User contributions for Eric Lengyel

15 July 2023

• 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..."
• 06:1406:14, 15 July 2023 diff hist +2,029‎ N Reverses ‎ Created page with "''Reverses'' are unary operations in geometric algebra that are analogs of conjugate or transpose operations. For any element $$\mathbf x$$ that is the wedge product of $$k$$ vectors, the ''reverse'' of $$\mathbf x$$, which we denote by $$\mathbf{\tilde x}$$, is the result of multiplying those same $$k$$ vectors in reverse order. For example, the reverse of $$\mathbf e_{234}$$ is $$\mathbf e_4 \wedge \mathbf e_3 \wedge \mathbf e_2$$, which we would write as $$-\math..."
• 06:1406:14, 15 July 2023 diff hist 0‎ N File:Reverses.svg ‎No edit summary
• 06:1106:11, 15 July 2023 diff hist +7,517‎ N Projections ‎ Created page with "Projections and antiprojections of one geometric object onto another can be accomplished using interior products as described below. The formulas on this page are general and do not require the geometric objects to be unitized. Most of them become simpler if unitization can be assumed. == Projection == The geometric projection of an object $$\mathbf x$$ onto an object $$\mathbf y$$ is given by the general formula $$(\mathbf y_\unicode{x25CB} \mathbin{\unicode{..."
• 06:1006:10, 15 July 2023 diff hist +3,533‎ N Euclidean distance ‎ Created page with "The Euclidean distance between geometric objects can be measured by using commutators to calculate homogeneous magnitudes. The following table lists formulas for Euclidean distances between the main types of geometric objects in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. These formulas are general and do not require the geometric objects to be unitized. Most of them become simpler if unitization can be assumed. The points, lines, and p..."
• 06:1006:10, 15 July 2023 diff hist 0‎ N File:Distance line line.svg ‎No edit summary current
• 06:1006:10, 15 July 2023 diff hist 0‎ N File:Distance point plane.svg ‎No edit summary current
• 06:0906:09, 15 July 2023 diff hist 0‎ N File:Distance point line.svg ‎No edit summary current
• 06:0906:09, 15 July 2023 diff hist 0‎ N File:Distance point point.svg ‎No edit summary current
• 06:0606:06, 15 July 2023 diff hist +12,199‎ N Flector ‎ Created page with "400px|thumb|right|'''Figure 1.''' A flector represents an improper Euclidean isometry, which can always be regarded as a rotation about a line $$\boldsymbol l$$ and a reflection across a plane perpendicular to the same line. A ''flector'' is an operator that performs an improper isometry in Euclidean space. Such isometries encompass all possible combinations of an odd number of reflections, inversions, transflections, and rotorefle..."
• 06:0606:06, 15 July 2023 diff hist 0‎ N File:Improper isom.svg ‎No edit summary
• 06:0406:04, 15 July 2023 diff hist 0‎ N File:Complements.svg ‎No edit summary
• 06:0406:04, 15 July 2023 diff hist +7,055‎ N Complements ‎ Created page with "''Complements'' are unary operations in geometric algebra that perform a specific type of dualization. Every basis element $$\mathbf x$$ has a ''right complement'', which we denote by $$\overline{\mathbf x}$$, that satisfies the equation :$$\mathbf x \wedge \overline{\mathbf x} = {\large\unicode{x1D7D9}}$$ . There is also a ''left complement'', which we denote by $$\underline{\mathbf x}$$, that satisfies the equation :$$\underline{\mathbf x} \wedge \mathbf x = {\larg..."
• 06:0306:03, 15 July 2023 diff hist +2,788‎ N Geometric constraint ‎ Created page with "An element $$\mathbf x$$ of a geometric algebra possesses the ''geometric property'' if and only if the geometric product between $$\mathbf x$$ and its own reverse is a scalar, which is given by the dot product, and the geometric antiproduct between $$\mathbf x$$ and its own antireverse is an antiscalar, which is given by the antidot product. That is, :$$\mathbf x \mathbin{\unicode{x27D1}} \mathbf{\tilde x} = \mathbf x \mathbin{\unicode{x25CF}} \mathbf{\..."
• 06:0306:03, 15 July 2023 diff hist +2,586‎ N Unitization ‎ Created page with "''Unitization'' is the process of scaling an element of a projective geometric algebra so that its weight norm becomes the antiscalar $$\large\unicode{x1D7D9}$$. An element that has a weight norm of $$\large\unicode{x1D7D9}$$ is said to be ''unitized''. An element $$\mathbf x$$ is unitized by calculating :$$\mathbf{\hat x} = \dfrac{\mathbf x}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} = \dfrac{\mathbf x}{\sqrt{\mathbf x \mathbin{\unicode{x25CB}} \smash{\ma..."
• 06:0206:02, 15 July 2023 diff hist +10,552‎ N Geometric norm ‎ Created page with "The ''geometric norm'' is a measure of the magnitude of an element. It has two components called the bulk norm and the weight norm. For points, lines, and planes, the geometric norm is equal to the shortest Euclidean distance between the geometry and the origin. For motors and flectors, the geometric norm is equal to half the distance that the origin is moved by the isometry operator. == Bulk Norm == The ''bulk norm'' of an element $$\mathbf x$$, d..."
• 05:5905:59, 15 July 2023 diff hist +5,971‎ N Complement rotation ‎ Created page with "A ''dual rotation'' is a proper isometry of dual Euclidean space. For a bulk normalized line $$\boldsymbol l$$, the specific kind of dual motor :$$\mathbf R = \boldsymbol l\sin\phi + \mathbf 1\cos\phi$$ , performs a dual rotation of an object $$\mathbf x$$ by twice the angle $$\phi$$ with the sandwich product $$\mathbf R \mathbin{\unicode{x27D1}} \mathbf x \mathbin{\unicode{x27D1}} \mathbf{\tilde R}$$. The line $$\boldsymbol l$$ and its bulk complement..."
• 05:5905:59, 15 July 2023 diff hist 0‎ N File:DualRotation.svg ‎No edit summary current
• 05:5905:59, 15 July 2023 diff hist 0‎ N File:Rotation.svg ‎No edit summary current
• 05:5905:59, 15 July 2023 diff hist +2,936‎ N Complement translation ‎ Created page with "__NOTOC__ A ''dual translation'' is a proper isometry of dual Euclidean space. The specific kind of dual motor :$$\mathbf T = t_x \mathbf e_{41} + t_y \mathbf e_{42} + t_z \mathbf e_{43} + \mathbf 1$$ performs a perspective projection in the direction of $$\mathbf t = (t_x, t_y, t_z)$$ with the focal length given by :$$g = \dfrac{1}{2\Vert \mathbf t \Vert}$$ . == Example == The left image below shows the flow field in the ''x''-''z'' plane for the translation $..."
• 05:5805:58, 15 July 2023 diff hist 0‎ N File:DualTranslation.svg ‎No edit summary current