User contributions for Eric Lengyel
Jump to navigation
Jump to search
15 July 2023
- 06:4906:49, 15 July 2023 diff hist +1 Geometric antiproduct Changed redirect target from Geometric product to Geometric products current Tag: Redirect target changed
- 06:4806:48, 15 July 2023 diff hist +31 N Geometric antiproduct Redirected page to Geometric product Tag: New redirect
- 06:4806:48, 15 July 2023 diff hist +37 N Scalar Redirected page to Scalars and antiscalars current Tag: New redirect
- 06:4806:48, 15 July 2023 diff hist +28 N Bulk norm Redirected page to Geometric norm current Tag: New redirect
- 06:4706:47, 15 July 2023 diff hist +28 N Weight norm Redirected page to Geometric norm current Tag: New redirect
- 06:4706:47, 15 July 2023 diff hist +37 N Antiscalar Redirected page to Scalars and antiscalars current Tag: New redirect
- 06:4706:47, 15 July 2023 diff hist +25 N Unitized Redirected page to Unitization current Tag: New redirect
- 06:4606:46, 15 July 2023 diff hist +29 N Weight Redirected page to Bulk and weight current Tag: New redirect
- 06:4606:46, 15 July 2023 diff hist +29 N Bulk Redirected page to Bulk and weight current Tag: New redirect
- 06:4606:46, 15 July 2023 diff hist +22 N Reverse Redirected page to Reverses current Tag: New redirect
- 06:4506:45, 15 July 2023 diff hist +25 N Complement Redirected page to Complements current Tag: New redirect
- 06:4506:45, 15 July 2023 diff hist +33 N Grades Redirected page to Grade and antigrade current Tag: New redirect
- 06:4506:45, 15 July 2023 diff hist +23 N Trivectors Redirected page to Trivector current Tag: New redirect
- 06:4406:44, 15 July 2023 diff hist +22 N Bivectors Redirected page to Bivector current Tag: New redirect
- 06:4406:44, 15 July 2023 diff hist +25 N Translations Redirected page to Translation current Tag: New redirect
- 06:4406:44, 15 July 2023 diff hist +24 N Reflections Redirected page to Reflection current Tag: New redirect
- 06:4406:44, 15 July 2023 diff hist +22 N Rotations Redirected page to Rotation current Tag: New redirect
- 06:4406:44, 15 July 2023 diff hist +19 N Planes Redirected page to Plane current Tag: New redirect
- 06:4406:44, 15 July 2023 diff hist +18 N Lines Redirected page to Line current Tag: New redirect
- 06:4306:43, 15 July 2023 diff hist +19 N Points Redirected page to Point current Tag: New redirect
- 06:4306:43, 15 July 2023 diff hist +166 N Trivector Created page with "A ''trivector'' in a geometric algebra is an element composed entirely of components having grade 3. == See Also == * Vector * Bivector * Antivector" current
- 06:4206:42, 15 July 2023 diff hist +166 N Bivector Created page with "A ''bivector'' in a geometric algebra is an element composed entirely of components having grade 2. == See Also == * Vector * Trivector * Antivector" current
- 06:4206:42, 15 July 2023 diff hist +261 N Antivector Created page with "An ''antivector'' in a geometric algebra is an element composed entirely of components having antigrade 1. In an ''n''-dimensional geometric algebra, these are the elements having grade $$n - 1$$. == See Also == * Vector * Bivector * Trivector" current
- 06:4106:41, 15 July 2023 diff hist +24 N Antivectors Redirected page to Antivector current Tag: New redirect
- 06:4006:40, 15 July 2023 diff hist +162 N Vector Created page with "A ''vector'' in a geometric algebra is an element composed entirely of components having grade 1. == See Also == * Bivector * Trivector * Antivector" current
- 06:4006:40, 15 July 2023 diff hist +20 N Vectors Redirected page to Vector current Tag: New redirect
- 06:3906:39, 15 July 2023 diff hist +37 N Antiscalars Redirected page to Scalars and antiscalars current Tag: New redirect
- 06:3906:39, 15 July 2023 diff hist +37 N Scalars Redirected page to Scalars and antiscalars current Tag: New redirect
- 06:3706:37, 15 July 2023 diff hist +33 N Antigrade Redirected page to Grade and antigrade current Tag: New redirect
- 06:3606:36, 15 July 2023 diff hist +33 N Grade Redirected page to Grade and antigrade current Tag: New redirect
- 06:3606:36, 15 July 2023 diff hist +1,113 N Commutators Created page with "In geometric algebra, there are four ''commutator'' products defined as follows. :$$[\mathbf a, \mathbf b]^{\Large\unicode{x27D1}}_- = \dfrac{1}{2}\left(\mathbf a \mathbin{\unicode{x27D1}} \mathbf b - \mathbf b \mathbin{\unicode{x27D1}} \mathbf a\right)$$ :$$[\mathbf a, \mathbf b]^{\Large\unicode{x27D1}}_+ = \dfrac{1}{2}\left(\mathbf a \mathbin{\unicode{x27D1}} \mathbf b + \mathbf b \mathbin{\unicode{x27D1}} \mathbf a\right)$$ :$$[\mathbf a, \mathbf b]^{\Large\unicode..." current
- 06:3506:35, 15 July 2023 diff hist +1,490 N Point Created page with "400px|thumb|right|'''Figure 1.''' A point is the intersection of a 4D vector with the 3D subspace where $$w = 1$$. In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a ''point'' $$\mathbf p$$ is a vector having the general form :$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ . All points possess the geometric property. The bulk of a point is given by its $$x$$, $$y$$, and $$z$$ coordinates, and..."
- 06:3506:35, 15 July 2023 diff hist 0 N File:Point.svg No edit summary
- 06:3306:33, 15 July 2023 diff hist +3,246 N Duality Created page with "480px|thumb|right|'''Figure 1.''' The coordinates $$(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)$$-dimensional span of all orthogonal vectors representing a homogeneous plane. Geometrically, these two interpretations are dual to each other, and their distances to the origin are reciprocals of each other. The concept of duality can be understood geometric..."
- 06:3306:33, 15 July 2023 diff hist 0 N File:Duality.svg No edit summary
- 06:3006:30, 15 July 2023 diff hist +31 N Wedge products Redirected page to Exterior products current Tag: New redirect
- 06:2906:29, 15 July 2023 diff hist +5,062 N Geometric products Created page with "The ''geometric product'' is the fundamental product of geometric algebra. There are two products with symmetric properties called the geometric product and geometric antiproduct. == Geometric Product == The geometric product between two elements $$\mathbf a$$ and $$\mathbf b$$ has often been written by simply juxtaposing its operands as $$\mathbf{ab}$$ without the use of any infix operator. However, this clearly becomes impractical when both a product and antiproduct..."
- 06:2906:29, 15 July 2023 diff hist 0 N File:GeometricAntiproduct.svg No edit summary
- 06:2906:29, 15 July 2023 diff hist 0 N File:GeometricProduct.svg No edit summary
- 06:2806:28, 15 July 2023 diff hist +6,603 N Exterior products Created page with "The ''exterior product'' is the fundamental product of Grassmann Algebra, and it forms part of the geometric product in geometric algebra. There are two products with symmetric properties called the exterior product and exterior antiproduct. The exterior product between two elements $$\mathbf a$$ and $$\mathbf b$$ generally combines their spatial extents, and the magnitude of the result indicates how close they are to being orthogonal. If the spatial extents of $$\m..."
- 06:2806:28, 15 July 2023 diff hist 0 N File:AntiwedgeProduct.svg No edit summary
- 06:2806:28, 15 July 2023 diff hist 0 N File:WedgeProduct.svg No edit summary
- 06:2706:27, 15 July 2023 diff hist +3,406 N Interior products Created page with "The left and right ''interior products'' are special products in geometric algebra that are useful for performing projections. These products cancel common factors in their operands and thus reduce grade. Depending on the choice of dualization function, there are several possible interior products. We define the interior products in terms of the left and right complements. Interior products are also known as contraction products. == Left and Right Interior Prod..."
- 06:2606:26, 15 July 2023 diff hist +1,779 N Dot products Created page with "The ''dot product'' is the inner product in geometric algebra, and it makes up the scalar part of the geometric product. There are two products with symmetric properties called the dot product and antidot product. The dot product and antidot product are important for the calculation of norms. == Dot Product == The dot product between two elements $$\mathbf a$$ and $$\mathbf b$$ is written $$\mathbf a \mathbin{\unicode{x25CF}} \mathbf b$$ and r..."
- 06:2406:24, 15 July 2023 diff hist +24 N Quaternions Redirected page to Quaternion current Tag: New redirect
- 06:2306:23, 15 July 2023 diff hist +3,118 N Quaternion Created page with "__NOTOC__ A ''quaternion'' is an operator that performs a rotation about the origin in 3D space. Conventionally, a quaternion $$\mathbf q$$ is written as :$$\mathbf q = q_w + q_x \mathbf i + q_y \mathbf j + q_z \mathbf k$$ , where the "imaginary" units $$\mathbf i$$, $$\mathbf j$$, and $$\mathbf k$$ all square to $$-1$$ and multiply according to the rules :$$\mathbf{ij} = -\mathbf{ji} = \mathbf k$$ :$$\mathbf{jk} = -\mathbf{kj} = \mathbf i$$ :$$\mathbf{ki} = -\mathbf{..." current
- 06:2006:20, 15 July 2023 diff hist +1,034 N Scalars and antiscalars Created page with "A ''scalar'' in a geometric algebra is an element having grade 0. Scalars are just ordinary real numbers, and they do not involve any basis vectors. The basis element representing the unit scalar is denoted by $$\mathbf 1$$, a boldface number one. The unit scalar $$\mathbf 1$$ is the multiplicative identity of the geometric product. For a general element $$\mathbf a$$, the notation $$a_{\mathbf 1}$$ means the scalar component of $$\mathbf a$$. An ''antiscalar'..." current
- 06:1606:16, 15 July 2023 diff hist +1,553 N Plane Created page with "400px|thumb|right|'''Figure 1.''' A plane is the intersection of a 4D trivector with the 3D subspace where $$w = 1$$. In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a ''plane'' $$\mathbf g$$ is a trivector having the general form :$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$ . All planes possess the geometric property. The bulk of a plane is given by its $$w$$ coordinate, a..."
- 06:1606:16, 15 July 2023 diff hist 0 N File:Plane.svg No edit summary
- 06:1506:15, 15 July 2023 diff hist +863 N Grade and antigrade Created page with "The ''grade'' of a basis element in a geometric algebra is equal to the number of basis vectors present in its factorization. An arbitrary element whose components all have the same grade is also said to have that grade. The ''antigrade'' of a basis element is equal to the number of basis vectors absent from its factorization. The grade of an element $$\mathbf x$$ is denoted by $$\operatorname{gr}(\mathbf x)$$, and the antigrade is denoted by $$\operatorname{ag}(\mathb..."