User contributions for Eric Lengyel
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15 July 2023
- 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