical meaning. It is a perspective projection onto the line through the origin perpendicular to the direction of translation.

In the three-dimensional projective geometric algebra $$\mathbb R_{2,0,1}$$, a homogeneous representation of two-dimensional space, a regular rotation about a center point $$\mathbf c$$ is given by

$$\mathbf Q = c_x \mathbf e_1 + c_y \mathbf e_2 + c_z \mathbf e_3 + r {\large\unicode{x1D7D9}}$$,

and this becomes a translation when $$c_z = 0$$. A regular transflection across the line $$\mathbf h$$ is given by

$$\mathbf F = s + h_x \mathbf e_{23} + h_y \mathbf e_{31} + h_z \mathbf e_{12}$$,

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 $$\unicode{x27C7}$$, arbitrary products of these operators form the group $$\mathrm E(2)$$ with $${\large\unicode{x1D7D9}}$$ as the identity, and they covariantly transform any object $$a$$ in the algebra through the sandwich products

$$a' = \mathbf Q \mathbin{\unicode{x27C7}} a \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}$$

and

$$a' = \mathbf F \mathbin{\unicode{x27C7}} a \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}}$$.

Symmetrically, a dual rotation about the line $$\mathbf c$$ is given by

$$\mathcal Q = c_x \mathbf e_{23} + c_y \mathbf e_{31} + c_z \mathbf e_{12} - r$$,

and a dual transflection across the point $$\mathbf h$$ is given by

$$\mathcal F = h_x \mathbf e_1 + h_y \mathbf e_2 + h_z \mathbf e_3 - s {\large\unicode{x1D7D9}}$$.

These two operators generate a different group of transformations under the geometric product $$\unicode{x27D1}$$. Arbitrary products of these operators form the dual Euclidean group $$\mathrm{RE}(2)$$ with $$\mathbf 1$$ as the identity, and they covariantly transform any object $$a$$ in the algebra through the sandwich products

$$a' = \mathcal Q \mathbin{\unicode{x27D1}} a \mathbin{\unicode{x27D1}} \mathcal{\tilde Q}$$

and

$$a' = \mathcal F \mathbin{\unicode{x27D1}} a \mathbin{\unicode{x27D1}} \mathcal{\tilde F}$$

The groups $$\mathrm E(2)$$ and $$\mathrm{RE}(2)$$ 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 transforms associated with members of $$\mathrm E(2)$$ and $$\mathrm{RE}(2)$$.

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 the corresponding dual transforms always fix a coinvariant containing the origin. In general, if $$x$$ is the primary invariant of a Euclidean isometry, then the complement of the weight of $$x$$ gives the coinvariant. Symmetrically, if $$x$$ is the primary invariant of a dual transform, then the complement of the bulk of $$x$$ gives the coinvariant. When the primary invariant of a Euclidean isometry contains the origin, there is a corresponding dual transform that performs the same operation. Symmetrically, when the primary invariant of a projective transform is contained in the horizon, there is a corresponding Euclidean isometry that performs the same operation. These are where $$\mathrm E(2)$$ and $$\mathrm{RE}(2)$$ intersect at $$\mathrm O(2)$$.

Table 1. These are the invariants of transforms occurring in the 3D projective geometric algebra representing the 2D plane. The primary invariant of any regular transform (a Euclidean isometry) or dual transform is given by the vector or bivector components of the operator itself. The coinvariant is given by the weight complement of the primary invariant in the case of regular transforms and by the bulk complement of the primary invariant in the case of dual transforms.
Transform Primary Invariant Coinvariant
Regular rotation $$\mathbf Q = c_x \mathbf e_1 + c_y \mathbf e_2 + c_z \mathbf e_3 + r {\large\unicode{x1D7D9}}$$ Point $$c_x \mathbf e_1 + c_y \mathbf e_2 + c_z \mathbf e_3$$ Horizon line $$\mathbf e_{12}$$
Regular transflection $$\mathbf F = s + h_x \mathbf e_{23} + h_y \mathbf e_{31} + h_z \mathbf e_{12}$$ Line $$h_x \mathbf e_{23} + h_y \mathbf e_{31} + h_z \mathbf e_{12}$$ Point in horizon $$h_x \mathbf e_1 + h_y \mathbf e_2$$
Dual 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 $$\mathbf e_3$$
Dual transflection $$\mathcal F = h_x \mathbf e_1 + h_y \mathbf e_2 + h_z \mathbf e_3 - s {\large\unicode{x1D7D9}}$$ Point $$h_x \mathbf e_1 + h_y \mathbf e_2 + h_z \mathbf e_3$$ Line through origin $$h_x \mathbf e_{23} + h_y \mathbf e_{31}$$