User contributions for Eric Lengyel
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15 July 2023
- 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"..."
- 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
