User contributions for Eric Lengyel

15 July 2023

• 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}$$ . == Example == The left image below shows the flow field in the ''x''-''z'' plane for the translation $..."
• 05:5805:58, 15 July 2023 diff hist 0‎ N File:DualTranslation.svg ‎No edit summary current
• 05:5805:58, 15 July 2023 diff hist 0‎ N File:Translation.svg ‎No edit summary current
• 05:5705:57, 15 July 2023 diff hist +2,900‎ N Translation ‎ Created page with "__NOTOC__ A ''translation'' is a proper isometry of Euclidean space. The specific kind of motor :$$\mathbf T = {t_x \mathbf e_{23} + t_y \mathbf e_{31} + t_z \mathbf e_{12} + \large\unicode{x1d7d9}}$$ performs a translation by twice the displacement vector $$\mathbf t = (t_x, t_y, t_z)$$ when used as an operator in the sandwich antiproduct. This can be interpreted as a rotation about the line at infinity perpendicular to the direction $$\mathbf t$$. === Trans..."
• 05:5705:57, 15 July 2023 diff hist +4,695‎ N Rotation ‎ Created page with "A ''rotation'' is a proper isometry of Euclidean space. For a unitized line $$\boldsymbol l$$, the specific kind of motor :$$\mathbf R = \boldsymbol l\sin\phi + {\large\unicode{x1d7d9}}\cos\phi$$ , performs a rotation by twice the angle $$\phi$$ about the line $$\boldsymbol l$$ with the sandwich product $$\mathbf R \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \mathbf{\underset{\Large\unicode{x7E}}{R}}$$. The operator $$\mathb..."
• 05:5605:56, 15 July 2023 diff hist +3,321‎ N Reflection ‎ Created page with "A ''reflection'' is an improper isometry of Euclidean space. When used as an operator in the sandwich antiproduct, a unitized plane $$\mathbf F = F_{gx}\mathbf e_{423} + F_{gy}\mathbf e_{431} + F_{gz}\mathbf e_{412} + F_{gw}\mathbf e_{321}$$ is a specific kind of flector that performs a reflection through $$\mathbf F$$. == Calculation == The exact reflection calculations for points, lines, and planes are shown in the following table. {| class="wikitable"..." current
• 05:5505:55, 15 July 2023 diff hist +2,029‎ N Inversion ‎ Created page with "An ''inversion'' is an improper isometry of Euclidean space. When used as an operator in the sandwich antiproduct, a unitized point $$\mathbf F = F_{px}\mathbf e_1 + F_{py}\mathbf e_2 + F_{pz}\mathbf e_3 + \mathbf e_4$$ is a specific kind of flector that performs an inversion through $$\mathbf F$$. == Calculation == The exact inversion calculations for points, lines, and planes are shown in the following table. {| class="wikitable" ! Type || Inversion |-..." current
• 05:5405:54, 15 July 2023 diff hist +4,564‎ N Transflection ‎ Created page with "A ''transflection'' is an improper isometry of Euclidean space consisting of a reflection through a plane and a translation parallel to the same plane. All combinations of a reflection and a translation, even if the original translation vector is not parallel to the original reflection plane, can be formulated as a transflection with respect to some plane. The specific kind of flector :$$\mathbf F = F_{px} \mathbf e_{1} + F_{py} \mathbf e_{2} + F_{pz} \math..."
• 05:5205:52, 15 July 2023 diff hist 0‎ N File:Groups.svg ‎No edit summary
• 05:5205:52, 15 July 2023 diff hist +6,554‎ N Transformation groups ‎ Created page with "In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, every Euclidean isometry of 3D space can be represented by a motor $$\mathbf Q$$ of the form :$$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$ or by a flector $$\mathbf F$$ of the form :$$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_..."
• 05:5005:50, 15 July 2023 diff hist +1,064‎ N Magnitude ‎ Created page with "A ''magnitude'' is a quantity that represents a concrete distance of some kind. In rigid geometric algebra, a magnitude $$\mathbf z$$ is composed of two components, a scalar and an antiscalar, as follows: :$$\mathbf z = x\mathbf 1 + y {\large\unicode{x1d7d9}}$$ Magnitudes are homogeneous just like everything else in a projective geometric algebra. This means it has both a bulk and a weight, and it is unitized by making the magnitude of its weight one. ===..."
• 05:4605:46, 15 July 2023 diff hist 0‎ N File:GeometricAntiproduct201.svg ‎No edit summary
• 05:4605:46, 15 July 2023 diff hist 0‎ N File:GeometricProduct201.svg ‎No edit summary
• 05:4605:46, 15 July 2023 diff hist 0‎ N File:Unary201.svg ‎No edit summary current
• 05:4605:46, 15 July 2023 diff hist 0‎ N File:Basis201.svg ‎No edit summary current
• 05:4605:46, 15 July 2023 diff hist +17,445‎ N Rigid Geometric Algebra for 2D Space ‎ Created page with "== Introduction == thumb|right|400px|'''Table 1.''' The 8 basis elements of the 3D rigid geometric algebra. In the three-dimensional rigid geometric algebra, there are 8 graded basis elements. These are listed in Table 1. There is a single ''scalar'' basis element $$\mathbf 1$$, and its multiples correspond to the real numbers, which are values that have no dimensions. There are three ''vector'' basis elements named $$\mathbf e_1$$, $$\mathbf e_..."
• 05:3905:39, 15 July 2023 diff hist 0‎ N File:Line meet plane.svg ‎No edit summary current
• 05:3905:39, 15 July 2023 diff hist 0‎ N File:Plane meet plane.svg ‎No edit summary current
• 05:3905:39, 15 July 2023 diff hist 0‎ N File:Line join point.svg ‎No edit summary current
• 05:3905:39, 15 July 2023 diff hist 0‎ N File:Point join point.svg ‎No edit summary current
• 05:3805:38, 15 July 2023 diff hist +5,522‎ N Join and meet ‎ Created page with "The ''join'' is a binary operation that calculates the higher-dimensional geometry containing its two operands, similar to a union. The ''meet'' is another binary operation that calculates the lower-dimensional geometry shared by its two operands, similar to an intersection. The points, lines, and planes appearing in the following tables are defined as follows: :$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ :$$\mathbf..."
• 05:3405:34, 15 July 2023 diff hist 0‎ N File:Skew lines.svg ‎No edit summary current
• 05:3405:34, 15 July 2023 diff hist 0‎ N File:Line infinity.svg ‎No edit summary
• 05:3405:34, 15 July 2023 diff hist 0‎ N File:Line.svg ‎No edit summary
• 05:3405:34, 15 July 2023 diff hist +4,278‎ N Line ‎ Created page with "400px|thumb|right|'''Figure 1.''' A line is the intersection of a 4D bivector with the 3D subspace where $$w = 1$$. In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a ''line'' $$\boldsymbol l$$ is a bivector having the general form :$$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$ . The components $$(l_{vx}, l_{vy}, l_{vz})$$ corr..."
• 05:3305:33, 15 July 2023 diff hist +695‎ Bulk and weight ‎No edit summary
• 05:3105:31, 15 July 2023 diff hist +3,283‎ N Bulk and weight ‎ Created page with "The bulk generally contains information about the position of an element relative to the origin, and the weight generally contains information about the attitude and orientation of an element. An object with zero bulk contains the origin. An object with zero weight is contained by the horizon. An element is unitized when the magnitude of its weight is one. The following table lists the bulk and weight for the main types in the 4D rigid geometric algebra $$\mathcal..."
• 05:2905:29, 15 July 2023 diff hist +1,645‎ N Attitude ‎ Created page with "The attitude function, denoted by $$\operatorname{att}$$, extracts the attitude of a geometry and returns a purely directional object. The attitude function is defined as :$$\operatorname{att}(\mathbf x) = \mathbf x \vee \overline{\mathbf e_4}$$ . The attitude of a line is the line's direction as a vector, and the attitude of a plane is the plane's normal as a bivector. The following table lists the attitude for the main types in the 4D rigid geometric algebra..."
• 05:2705:27, 15 July 2023 diff hist 0‎ N File:Proper isom.svg ‎No edit summary
• 05:2705:27, 15 July 2023 diff hist +21,325‎ N Motor ‎ Created page with "400px|thumb|right|'''Figure 1.''' A motor represents a proper Euclidean isometry, which can always be regarded as a rotation about a line $$\mathbf L$$ and a displacement along the same line. A ''motor'' is an operator that performs a proper isometry in Euclidean space. Such isometries encompass all possible combinations of any number of rotations and translations. The name motor is a portmanteau of ''motion operator'' or ''moment vector..."
• 05:2405:24, 15 July 2023 diff hist 0‎ N File:Basis.svg ‎No edit summary
• 05:2305:23, 15 July 2023 diff hist +7,232‎ Main Page ‎No edit summary