User contributions for Eric Lengyel

15 July 2023

• 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