__NOTOC__
This is the errata page for the ''Projective Geometric Algebra Illuminated''. To find out which printing you have, look on the copyright page. If you believe you have found an error that should be added to this list, please email the author (lengyel@terathon.com).

== Fourth Printing and Earlier ==

* '''Page 12.''' Right before Equation (1.25), the expression for the magnitude of the projection of $$\mathbf u$$ onto the line should be $$|\mathbf u \cdot \mathbf v| / \Vert\mathbf v\Vert$$.

== Third Printing and Earlier ==

* '''Page 70.''' In the line following Equation (2.71), the product of the bulks should use the antiwedge product and appear as $$\mathbf a_\unicode["segoe ui symbol"]{x25CF} \vee \mathbf b_\unicode["segoe ui symbol"]{x25CF}$$.

== Second Printing and Earlier ==

* '''Page 248.''' In Equation (5.27), the operator $$\mathbf D$$ should be dualized as $$\mathbf D^\unicode["segoe ui symbol"]{x2605}$$. It should also be dualized in the first factor of the sandwich product in the sentence that follows this equation.

== First Printing ==

* '''Page 84.''' In Table 2.16, both bulk duals of $$\mathbf e_4$$ should be 0. The table should look like the one on the wiki page for [[duals]].

* '''Page 156.''' The number of multiply-adds required for transforming a line with a motor did not include the work needed to calculate $$\mathbf a$$, $$\mathbf b$$, and $$\mathbf c$$. Each cross product adds 6 multiply-adds, so the correct total for transforming a line is 54.

* '''Page 156.''' Similarly, the number of multiply-adds required for transforming a plane with a motor did not include the work needed to calculate $$\mathbf a$$. The correct total for transforming a plane is 35.

== See Also ==

* [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''PGA Illuminated'' on Amazon.com]
* [[Contents]]
* [[Exercises]]

Exercises

2025-06-28T07:46:58Z

Eric Lengyel: /* See Also */

These are exercises accompanying the book [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''Projective Geometric Algebra Illuminated''].

== Exercises for Chapter 2 ==

'''1.''' Show that Equation (2.35) properly constructs a line containing two points $$\mathbf p$$ and $$\mathbf q$$ with non-unit weights by considering $$\mathbf p / p_w \wedge \mathbf q / q_w$$ and then scaling by $$p_wq_w$$.

'''2.''' Let $$\mathbf u$$ be a basis element of the 4D projective algebra. Prove that if $$\mathbf u \wedge \overline{\mathbf u} = {\large\unicode{x1D7D9}}$$ and $$\underline{\mathbf u} \wedge \mathbf u = {\large\unicode{x1D7D9}}$$, then it must also be true that $$\mathbf u \vee \overline{\mathbf u} = \mathbf 1$$ and $$\underline{\mathbf u} \vee \mathbf u = \mathbf 1$$. That is, show that right and left complements under the wedge product are also the right and left complements under the antiwedge product.

'''3.''' Suppose that the 4D trivectors $$\mathbf g$$ and $$\mathbf h$$ represent parallel planes in 3D space. Show that the magnitude of the moment of $$\mathbf g \vee \mathbf h$$ is the distance between the planes multiplied by both their weights.

'''4.''' Let $$\mathbf m$$ be a $$4 \times 4$$ matrix that performs a rotation about the $$z$$ axis in homogeneous coordinates. Calculate the $$16 \times 16$$ exomorphism matrix $$\mathbf M$$ corresponding to $$\mathbf m$$.

'''5.''' Suppose that $$\mathbf G$$ is a metric exomorphism. Use the fact that $$\mathbf G$$ is an exomorphism to prove that the associated antimetric $$\mathbb G$$ must satisfy $$\mathbb G(\mathbf a \vee \mathbf b) = \mathbb G\mathbf a \vee \mathbb G\mathbf b$$ for any $$\mathbf a$$ and $$\mathbf b$$.

'''6.''' Suppose that the metric tensor $$\mathfrak g$$ is invertible. Show that the wedge and antiwedge products satisfy the relationship $$\mathbf a \vee \mathbf b = (\mathbf a^\unicode["segoe ui symbol"]{x2605} \wedge \mathbf b^\unicode["segoe ui symbol"]{x2605})^\unicode["segoe ui symbol"]{x2606}$$.

'''7.''' Suppose that $$\mathbf a$$ and $$\mathbf b$$ are basis elements of an $$n$$-dimensional exterior algebra and $$\operatorname{gr}(\mathbf a) + \operatorname{gr}(\mathbf b) = n$$. Show that $$(\mathbf a \wedge \mathbf b)^\unicode["segoe ui symbol"]{x2605} = \mathbf a^\unicode["segoe ui symbol"]{x2605} \mathbin{\unicode{x25CF}} \mathbf b$$.

'''8.''' Show that the geometric norm is idempotent. That is, show that $$\Vert \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}} \Vert = \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}}$$.

'''9.''' Derive the relationship between left and right interior products shown in Equation (2.110).

'''10.''' Derive Equation (2.159), which is the expansion analog of Equation (2.129).

'''11.''' Derive a formula for $$\mathbf u^{\unicode["segoe ui symbol"]{x2605}\unicode["segoe ui symbol"]{x2605}}$$, the double bulk dual of $$\mathbf u$$, that uses only $$\operatorname{gr}(\mathbf u)$$, $$\operatorname{ag}(\mathbf u)$$, and the determinant of the metric tensor $$\mathfrak g$$.

'''12.''' Assuming that the antivector basis elements are written in the same order as their vector complements, prove that the $$(n - 1)$$-th compound matrix $$C_{n - 1}(\mathbf m)$$ of a matrix $$\mathbf m$$ is always equal to the adjugate transpose of $$\mathbf m$$.

== See Also ==

* [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''PGA Illuminated'' on Amazon.com]
* [[Contents]]

2025-06-28T07:46:45Z

Eric Lengyel: /* See Also */

This is the table of contents for ''Projective Geometric Algebra Illuminated''.

* Preface
* 1 Conventional Mathematics
** 1.1 The Cross Product
** 1.2 Homogeneous Coordinates
** 1.3 Lines and Planes
*** 1.3.1 Parametric Forms
*** 1.3.2 Implicit Forms
*** 1.3.3 Distance Between a Point and a Line
*** 1.3.4 Intersection of a Line and a Plane
*** 1.3.5 Intersection of Multiple Planes
*** 1.3.6 Reflection Across a Plane
*** 1.3.7 Homogeneous Formulas
*** 1.3.8 Plane Transformation
*** 1.3.9 Line Transformation
** 1.4 Quaternions
*** 1.4.1 Quaternion Fundamentals
*** 1.4.2 Rotations With Quaternions
*** 1.4.3 Interpolating Quaternions
*** 1.4.4 Dual Quaternions
**Historical Remarks
* 2 Flat Projective Geometry
** 2.1 Algebraic Structure
*** 2.1.1 The Wedge Product
*** 2.1.2 Bivectors
*** 2.1.3 Trivectors
*** 2.1.4 Basis Elements
** 2.2 Complements
** 2.3 Antiproducts
** 2.4 3D Flat Geometry
*** 2.4.1 Points
*** 2.4.2 Lines
*** 2.4.3 Planes
** 2.5 Join and Meet
** 2.6 Duality
** 2.7 Exomorphisms
** 2.8 Metric Transformations
*** 2.8.1 The Metric
*** 2.8.2 The Antimetric
*** 2.8.3 Bulk and Weight
*** 2.8.4 Attitude
** 2.9 Inner Products
** 2.10 Norms
*** 2.10.1 Bulk and Weight Norms
*** 2.10.2 Unitization
*** 2.10.3 The Geometric Norm
** 2.11 Euclidean Distances
** 2.12 Duals
** 2.13 Interior Products
*** 2.13.1 Contractions
*** 2.13.2 Projection and Rejection
*** 2.13.3 Euclidean Angles
*** 2.13.4 Parametric Forms
*** 2.13.5 Expansions
*** 2.13.6 Geometric Projection
** 2.14 2D Flat Geometry
** 2.15 Dependencies
** Historical Remarks
* 3 Rigid Transformations
** 3.1 The Geometric Product
** 3.2 Dual Numbers
** 3.3 Reflection and Rotation
** 3.4 Reversion
*** 3.4.1 Reverse and Antireverse
*** 3.4.2 Dual Identities
*** 3.4.3 Geometric Constraint
** 3.5 Euclidean Isometries
*** 3.5.1 Reflection
*** 3.5.2 Rotation
*** 3.5.3 Translation
*** 3.5.4 Inversion
*** 3.5.5 Transflection
** 3.6 Motors
*** 3.6.1 Motion Operator
*** 3.6.2 Parameterization
*** 3.6.3 Line to Line Motion
*** 3.6.4 Matrix Conversion
*** 3.6.5 Implementation
** 3.7 Flectors
*** 3.7.1 Reflection Operator
*** 3.7.2 Matrix Conversion
*** 3.7.3 Implementation
** 3.8 2D Rigid Transformations
** 3.9 Operator Duality
*** 3.9.1 Complement Isometries
*** 3.9.2 Transformation Groups
*** 3.9.3 Quaternions Revisited
** Historical Remarks
* 4 Round Projective Geometry
** 4.1 Construction
** 4.2 3D Round Geometry
*** 4.2.1 Representations
*** 4.2.2 Duals
*** 4.2.3 Carriers
*** 4.2.4 Centers
*** 4.2.5 Containers
*** 4.2.6 Partners
*** 4.2.7 Attitude
** 4.3 Norms
** 4.4 Alignment
** 4.5 Dot Products
*** 4.5.1 Round Points
*** 4.5.2 Spheres
*** 4.5.3 Partners
*** 4.5.4 Conjugates
** 4.6 Containment
** 4.7 Join and Meet
** 4.8 Expansions
** 4.9 2D Round Geometry
** 4.10 Degrees of Freedom
* 5 Conformal Transformations
** 5.1 Generalized Operators
*** 5.1.1 Rigid Transformations
*** 5.1.2 Sphere Inversion
*** 5.1.3 Circle Rotation
** 5.2 Dilation
** 5.3 Duals and Complements
** 5.4 2D Conformal Transformations
* A Multiplication Tables
* B Geometric Properties
* C Notation Reference
* Bibliography
* Index

== See Also ==

* [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''PGA Illuminated'' on Amazon.com]
* [[Exercises]]

Exercises

2025-06-28T07:46:22Z

Eric Lengyel:

Errata

2025-06-28T07:46:04Z

Eric Lengyel:

Errata

2025-06-28T07:45:47Z

Eric Lengyel:

Main Page

2025-06-28T07:45:01Z

Eric Lengyel:

__NOTOC__
== Rigid Geometric Algebra ==

This wiki is a repository of information about Rigid Geometric Algebra (RGA), and specifically the four-dimensional Clifford algebra $$\mathcal G_{3,0,1}$$. This wiki is associated with the following websites:

* [http://projectivegeometricalgebra.org Projective Geometric Algebra overview site]
* [http://conformalgeometricalgebra.org/wiki/index.php?title=Main_Page Conformal Geometric Algebra companion site]

Rigid geometric algebra 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. [http://conformalgeometricalgebra.org/wiki/index.php?title=Main_Page Conformal Geometric Algebra] (CGA) is a larger algebra that contains the complete RGA and also includes round objects like circles and spheres.

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 ==

[[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 $$\mathbf e_{23}$$, $$\mathbf e_{31}$$, $$\mathbf e_{12}$$, $$\mathbf e_{41}$$, $$\mathbf e_{42}$$, and $$\mathbf e_{43}$$ 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 $$\mathbf e_{423}$$, $$\mathbf e_{431}$$, $$\mathbf e_{412}$$, 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 $$\mathbf u$$, denoted by $$\operatorname{gr}(\mathbf u)$$, 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 $$\mathbf u$$. Denoted by $$\operatorname{ag}(\mathbf u)$$, the antigrade of $$\mathbf u$$ 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

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

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 ===

* [[Point]]
* [[Line]]
* [[Plane]]
* [[Motor]]
* [[Flector]]

=== Various properties and unary operations ===

* [[Grade and antigrade]]
* [[Complements]]
* [[Metrics]]
* [[Bulk and weight]]
* [[Duals]]
* [[Reverses]]
* [[Attitude]]
* [[Geometric norm]]
* [[Geometric constraint]]
* [[Unitization]]
* [[Duality]]

=== Products and other binary operations ===

* [[Exterior products]]
* [[Dot products]]
* [[Interior products]]
* [[Geometric products]]
* [[Transwedge products]]
* [[Join and meet]]
* [[Projections]]
* [[Euclidean distance]]
* [[Euclidean angle]]

=== Isometries of 3D space ===

* [[Transformation groups]]
* [[Translation]]
* [[Rotation]]
* [[Reflection]]
* [[Inversion]]
* [[Transflection]]

=== Projective Geometric Algebra Illuminated ===

* [[Contents]]
* [[Exercises]]
* [[Errata]]

Main Page

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2025-05-21T21:17:22Z

Eric Lengyel: Eric Lengyel uploaded a new version of File:TranswedgeAntiproducts.svg

File:TranswedgeProducts.svg

2025-05-21T21:17:08Z

Eric Lengyel: Eric Lengyel uploaded a new version of File:TranswedgeProducts.svg

Transwedge products

2025-05-21T21:10:05Z

Eric Lengyel:

The ''transwedge product'' is a generalization of the [[exterior product]] and [[interior product]] that also includes a transitional sequence of liminal products between exterior and interior.

== Transwedge Product ==

The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as

:$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf c^{\unicode{x2605}})}$$.

The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,

:$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.

When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, which is an interior product. If $$k > \operatorname{gr}(\mathbf a)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.

An equivalent definition for the transwedge product is given by

:$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf c_{\unicode{x2605}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,

where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.

For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it’s nonzero.

The sum of all possible transwedge products, negating for orders 2 and 3 modulo 4, yields the geometric product. That is,

:$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,

where $$n$$ is the dimension of the algebra.

The [[geometric product]] of each pair of the basis elements in a geometric algebra is given by exactly one of the transwedge products. These are shown for the 16 basis elements of the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ in the following table, which color codes the transwedge products of order 0, 1, 2, 3, and 4. (Some of the products are zero in this algebra due to the degenerate metric.)

[[Image:TranswedgeProducts.svg|720px]]

For vectors $$\mathbf a$$ and $$\mathbf b$$, we have

:$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$,

where the [[dot product]] on the right is the result of the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$.

For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have

:$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b - \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$,

where the [[dot product]] is again attributable to the right contraction $$\mathbf b \vee \mathbf{\tilde a}^{\unicode{x2605}}$$. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.

== Transwedge Antiproduct ==

The ''transwedge antiproduct'' of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as

:$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf c^{\unicode{x2606}})} = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf c_{\unicode{x2606}} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\overline c})}$$,

where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], and we are now summing over all basis elements of [[antigrade]] $$k$$.

The sum of all possible transwedge antiproducts, again negating for orders 2 and 3 modulo 4, yields the geometric antiproduct. That is,

:$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,

The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, 3, and 4.

[[Image:TranswedgeAntiproducts.svg|720px]]

== See Also ==

* [[Exterior products]]
* [[Interior products]]

Transwedge product

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Eric Lengyel: Redirected page to Transwedge products

#REDIRECT [[Transwedge products]]

Geometric products

2025-05-20T20:28:52Z

Eric Lengyel:

The ''geometric product'' is a method of multiplication in geometric algebra derived from the [[exterior product]] and the [[metric]]. There are two products with symmetric properties called the geometric product and geometric antiproduct.

== Geometric Product ==

The geometric product between two elements $$\mathbf a$$ and $$\mathbf b$$ has often been written by simply juxtaposing its operands as $$\mathbf{ab}$$ without the use of any infix operator. However, this clearly becomes impractical when both a product and antiproduct are present in the same context, which is now known to be necessary for a proper understanding of the algebra. To remedy the situation, we write the geometric product between elements $$\mathbf a$$ and $$\mathbf b$$ as $$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b$$ and read it as "$$\mathbf a$$ wedge-dot $$\mathbf b$$".

The geometric product is characterized by a metric that defines the products of the basis vectors with themselves. The subscript in $$\mathcal G_{3,0,1}$$ means that three basis vectors square to +'''1''', zero basis vectors square to −'''1''', and one basis vector squares to 0. The geometric product between two different basis vectors is given by the [[wedge product]]. We can write these rules as follows.

:$$\mathbf e_1 \mathbin{\unicode{x27D1}} \mathbf e_1 = \mathbf 1$$

:$$\mathbf e_2 \mathbin{\unicode{x27D1}} \mathbf e_2 = \mathbf 1$$

:$$\mathbf e_3 \mathbin{\unicode{x27D1}} \mathbf e_3 = \mathbf 1$$

:$$\mathbf e_4 \mathbin{\unicode{x27D1}} \mathbf e_4 = 0$$

:$$\mathbf e_i \mathbin{\unicode{x27D1}} \mathbf e_j = \mathbf e_i \wedge \mathbf e_j$$, for $$i \neq j$$.

The geometric product is equal to the sum of the [[transwedge products]] of orders 0 through $$n$$, where $$n$$ is the dimension of the algebra, in this case 4.

The following Cayley table shows the geometric products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric product is the [[scalar]] basis element $$\mathbf 1$$.

Cells highlighted in green correspond to the contribution from the [[wedge product]].

[[Image:GeometricProduct.svg|720px]]

== Geometric Antiproduct ==

The geometric antiproduct is dual to the geometric product. The geometric antiproduct between elements $$\mathbf a$$ and $$\mathbf b$$ is written $$\mathbf a \mathbin{\unicode{x27C7}} \mathbf b$$ and is read as "$$\mathbf a$$ antiwedge-dot $$\mathbf b$$".

The same metric that defines products of basis vectors under the geometric product also applies to the geometric antiproduct, except that now it defines products of basis [[antivectors]]. Three basis antivectors square to $$+{\large\unicode{x1D7D9}}$$, zero basis antivectors square to $$-{\large\unicode{x1D7D9}}$$, and one basis antivector squares to 0. The geometric antiproduct between two different basis antivectors is given by the [[antiwedge product]]. We can write these rules as follows.

:$$\overline{\mathbf e_1} \mathbin{\unicode{x27C7}} \overline{\mathbf e_1} = {\large\unicode{x1D7D9}}$$

:$$\overline{\mathbf e_2} \mathbin{\unicode{x27C7}} \overline{\mathbf e_2} = {\large\unicode{x1D7D9}}$$

:$$\overline{\mathbf e_3} \mathbin{\unicode{x27C7}} \overline{\mathbf e_3} = {\large\unicode{x1D7D9}}$$

:$$\overline{\mathbf e_4} \mathbin{\unicode{x27C7}} \overline{\mathbf e_4} = 0$$

:$$\overline{\mathbf e_i} \mathbin{\unicode{x27C7}} \overline{\mathbf e_j} = \overline{\mathbf e_i} \vee \overline{\mathbf e_j}$$, for $$i \neq j$$.

The following Cayley table shows the geometric antiproducts between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric antiproduct is the [[antiscalar]] basis element $${\large\unicode{x1D7D9}}$$.

Cells highlighted in green correspond to the contribution from the [[antiwedge product]].

[[Image:GeometricAntiproduct.svg|720px]]

== De Morgan Laws ==

The relationship between the product and antiproduct is based on an exchange of full and empty dimensions. We can then express each product in terms of the other through an analog of De Morgan's laws as follows.

:$$\overline{\mathbf a \mathbin{\smash{\unicode{x27D1}}} \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27C7}} \overline{\mathbf b}$$

:$$\overline{\mathbf a \mathbin{\smash{\unicode{x27C7}}} \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27D1}} \overline{\mathbf b}$$

:$$\underline{\mathbf a \mathbin{\smash{\unicode{x27D1}}} \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27C7}} \underline{\mathbf b}$$

:$$\underline{\mathbf a \mathbin{\smash{\unicode{x27C7}}} \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27D1}} \underline{\mathbf b}$$

== In the Book ==

* The geometric product and antiproduct are introduced in Section 3.1.

== See Also ==

* [[Transwedge products]]
* [[Exterior products]]
* [[Dot products]]
* [[Complements]]

Geometric products

2025-05-20T19:56:06Z

Eric Lengyel: /* See Also */

The ''geometric product'' is the fundamental product of geometric algebra. There are two products with symmetric properties called the geometric product and geometric antiproduct.

== Geometric Product ==

The geometric product between two elements $$\mathbf a$$ and $$\mathbf b$$ has often been written by simply juxtaposing its operands as $$\mathbf{ab}$$ without the use of any infix operator. However, this clearly becomes impractical when both a product and antiproduct are present in the same context, which is now known to be necessary for a proper understanding of the algebra. To remedy the situation, we write the geometric product between elements $$\mathbf a$$ and $$\mathbf b$$ as $$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b$$ and read it as "$$\mathbf a$$ wedge-dot $$\mathbf b$$".

The geometric product is characterized by a metric that defines the products of the basis vectors with themselves. The subscript in $$\mathcal G_{3,0,1}$$ means that three basis vectors square to +'''1''', zero basis vectors square to −'''1''', and one basis vector squares to 0. The geometric product between two different basis vectors is given by the [[wedge product]]. We can write these rules as follows.

:$$\mathbf e_1 \mathbin{\unicode{x27D1}} \mathbf e_1 = \mathbf 1$$

:$$\mathbf e_2 \mathbin{\unicode{x27D1}} \mathbf e_2 = \mathbf 1$$

:$$\mathbf e_3 \mathbin{\unicode{x27D1}} \mathbf e_3 = \mathbf 1$$

:$$\mathbf e_4 \mathbin{\unicode{x27D1}} \mathbf e_4 = 0$$

:$$\mathbf e_i \mathbin{\unicode{x27D1}} \mathbf e_j = \mathbf e_i \wedge \mathbf e_j$$, for $$i \neq j$$.

The following Cayley table shows the geometric products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric product is the [[scalar]] basis element $$\mathbf 1$$.

Cells highlighted in green correspond to the contribution from the [[wedge product]].

[[Image:GeometricProduct.svg|720px]]

== Geometric Antiproduct ==

The geometric antiproduct is a dual to the geometric product. The geometric antiproduct between elements $$\mathbf a$$ and $$\mathbf b$$ is written $$\mathbf a \mathbin{\unicode{x27C7}} \mathbf b$$ and is read as "$$\mathbf a$$ antiwedge-dot $$\mathbf b$$".

The same metric that defines products of basis vectors under the geometric product also applies to the geometric antiproduct, except that now it defines products of basis [[antivectors]]. Three basis antivectors square to $$+{\large\unicode{x1D7D9}}$$, zero basis antivectors square to $$-{\large\unicode{x1D7D9}}$$, and one basis antivector squares to 0. The geometric antiproduct between two different basis antivectors is given by the [[antiwedge product]]. We can write these rules as follows.

:$$\overline{\mathbf e_1} \mathbin{\unicode{x27C7}} \overline{\mathbf e_1} = {\large\unicode{x1D7D9}}$$

:$$\overline{\mathbf e_2} \mathbin{\unicode{x27C7}} \overline{\mathbf e_2} = {\large\unicode{x1D7D9}}$$

:$$\overline{\mathbf e_3} \mathbin{\unicode{x27C7}} \overline{\mathbf e_3} = {\large\unicode{x1D7D9}}$$

:$$\overline{\mathbf e_4} \mathbin{\unicode{x27C7}} \overline{\mathbf e_4} = 0$$

:$$\overline{\mathbf e_i} \mathbin{\unicode{x27C7}} \overline{\mathbf e_j} = \overline{\mathbf e_i} \vee \overline{\mathbf e_j}$$, for $$i \neq j$$.

The following Cayley table shows the geometric antiproducts between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric antiproduct is the [[antiscalar]] basis element $${\large\unicode{x1D7D9}}$$.

Cells highlighted in green correspond to the contribution from the [[antiwedge product]].

[[Image:GeometricAntiproduct.svg|720px]]

== De Morgan Laws ==

The relationship between the product and antiproduct is based on an exchange of full and empty dimensions. We can then express each product in terms of the other through an analog of De Morgan's laws as follows.

:$$\overline{\mathbf a \mathbin{\smash{\unicode{x27D1}}} \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27C7}} \overline{\mathbf b}$$

:$$\overline{\mathbf a \mathbin{\smash{\unicode{x27C7}}} \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27D1}} \overline{\mathbf b}$$

:$$\underline{\mathbf a \mathbin{\smash{\unicode{x27D1}}} \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27C7}} \underline{\mathbf b}$$

:$$\underline{\mathbf a \mathbin{\smash{\unicode{x27C7}}} \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27D1}} \underline{\mathbf b}$$

== In the Book ==

* The geometric product and antiproduct are introduced in Section 3.1.

== See Also ==

* [[Transwedge products]]
* [[Exterior products]]
* [[Dot products]]
* [[Complements]]

Exterior products

2025-05-20T19:55:46Z

Eric Lengyel: /* See Also */

The ''exterior product'' is the fundamental product of Grassmann Algebra, and it forms part of the [[geometric product]] in geometric algebra. There are two products with symmetric properties called the exterior product and exterior antiproduct.

The exterior product between two elements $$\mathbf a$$ and $$\mathbf b$$ generally combines their spatial extents, and the magnitude of the result indicates how close they are to being orthogonal. If the spatial extents of $$\mathbf a$$ and $$\mathbf b$$ are parallel, then their exterior product is zero. (Compare this to the [[dot product]], which is zero whenever $$\mathbf a$$ and $$\mathbf b$$ are orthogonal.)

== Exterior Product ==

The exterior product is widely known as the ''wedge product'' because it is written with an upward pointing wedge. The exterior product between elements $$\mathbf a$$ and $$\mathbf b$$ is written $$\mathbf a \wedge \mathbf b$$ and read "$$\mathbf a$$ wedge $$\mathbf b$$". Grassmann called this the progressive combinatorial product.

The defining characteristic of the wedge product is that multiplying any vector $$\mathbf v$$ by itself produces zero: $$\mathbf v \wedge \mathbf v = 0$$. This implies that the wedge product is anticommutative for vectors, so we always have

:$$\mathbf v \wedge \mathbf w = -\mathbf w \wedge \mathbf v$$

for vectors $$\mathbf v$$ and $$\mathbf w$$. The wedge product is not anticommutative in general, however. For general basis elements $$\mathbf a$$ and $$\mathbf b$$, reversing the order of the operands satisfies the relationship

:$$\mathbf a \wedge \mathbf b = (-1)^{\operatorname{gr}(\mathbf a)\operatorname{gr}(\mathbf b)} \mathbf b \wedge \mathbf a$$ .

The wedge product adds the [[grades]] of its operands, so we have

:$$\operatorname{gr}(\mathbf a \wedge \mathbf b) = \operatorname{gr}(\mathbf a) + \operatorname{gr}(\mathbf b)$$ .

The following Cayley table shows the exterior products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.

[[Image:WedgeProduct.svg|720px]]

== Exterior Antiproduct ==

The exterior antiproduct is a dual to the exterior product. It is written with a downward pointing wedge and thus called the ''antiwedge product''. The exterior antiproduct $$\mathbf a \vee \mathbf b$$ is read "$$\mathbf a$$ antiwedge $$\mathbf b$$". Grassmann called this the regressive combinatorial product.

Whereas the wedge product combines the full dimensions of its operands, the antiwedge product combines the empty dimensions of its operands. The antiwedge product adds the [[antigrades]] of its operands, so we have

:$$\operatorname{ag}(\mathbf a \vee \mathbf b) = \operatorname{ag}(\mathbf a) + \operatorname{ag}(\mathbf b)$$ .

The following Cayley table shows the exterior antiproducts between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.

[[Image:AntiwedgeProduct.svg|720px]]

== De Morgan Laws ==

The relationship between the product and antiproduct is based on an exchange of full and empty dimensions. We can then express each product in terms of the other through an analog of De Morgan's laws as follows.

:$$\overline{\mathbf a \wedge \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \vee \overline{\mathbf b}$$

:$$\overline{\mathbf a \vee \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \wedge \overline{\mathbf b}$$

:$$\underline{\mathbf a \wedge \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \vee \underline{\mathbf b}$$

:$$\underline{\mathbf a \vee \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \wedge \underline{\mathbf b}$$

== General Properties ==

The following table lists several general properties of the wedge product and antiwedge product.

{| class="wikitable"
! Property !! Description
|-
| style="padding: 12px;" | $$\mathbf a \wedge \mathbf b = -\mathbf b \wedge \mathbf a$$
| style="padding: 12px;" | Anticommutativity of the wedge product for vectors $$\mathbf a$$ and $$\mathbf b$$.
|-
| style="padding: 12px;" | $$\mathbf a \vee \mathbf b = -\mathbf b \vee \mathbf a$$
| style="padding: 12px;" | Anticommutativity of the antiwedge product for antivectors $$\mathbf a$$ and $$\mathbf b$$.
|-
| style="padding: 12px;" | $$(\mathbf a \wedge \mathbf b) \wedge \mathbf c = \mathbf a \wedge (\mathbf b \wedge \mathbf c)$$
| style="padding: 12px;" | Associative law for the wedge product for any elements $$\mathbf a$$, $$\mathbf b$$, and $$\mathbf c$$.
|-
| style="padding: 12px;" | $$(\mathbf a \vee \mathbf b) \vee \mathbf c = \mathbf a \vee (\mathbf b \vee \mathbf c)$$
| style="padding: 12px;" | Associative law for the antiwedge product for any elements $$\mathbf a$$, $$\mathbf b$$, and $$\mathbf c$$.
|-
| style="padding: 12px;" | $$\mathbf a \wedge (\mathbf b + \mathbf c) = \mathbf a \wedge \mathbf b + \mathbf a \wedge \mathbf c$$
| style="padding: 12px;" | Distributive law for the wedge product for any elements $$\mathbf a$$, $$\mathbf b$$, and $$\mathbf c$$.
|-
| style="padding: 12px;" | $$\mathbf a \vee (\mathbf b + \mathbf c) = \mathbf a \vee \mathbf b + \mathbf a \vee \mathbf c$$
| style="padding: 12px;" | Distributive law for the antiwedge product for any elements $$\mathbf a$$, $$\mathbf b$$, and $$\mathbf c$$.
|-
| style="padding: 12px;" | $$(t\mathbf a) \wedge \mathbf b = \mathbf a \wedge (t\mathbf b) = t(\mathbf a \wedge \mathbf b)$$
| style="padding: 12px;" | Scalar factorization of the wedge product.
|-
| style="padding: 12px;" | $$(t\mathbf a) \vee \mathbf b = \mathbf a \vee (t\mathbf b) = t(\mathbf a \vee \mathbf b)$$
| style="padding: 12px;" | Scalar factorization of the antiwedge product.
|-
| style="padding: 12px;" | $$s \wedge \mathbf a = \mathbf a \wedge s = s\mathbf a$$
| style="padding: 12px;" | Wedge product of a [[scalar]] $$s$$ and any basis element $$\mathbf a$$.
|-
| style="padding: 12px;" | $$s \vee \mathbf a = \mathbf a \vee s = s\mathbf a$$
| style="padding: 12px;" | Antiwedge product of an [[antiscalar]] $$s$$ and any basis element $$\mathbf a$$.
|-
| style="padding: 12px;" | $$s \wedge t = st$$
| style="padding: 12px;" | Wedge product of [[scalars]] $$s$$ and $$t$$.
|-
| style="padding: 12px;" | $$s \vee t = st$$
| style="padding: 12px;" | Antiwedge product of [[antiscalars]] $$s$$ and $$t$$.
|}

== In the Book ==

* The exterior (wedge) product is introduced in Section 2.1.1.
* The exterior antiproduct is discussed in Section 2.3.

== See Also ==

* [[Transwedge products]]
* [[Geometric products]]
* [[Dot products]]
* [[Complements]]

Transwedge products

2025-05-20T00:50:16Z

Eric Lengyel: /* Transwedge Antiproduct */

The ''transwedge product'' is a generalization of the [[exterior product]] and [[interior product]] that also includes a transitional sequence of liminal products between exterior and interior.

== Transwedge Product ==

The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as

:$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\tilde c^{\unicode{x2605}}})}$$.

The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,

:$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.

When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf{\tilde a^{\unicode{x2605}}}$$, which is an interior product. If $$k > \operatorname{gr}(\mathbf a)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.

An equivalent definition for the transwedge product is given by

:$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\tilde c_{\unicode{x2605}}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,

where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.

For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it’s nonzero.

The sum of all possible transwedge products yields the geometric product. That is,

:$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,

where $$n$$ is the dimension of the algebra.

The [[geometric product]] of each pair of the basis elements in a geometric algebra is given by exactly one of the transwedge products. These are shown for the 16 basis elements of the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ in the following table, which color codes the transwedge products of order 0, 1, 2, 3, and 4. (Some of the products are zero in this algebra due to the degenerate metric.)

[[Image:TranswedgeProducts.svg|720px]]

For vectors $$\mathbf a$$ and $$\mathbf b$$, we have

:$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$,

where the [[dot product]] on the right is the result of the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, and we have dropped the reverse operation because $$\mathbf a$$ is a vector.

For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have

:$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$,

where the [[dot product]] is again attributable to the right contraction $$\mathbf b \vee \mathbf{\tilde a}^{\unicode{x2605}}$$. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.

== Transwedge Antiproduct ==

The ''transwedge antiproduct'' of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as

:$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\smash{\mathbf{\underset{\Large\unicode{x7E}}{c}}}^{\unicode{x2606}}})} = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\smash{\mathbf{\underset{\Large\unicode{x7E}}{c}}}_{\unicode{x2606}}} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\overline c})}$$,

where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], the [[reverse]] has become the [[antireverse]], and we are now summing over all basis elements of [[antigrade]] $$k$$.

The sum of all possible transwedge antiproducts yields the geometric antiproduct. That is,

:$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,

The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, 3, and 4.

[[Image:TranswedgeAntiproducts.svg|720px]]

== See Also ==

* [[Exterior products]]
* [[Interior products]]

Transwedge products

2025-05-20T00:49:53Z

Eric Lengyel:

The ''transwedge product'' is a generalization of the [[exterior product]] and [[interior product]] that also includes a transitional sequence of liminal products between exterior and interior.

== Transwedge Product ==

The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as

:$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\tilde c^{\unicode{x2605}}})}$$.

The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,

:$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.

When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf{\tilde a^{\unicode{x2605}}}$$, which is an interior product. If $$k > \operatorname{gr}(\mathbf a)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.

An equivalent definition for the transwedge product is given by

:$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\tilde c_{\unicode{x2605}}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,

where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.

For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it’s nonzero.

The sum of all possible transwedge products yields the geometric product. That is,

:$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,

where $$n$$ is the dimension of the algebra.

The [[geometric product]] of each pair of the basis elements in a geometric algebra is given by exactly one of the transwedge products. These are shown for the 16 basis elements of the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ in the following table, which color codes the transwedge products of order 0, 1, 2, 3, and 4. (Some of the products are zero in this algebra due to the degenerate metric.)

[[Image:TranswedgeProducts.svg|720px]]

For vectors $$\mathbf a$$ and $$\mathbf b$$, we have

:$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$,

where the [[dot product]] on the right is the result of the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, and we have dropped the reverse operation because $$\mathbf a$$ is a vector.

For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have

:$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$,

where the [[dot product]] is again attributable to the right contraction $$\mathbf b \vee \mathbf{\tilde a}^{\unicode{x2605}}$$. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.

== Transwedge Antiproduct ==

The ''transwedge antiproduct'' of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as

:$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\smash{\mathbf{\underset{\Large\unicode{x7E}}{c}}}^{\unicode{x2606}}})} = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\smash{\mathbf{\underset{\Large\unicode{x7E}}{c}}}_{\unicode{x2606}}} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\overline c})}$$,

where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], the [[reverse]] has become the [[antireverse]], and we are now summing over all basis elements of [[antigrade]] $$k$$.

The sum of all possible transwedge antiproducts yields the geometric antiproduct. That is,

:$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,

The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, and 3.

[[Image:TranswedgeAntiproducts.svg|720px]]

== See Also ==

* [[Exterior products]]
* [[Interior products]]

File:TranswedgeAntiproducts.svg

2025-05-20T00:48:00Z

Eric Lengyel: Eric Lengyel uploaded a new version of File:TranswedgeAntiproducts.svg

2025-05-17T05:37:31Z

Eric Lengyel:

Expansion

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Eric Lengyel: Redirected page to Interior products

#REDIRECT [[Interior products]]

Contraction

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Eric Lengyel: Redirected page to Interior products

#REDIRECT [[Interior products]]

Transwedge products

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Eric Lengyel:

Transwedge products

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Eric Lengyel:

Transwedge products

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Eric Lengyel: /* See Also */