User contributions for Eric Lengyel

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23 October 2023

27 August 2023

25 August 2023

6 August 2023

  • 07:1207:12, 6 August 2023 diff hist +4‎ Carriers ‎ →‎Anticarrier
  • 03:5803:58, 6 August 2023 diff hist +20‎ N Spheres ‎ Redirected page to Sphere current Tag: New redirect
  • 03:5803:58, 6 August 2023 diff hist +20‎ N Circles ‎ Redirected page to Circle current Tag: New redirect
  • 03:5803:58, 6 August 2023 diff hist +20‎ N Dipoles ‎ Redirected page to Dipole current Tag: New redirect
  • 03:5703:57, 6 August 2023 diff hist +19‎ N Planes ‎ Redirected page to Plane current Tag: New redirect
  • 03:5703:57, 6 August 2023 diff hist +18‎ N Lines ‎ Redirected page to Line current Tag: New redirect
  • 03:5703:57, 6 August 2023 diff hist +24‎ N Flat points ‎ Redirected page to Flat point current Tag: New redirect
  • 03:5703:57, 6 August 2023 diff hist +25‎ N Round points ‎ Redirected page to Round point current Tag: New redirect
  • 03:2603:26, 6 August 2023 diff hist +31‎ N Exterior antiproduct ‎ Redirected page to Exterior products current Tag: New redirect
  • 03:2503:25, 6 August 2023 diff hist +32‎ N Geometric antiproduct ‎ Redirected page to Geometric products current Tag: New redirect
  • 03:2503:25, 6 August 2023 diff hist +32‎ N Geometric product ‎ Redirected page to Geometric products current Tag: New redirect
  • 03:2503:25, 6 August 2023 diff hist +31‎ N Antiwedge product ‎ Redirected page to Exterior products current Tag: New redirect
  • 03:2503:25, 6 August 2023 diff hist +31‎ N Wedge product ‎ Redirected page to Exterior products current Tag: New redirect
  • 03:2503:25, 6 August 2023 diff hist +25‎ N Projection ‎ Redirected page to Projections current Tag: New redirect
  • 03:2403:24, 6 August 2023 diff hist +19‎ N Dual ‎ Redirected page to Duals current Tag: New redirect
  • 03:2403:24, 6 August 2023 diff hist +22‎ N Carrier ‎ Redirected page to Carriers current Tag: New redirect
  • 03:2303:23, 6 August 2023 diff hist +22‎ N Partner ‎ Redirected page to Partners current Tag: New redirect
  • 03:2303:23, 6 August 2023 diff hist +24‎ N Container ‎ Redirected page to Containers current Tag: New redirect
  • 03:2003:20, 6 August 2023 diff hist +688‎ N Transversion ‎ Created page with "__NOTOC__ A ''transversion'' is a reciprocal transformation performed by the operator :$$\mathfrak T = {\dfrac{\tau_{x\vphantom{y}}}{2} \mathbf e_{423} + \dfrac{\tau_y}{2} \mathbf e_{431} + \dfrac{\tau_z{\vphantom{y}}}{2} \mathbf e_{412} + \large\unicode{x1d7d9}}$$ . == Matrix Form == When a transversion $$\mathfrak T$$ is applied to a round point, it is equivalent to premultiplying the point by the $$5 \times 5$$ matrix :$$\begin{bmatrix} 1 & 0 & 0 & 0 & -\tau_x..." current
  • 03:2003:20, 6 August 2023 diff hist +2,238‎ N Dilation ‎ Created page with "__NOTOC__ A ''dilation'' is a conformal transformation of Euclidean space performed by the operator :$$\mathbf D = \dfrac{1 - \sigma}{2} (c_x \mathbf e_{235} + c_y \mathbf e_{315} + c_z \mathbf e_{125} - \mathbf e_{321}) + \dfrac{1 + \sigma}{2} {\large\unicode{x1d7d9}}$$ . This operator scales an object $$\mathbf x$$ by the factor $$\sigma$$ about the center point $$\mathbf c = (c_x, c_y, c_z)$$ when used with the sandwich antiproduct $$\mathbf D \mathbin{\unicode{x27C..." current
  • 03:1903:19, 6 August 2023 diff hist +764‎ N Rotation ‎ Created page with "__NOTOC__ A ''rotation'' is a proper isometry of Euclidean space performed by the operator :$$\mathbf R = \boldsymbol l\sin\dfrac{\phi}{2} + {\large\unicode{x1d7d9}}\cos\dfrac{\phi}{2}$$ , where $$\boldsymbol l$$ is a unitized line corresponding to the axis of rotation. This operator is identical to the [http://rigidgeometricalgebra.org/wiki/index.php?title=Rotation rotation operator in rigid geometric algebra] but with the extra factor of $$\mathbf e_5$$. It rotat..." current
  • 03:1903:19, 6 August 2023 diff hist +1,718‎ N Translation ‎ Created page with "__NOTOC__ A ''translation'' is a proper isometry of Euclidean space performed by the operator :$$\mathbf T = {\dfrac{\tau_{x\vphantom{y}}}{2} \mathbf e_{235} + \dfrac{\tau_y}{2} \mathbf e_{315} + \dfrac{\tau_z{\vphantom{y}}}{2} \mathbf e_{125} + \large\unicode{x1d7d9}}$$ . This operator is identical to the [http://rigidgeometricalgebra.org/wiki/index.php?title=Translation translation operator in rigid geometric algebra] but with the extra factor of $$\mathbf e_5$$. It..." current
  • 03:1903:19, 6 August 2023 diff hist +563‎ N Projections ‎ Created page with "Any geometric object $$\mathbf x$$ can be projected onto another geometric object $$\mathbf y$$ of higher grade by first calculating the connect of $$\mathbf x$$ with $$\mathbf y$$ and then using the meet operation to intersect the result with $$\mathbf y$$. That is, the projection of $$\mathbf x$$ onto $$\mathbf y$$ is given by :$$(\mathbf y^* \wedge \mathbf x) \vee \mathbf y$$ . This formula is general and works for flat points, lines, planes, r..."
  • 03:1903:19, 6 August 2023 diff hist +12,873‎ N Expansion ‎ Created page with "The ''connect'' operation is performed by taking the wedge product between the dual of an object ''A'' and another object ''B'' with lower grade. The result is an object ''C'' that is orthogonal to ''A'' and contains ''B'', allowing a projection of ''B'' onto ''A'' through a simple intersection of ''A'' and ''C''. The flat points, lines, planes, round points, dipoles, circles, and spheres appearing in the following tables are defined..."
  • 03:1803:18, 6 August 2023 diff hist +14,963‎ 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 flat points, lines, planes, round points, dipoles, circles, and spheres appearing in the following tables are defined as follows: :$$\mathbf p = p_x \mathbf e_{15} +..."
  • 03:1803:18, 6 August 2023 diff hist −1,292‎ Wedge products ‎ Redirected page to Exterior products current Tag: New redirect
  • 03:1803:18, 6 August 2023 diff hist +1,323‎ N Exterior products ‎ Created page with "The ''exterior product'' is the fundamental product of Grassmann Algebra, and it forms part of the geometric product in geometric algebra. There are two products with symmetric properties called the exterior product and exterior antiproduct. == Exterior Product == The following Cayley table shows the exterior products between all pairs of basis elements in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$. 1440px == Exterior Anti..."
  • 03:1703:17, 6 August 2023 diff hist +1,323‎ N Wedge products ‎ Created page with "The ''exterior product'' is the fundamental product of Grassmann Algebra, and it forms part of the geometric product in geometric algebra. There are two products with symmetric properties called the exterior product and exterior antiproduct. == Exterior Product == The following Cayley table shows the exterior products between all pairs of basis elements in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$. 1440px == Exterior Anti..."
  • 03:1703:17, 6 August 2023 diff hist +1,223‎ N Geometric products ‎ Created page with "The ''geometric product'' is the fundamental product of geometric algebra. There are two products with symmetric properties called the geometric product and geometric antiproduct. == Geometric Product == The following Cayley table shows the geometric products between all pairs of basis elements in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$. The identity of the geometric product is the scalar basis element $$\mathbf 1$$. Cells colored yellow correspond..."
  • 03:1603:16, 6 August 2023 diff hist +3,293‎ N Duals ‎ Created page with "The ''dual'' of an object $$\mathbf x$$, denoted by $$\mathbf x^*$$, is given by :$$\mathbf x^* = \mathbf{\tilde x} \mathbin{\unicode{x27D1}} {\large\unicode{x1d7d9}}$$ . The following table lists the duals for the geometric objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$. {| class="wikitable" ! Type !! Definition !! Dual |- | style="padding: 12px;" | Flat point | style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_{15} + p_y \mathbf e_{25} + p..."
  • 03:1603:16, 6 August 2023 diff hist +3,794‎ 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 following table lists the attitude for the geometric objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$. {| class="wikitable" ! Type !! Definition !! Attitude |- | style="padding: 12px;" | Flat poin..."
  • 03:1603:16, 6 August 2023 diff hist +3,776‎ N Partners ‎ Created page with "The ''partner'' of a round object (a round point, dipole, circle, or sphere) is the round object having the same center, same carrier, and same absolute size, but having a squared radius of the opposite sign. The partner of an object $$\mathbf x$$ is denoted by $$\operatorname{par}(\mathbf x)$$, and it is given by the meet of the carrier of $$\mathbf x$$ with the container of $$\mathbf x^*$$: :$$\operatorname{par}(\mathbf x) = \operatorna..."
  • 03:1603:16, 6 August 2023 diff hist +4,839‎ N Carriers ‎ Created page with "== Carrier == The ''carrier'' of a round object (a round point, dipole, circle, or sphere) is the lowest dimensional flat object (a flat point, line, or plane) that contains it. The carrier of an object $$\mathbf x$$ is denoted by $$\operatorname{car}(\mathbf x)$$, and it is calculated by simply multiplying $$\mathbf x$$ by $$\mathbf e_5$$ with the wedge product to extract the round part of $$\mathbf x$$ as a flat geometry: :$$\operatorn..."
  • 03:1603:16, 6 August 2023 diff hist +3,359‎ N Centers ‎ Created page with "The ''center'' of a round object (a round point, dipole, circle, or sphere) is the round point having the same center and radius. The center of an object $$\mathbf x$$ is denoted by $$\operatorname{cen}(\mathbf x)$$, and it is given by the meet of $$\mathbf x$$ and its own anticarrier: :$$\operatorname{cen}(\mathbf x) = -\operatorname{car}(\mathbf x^*) \vee \mathbf x$$ . (The negative sign is not strictly necessary, but is included so the fu..."
  • 03:1603:16, 6 August 2023 diff hist −3,338‎ Center ‎ Redirected page to Centers current Tag: New redirect
  • 03:1503:15, 6 August 2023 diff hist +3,279‎ N Containers ‎ Created page with "The ''container'' of a round object (a round point, dipole, circle, or sphere) is the smallest sphere that contains it. The container of an object $$\mathbf x$$ is denoted by $$\operatorname{con}(\mathbf x)$$, and it is given by the connect of $$\mathbf x$$ with its own carrier: :$$\operatorname{con}(\mathbf x) = \operatorname{car}(\mathbf x)^* \wedge \mathbf x$$ . The squared radius of an object's container has the same sign as the squared radi..."
  • 03:1503:15, 6 August 2023 diff hist +3,359‎ N Center ‎ Created page with "The ''center'' of a round object (a round point, dipole, circle, or sphere) is the round point having the same center and radius. The center of an object $$\mathbf x$$ is denoted by $$\operatorname{cen}(\mathbf x)$$, and it is given by the meet of $$\mathbf x$$ and its own anticarrier: :$$\operatorname{cen}(\mathbf x) = -\operatorname{car}(\mathbf x^*) \vee \mathbf x$$ . (The negative sign is not strictly necessary, but is included so the fu..."
  • 03:1503:15, 6 August 2023 diff hist +2,931‎ N Sphere ‎ Created page with "__NOTOC__ In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''sphere'' $$\mathbf s$$ is a quadrivector having the general form :$$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$ . If the $$s_u$$ component is zero, then the sphere contains the point at infinity, and it is thus a flat plane. Given a center $$\mathbf p = (p_x, p_y, p_z)$$ and a radius $$r$$, a sphere can be..."
  • 03:1503:15, 6 August 2023 diff hist +5,915‎ N Circle ‎ Created page with "__NOTOC__ In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''circle'' $$\mathbf c$$ is a trivector with ten components having the general form :$$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$ . If the $$gx$$, $$gy$$, $$gz$$, and $$gw$$ componen..."
  • 03:1403:14, 6 August 2023 diff hist +5,689‎ N Dipole ‎ Created page with "__NOTOC__ In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''dipole'' $$\mathbf d$$ is a bivector with ten components having the general form :$$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$ . If the $$vx$$, $$vy$$, $$vz$$, $$mx$$, $$my$$, and $$mz$$ com..."
  • 03:1403:14, 6 August 2023 diff hist +1,626‎ N Round point ‎ Created page with "__NOTOC__ In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''round point'' $$\mathbf a$$ is a vector having the general form :$$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$ . Given a position $$\mathbf p = (p_x, p_y, p_z)$$ and a radius $$r$$, a round point can be formulated as :$$\mathbf a = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + \mathbf e_4 + \dfrac{p^2 + r^2}{2} \mathbf e_5$$ . Th..."
  • 03:1403:14, 6 August 2023 diff hist +837‎ N Plane ‎ Created page with "400px|thumb|right|'''Figure 1.''' The various properties of a plane. In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''plane'' $$\mathbf g$$ is a quadrivector having the general form :$$\mathbf g = g_x \mathbf e_{4235} + g_y \mathbf e_{4315} + g_z \mathbf e_{4125} + g_w \mathbf e_{3215}$$ . A plane can be viewed as an infinitely large sphere containing the point at infinity. A plane in conformal geometric algebra is the precise..." current
  • 03:1403:14, 6 August 2023 diff hist +878‎ N Line ‎ Created page with "400px|thumb|right|'''Figure 1.''' The various properties of a line. In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''line'' $$\boldsymbol l$$ is a trivector having the general form :$$\boldsymbol l = v_x \mathbf e_{415} + v_y \mathbf e_{425} + v_z \mathbf e_{435} + m_x \mathbf e_{235} + m_y \mathbf e_{315} + m_z \mathbf e_{125}$$ . A line can be viewed as an infinitely large circle that contains the point at infinity. A line in..." current
  • 03:1303:13, 6 August 2023 diff hist +836‎ N Flat point ‎ Created page with "400px|thumb|right|'''Figure 1.''' The various properties of a flat point. In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''flat point'' $$\mathbf p$$ is a bivector having the general form :$$\mathbf p = p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + p_w \mathbf e_{45}$$ . A flat point can be viewed as a dipole that has one end at the point at infinity. A flat point in conformal geometric algebra is the precise..." current
  • 03:1203:12, 6 August 2023 diff hist 0‎ N File:Wedge product.svg ‎ Uploaded with SimpleBatchUpload current
  • 03:1203:12, 6 August 2023 diff hist 0‎ N File:Sphere meet sphere.svg ‎ Uploaded with SimpleBatchUpload current
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