Difference between pages "Duality" and "Main Page"

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[[Image:Duality.svg|480px|thumb|right|'''Figure 1.''' The coordinates $$(p_x, p_y, p_z, p_w)$$ 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 plane. Geometrically, these two interpretations are dual to each other, and their distances to the origin are reciprocals of each other.]]
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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. The example shown in '''Figure 1''' demonstrates the duality between homogeneous points and planes in a four-dimensional projective space. The quadruplet of coordinates $$(p_x, p_y, p_z, p_w)$$ can be interpreted as a vector pointing from the origin toward a specific location on the projection subspace $$w = 1$$. 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 $$(p_x, p_y, p_z, p_w)$$. As illustrated by the figure, these directions span an $$(n - 1)$$-dimensional subspace that intersects the projection subspace at a plane when $$n = 4$$. In this way, the coordinates $$(p_x, p_y, p_z, p_w)$$ can be interpreted as both a point and a plane, and they are duals of each other.
== Rigid Geometric Algebra ==


When we express the coordinates $$(p_x, p_y, p_z, p_w)$$ on the [[vector]] basis as $$p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$, 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 [[antivector]] basis as $$p_x \mathbf e_{234} + p_y \mathbf e_{314} + p_z \mathbf e_{124} + p_w \mathbf e_{321}$$, then we are working with the three orthogonal spatial dimensions representing a plane. 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 $$p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ is never only a point, but both a point and a plane simultaneously, where the point exists in space, and the plane exists in antispace. Likewise, an antivector $$p_x \mathbf e_{234} + p_y \mathbf e_{314} + p_z \mathbf e_{124} + p_w \mathbf e_{321}$$ is never only a plane, but both a plane and a point simultaneously, where the plane exists in space, and the point exists in antispace. If we study only the spatial facet of these objects, then we are missing half of a bigger picture.
This wiki is a repository of information about Rigid Geometric Algebra, and specifically the four-dimensional Clifford algebra $$\mathcal G_{3,0,1}$$. This is a mathematical model that naturally incorporates representations for Euclidean [[points]], [[lines]], and [[planes]] in 3D space as well as operations for performing [[rotations]], [[reflections]], and [[translations]] in a single algebraic structure. It completely subsumes conventional models that include homogeneous coordinates, Plücker coordinates, [[quaternions]], and screw theory (which makes use of dual quaternions). This makes rigid geometric algebra a natural fit for areas of computer science that routinely use these mathematical concepts, especially computer graphics and robotics.


== See Also ==
Rigid geometric algebra is an area of active research, and new information is frequently being added to this wiki.


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== Introduction ==
[[Image:basis.svg|thumb|right|400px|'''Table 1.''' The 16 basis elements of the 4D rigid geometric algebra.]]
In the four-dimensional rigid geometric algebra, there are 16 graded basis elements. These are listed in Table 1.
There is a single ''[[scalar]]'' basis element that we denote by $$\mathbf 1$$, in bold, and its multiples correspond to the real numbers, which are values that have no dimensions.
There are four ''[[vector]]'' basis elements named $$\mathbf e_1$$, $$\mathbf e_2$$, $$\mathbf e_3$$, and $$\mathbf e_4$$ that have one-dimensional extents. A general vector $$\mathbf v = (v_x, v_y, v_z, v_w)$$ has the form
:$$\mathbf v = v_x \mathbf e_1 + v_y \mathbf e_2 + v_z \mathbf e_3 + v_w \mathbf e_4$$ .
There are six ''[[bivector]]'' basis elements named $$\mathbf e_{41}$$, $$\mathbf e_{42}$$, $$\mathbf e_{43}$$, $$\mathbf e_{23}$$, $$\mathbf e_{31}$$, and $$\mathbf e_{12}$$ having two-dimensional extents. These correspond to all possible [[wedge products]] between pairs of vector basis elements up to order. We use the multiple subscript notation $$\mathbf e_{ij}$$ as shorthand for the wedge product $$\mathbf e_i \wedge \mathbf e_j$$. Numerical subscripts for the bivector basis elements are always written in the order shown in Table 1, and the bivectors are negated when basis vectors are multiplied in the opposite order. For example, $$\mathbf e_3 \wedge \mathbf e_2 = -\mathbf e_{23}$$.
There are four ''[[trivector]]'' basis elements named $$\mathbf e_{234}$$, $$\mathbf e_{314}$$, $$\mathbf e_{124}$$, and $$\mathbf e_{321}$$ having three-dimensional extents. These correspond to all possible wedge products of three different vector basis elements. Again, numerical subscripts will always be written exactly as shown in the table, and negation will be applied for any odd permutation of the multiplication order.
Finally, there is a single ''quadrivector'' basis element $$\mathbf e_1 \wedge \mathbf e_2 \wedge \mathbf e_3 \wedge \mathbf e_4$$ having four-dimensional extents. Because the quadrivector element has only one component, it is often called the ''pseudoscalar'', and it is often denoted by $$\mathbf I_4$$. The subscript 4 corresponds to the number of dimensions, and it is usually dropped when the dimensionality is clear from the context. Because the quadrivector contains all four dimensions, it is also called the ''volume element'' of the algebra, and this is often denoted by $$\mathbf E_4$$. We use the notation
:$${\large\unicode{x1D7D9}} = \mathbf e_1 \wedge \mathbf e_2 \wedge \mathbf e_3 \wedge \mathbf e_4$$ ,
with a blackboard bold $${\large\unicode{x1D7D9}}$$, to emphasize that the volume element is in symmetric opposition to the scalar basis element $$\mathbf 1$$ and is equally functional within the algebra. We refer to multiples of the basis element $${\large\unicode{x1D7D9}}$$ as ''[[antiscalars]]''. Scalars and antiscalars are two sides of the same coin, and neither has a place of greater importance. We eschew the term pseudoscalar due to its portrayal of the element $${\large\unicode{x1D7D9}}$$ as different from and perhaps somewhat less significant than the element $$\mathbf 1$$. It is not.
As shown in the rightmost column in the table, each of the basis elements can be identified by which specific multiplicative combination of the four available dimensions it represents. This is essentially a four-bit code in which black bars correspond to the dimensions that are present or ''full'', and white bars correspond to the dimensions that are absent or ''empty''. The ''[[grade]]'' of a basis element $$\mathbf a$$, denoted by $$\operatorname{gr}(\mathbf a)$$, is the number of black bars it has, which is the same as the number of vector basis elements in its factorization.
For a thorough understanding of the algebraic structure, it is critically important to recognize that there is a fundamental symmetry at work. We have assigned a dimensionality to each basis element according to the number of full dimensions it has, but it is equally valid to assign a dimensionality according to the number of empty dimensions each one has. Vectors, bivectors, and trivectors have dimensions one, two, and three when we count the black bars. However, from the opposite perspective, vectors, bivectors, and trivectors have dimensions three, two, and one when we count the white bars. Both of these interpretations are simultaneously correct, and together they establish the concept of ''[[duality]]''. [[Duality]] is always present, and it pervades geometric algebra. It can be found not only in the elements of the algebra but in the operations that act on those elements.
In addition to the grade, we can assign an ''[[antigrade]]'' to each basis element $$\mathbf a$$. Denoted by $$\operatorname{ag}(\mathbf a)$$, the antigrade of $$\mathbf a$$ is the number of vector basis elements missing from its factorization, which is the number of white bars in the table. Of course, it is always the case that
:$$\operatorname{gr}(\mathbf a) + \operatorname{ag}(\mathbf a) = n$$ ,
where $$n$$ is the total number of dimensions in the algebra. Whenever we can make a statement about how an operation relates to the grade of its inputs and outputs, we can make the same statement about how the dual operation relates to the antigrade of its inputs and outputs.
In an $$n$$-dimensional algebra, the elements with grade $$n - 1$$ are called ''[[antivectors]]''. Antivectors have the same number of components as vectors, and the two can be regarded as the dimensional inverses of each other. Vectors have grade one because they have one full dimension, and antivectors have antigrade one because they have one empty dimension.
== Pages ==
{| style="width: 50%; border: 0px;"
| style="width: 50%; vertical-align: top; padding-right: 10%;" |
=== The five main types of rigid geometric objects ===
* [[Point]]
* [[Line]]
* [[Plane]]
* [[Motor]]
* [[Flector]]
=== Various properties and unary operations ===
* [[Grade and antigrade]]
* [[Bulk and weight]]
* [[Complements]]
* [[Complements]]
* [[Reverses]]
* [[Geometric norm]]
* [[Geometric property]]
* [[Unitization]]
| style="width: 50%; vertical-align: top; padding-right: 10%;" |
=== Products and other binary operations ===
* [[Geometric products]]
* [[Exterior products]]
* [[Interior products]]
* [[Dot products]]
* [[Join and meet]]
* [[Projections]]
* [[Commutators]]
* [[Euclidean distance]]
=== Isometries of 3D space ===
* [[Translation]]
* [[Rotation]]
* [[Reflection]]
* [[Inversion]]
* [[Transflection]]
|}
== Two-Dimensional Geometry ==
The three-dimensional algebra $$\mathcal G_{2,0,1}$$, which deals with points, lines, and isometries in 2D space, is discussed on the following page.
* [[Rigid Geometric Algebra for 2D Space]]

Latest revision as of 04:20, 17 June 2022

Rigid Geometric Algebra

This wiki is a repository of information about Rigid Geometric Algebra, and specifically the four-dimensional Clifford algebra $$\mathcal G_{3,0,1}$$. This is a mathematical model that naturally incorporates representations for Euclidean points, lines, and planes in 3D space as well as operations for performing rotations, reflections, and translations in a single algebraic structure. It completely subsumes conventional models that include homogeneous coordinates, Plücker coordinates, quaternions, and screw theory (which makes use of dual quaternions). This makes rigid geometric algebra a natural fit for areas of computer science that routinely use these mathematical concepts, especially computer graphics and robotics.

Rigid geometric algebra is an area of active research, and new information is frequently being added to this wiki.

If you are experiencing problems with the LaTeX on this site, please clear the cookies for rigidgeometricalgebra.org and reload.

Introduction

Table 1. The 16 basis elements of the 4D rigid geometric algebra.

In the four-dimensional rigid geometric algebra, there are 16 graded basis elements. These are listed in Table 1.

There is a single scalar basis element that we denote by $$\mathbf 1$$, in bold, and its multiples correspond to the real numbers, which are values that have no dimensions.

There are four vector basis elements named $$\mathbf e_1$$, $$\mathbf e_2$$, $$\mathbf e_3$$, and $$\mathbf e_4$$ that have one-dimensional extents. A general vector $$\mathbf v = (v_x, v_y, v_z, v_w)$$ has the form

$$\mathbf v = v_x \mathbf e_1 + v_y \mathbf e_2 + v_z \mathbf e_3 + v_w \mathbf e_4$$ .

There are six bivector basis elements named $$\mathbf e_{41}$$, $$\mathbf e_{42}$$, $$\mathbf e_{43}$$, $$\mathbf e_{23}$$, $$\mathbf e_{31}$$, and $$\mathbf e_{12}$$ having two-dimensional extents. These correspond to all possible wedge products between pairs of vector basis elements up to order. We use the multiple subscript notation $$\mathbf e_{ij}$$ as shorthand for the wedge product $$\mathbf e_i \wedge \mathbf e_j$$. Numerical subscripts for the bivector basis elements are always written in the order shown in Table 1, and the bivectors are negated when basis vectors are multiplied in the opposite order. For example, $$\mathbf e_3 \wedge \mathbf e_2 = -\mathbf e_{23}$$.

There are four trivector basis elements named $$\mathbf e_{234}$$, $$\mathbf e_{314}$$, $$\mathbf e_{124}$$, and $$\mathbf e_{321}$$ having three-dimensional extents. These correspond to all possible wedge products of three different vector basis elements. Again, numerical subscripts will always be written exactly as shown in the table, and negation will be applied for any odd permutation of the multiplication order.

Finally, there is a single quadrivector basis element $$\mathbf e_1 \wedge \mathbf e_2 \wedge \mathbf e_3 \wedge \mathbf e_4$$ having four-dimensional extents. Because the quadrivector element has only one component, it is often called the pseudoscalar, and it is often denoted by $$\mathbf I_4$$. The subscript 4 corresponds to the number of dimensions, and it is usually dropped when the dimensionality is clear from the context. Because the quadrivector contains all four dimensions, it is also called the volume element of the algebra, and this is often denoted by $$\mathbf E_4$$. We use the notation

$${\large\unicode{x1D7D9}} = \mathbf e_1 \wedge \mathbf e_2 \wedge \mathbf e_3 \wedge \mathbf e_4$$ ,

with a blackboard bold $${\large\unicode{x1D7D9}}$$, to emphasize that the volume element is in symmetric opposition to the scalar basis element $$\mathbf 1$$ and is equally functional within the algebra. We refer to multiples of the basis element $${\large\unicode{x1D7D9}}$$ as antiscalars. Scalars and antiscalars are two sides of the same coin, and neither has a place of greater importance. We eschew the term pseudoscalar due to its portrayal of the element $${\large\unicode{x1D7D9}}$$ as different from and perhaps somewhat less significant than the element $$\mathbf 1$$. It is not.

As shown in the rightmost column in the table, each of the basis elements can be identified by which specific multiplicative combination of the four available dimensions it represents. This is essentially a four-bit code in which black bars correspond to the dimensions that are present or full, and white bars correspond to the dimensions that are absent or empty. The grade of a basis element $$\mathbf a$$, denoted by $$\operatorname{gr}(\mathbf a)$$, is the number of black bars it has, which is the same as the number of vector basis elements in its factorization.

For a thorough understanding of the algebraic structure, it is critically important to recognize that there is a fundamental symmetry at work. We have assigned a dimensionality to each basis element according to the number of full dimensions it has, but it is equally valid to assign a dimensionality according to the number of empty dimensions each one has. Vectors, bivectors, and trivectors have dimensions one, two, and three when we count the black bars. However, from the opposite perspective, vectors, bivectors, and trivectors have dimensions three, two, and one when we count the white bars. Both of these interpretations are simultaneously correct, and together they establish the concept of duality. Duality is always present, and it pervades geometric algebra. It can be found not only in the elements of the algebra but in the operations that act on those elements.

In addition to the grade, we can assign an antigrade to each basis element $$\mathbf a$$. Denoted by $$\operatorname{ag}(\mathbf a)$$, the antigrade of $$\mathbf a$$ is the number of vector basis elements missing from its factorization, which is the number of white bars in the table. Of course, it is always the case that

$$\operatorname{gr}(\mathbf a) + \operatorname{ag}(\mathbf a) = n$$ ,

where $$n$$ is the total number of dimensions in the algebra. Whenever we can make a statement about how an operation relates to the grade of its inputs and outputs, we can make the same statement about how the dual operation relates to the antigrade of its inputs and outputs.

In an $$n$$-dimensional algebra, the elements with grade $$n - 1$$ are called antivectors. Antivectors have the same number of components as vectors, and the two can be regarded as the dimensional inverses of each other. Vectors have grade one because they have one full dimension, and antivectors have antigrade one because they have one empty dimension.

Pages

The five main types of rigid geometric objects

Various properties and unary operations

Products and other binary operations

Isometries of 3D space

Two-Dimensional Geometry

The three-dimensional algebra $$\mathcal G_{2,0,1}$$, which deals with points, lines, and isometries in 2D space, is discussed on the following page.