User contributions for Eric Lengyel

Jump to navigation Jump to search
Search for contributionsExpandCollapse
⧼contribs-top⧽
⧼contribs-date⧽
(newest | oldest) View ( | ) (20 | 50 | 100 | 250 | 500)

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 SpheresRedirected page to Sphere current Tag: New redirect
  • 03:5803:58, 6 August 2023 diff hist +20 N CirclesRedirected page to Circle current Tag: New redirect
  • 03:5803:58, 6 August 2023 diff hist +20 N DipolesRedirected page to Dipole current Tag: New redirect
  • 03:5703:57, 6 August 2023 diff hist +19 N PlanesRedirected page to Plane current Tag: New redirect
  • 03:5703:57, 6 August 2023 diff hist +18 N LinesRedirected page to Line current Tag: New redirect
  • 03:5703:57, 6 August 2023 diff hist +24 N Flat pointsRedirected page to Flat point current Tag: New redirect
  • 03:5703:57, 6 August 2023 diff hist +25 N Round pointsRedirected page to Round point current Tag: New redirect
  • 03:2603:26, 6 August 2023 diff hist +31 N Exterior antiproductRedirected page to Exterior products current Tag: New redirect
  • 03:2503:25, 6 August 2023 diff hist +32 N Geometric antiproductRedirected page to Geometric products current Tag: New redirect
  • 03:2503:25, 6 August 2023 diff hist +32 N Geometric productRedirected page to Geometric products current Tag: New redirect
  • 03:2503:25, 6 August 2023 diff hist +31 N Antiwedge productRedirected page to Exterior products current Tag: New redirect
  • 03:2503:25, 6 August 2023 diff hist +31 N Wedge productRedirected page to Exterior products current Tag: New redirect
  • 03:2503:25, 6 August 2023 diff hist +25 N ProjectionRedirected page to Projections current Tag: New redirect
  • 03:2403:24, 6 August 2023 diff hist +19 N DualRedirected page to Duals current Tag: New redirect
  • 03:2403:24, 6 August 2023 diff hist +22 N CarrierRedirected page to Carriers current Tag: New redirect
  • 03:2303:23, 6 August 2023 diff hist +22 N PartnerRedirected page to Partners current Tag: New redirect
  • 03:2303:23, 6 August 2023 diff hist +24 N ContainerRedirected page to Containers current Tag: New redirect
  • 03:2003:20, 6 August 2023 diff hist +688 N TransversionCreated 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 :$$\begin{bmatrix} 1 & 0 & 0 & 0 & -\tau_x..." current
  • 03:2003:20, 6 August 2023 diff hist +2,238 N DilationCreated page with "__NOTOC__ A ''dilation'' is a conformal transformation of Euclidean space performed by the operator :$$\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 RotationCreated 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 TranslationCreated 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 ProjectionsCreated 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 ExpansionCreated 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 meetCreated 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: :$$\mathbf p = p_x \mathbf e_{15} +..."
  • 03:1803:18, 6 August 2023 diff hist −1,292 Wedge productsRedirected page to Exterior products current Tag: New redirect
  • 03:1803:18, 6 August 2023 diff hist +1,323 N Exterior productsCreated page with "The ''exterior product'' is the fundamental product of Grassmann Algebra, and it forms part of the geometric product in geometric algebra. There are two products with symmetric properties called the exterior product and exterior antiproduct. == Exterior Product == The following Cayley table shows the exterior products between all pairs of basis elements in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$. 1440px == Exterior Anti..."
  • 03:1703:17, 6 August 2023 diff hist +1,323 N Wedge productsCreated page with "The ''exterior product'' is the fundamental product of Grassmann Algebra, and it forms part of the geometric product in geometric algebra. There are two products with symmetric properties called the exterior product and exterior antiproduct. == Exterior Product == The following Cayley table shows the exterior products between all pairs of basis elements in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$. 1440px == Exterior Anti..."
  • 03:1703:17, 6 August 2023 diff hist +1,223 N Geometric productsCreated page with "The ''geometric product'' is the fundamental product of geometric algebra. There are two products with symmetric properties called the geometric product and geometric antiproduct. == Geometric Product == The following Cayley table shows the geometric products between all pairs of basis elements in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$. The identity of the geometric product is the scalar basis element $$\mathbf 1$$. Cells colored yellow correspond..."
  • 03:1603:16, 6 August 2023 diff hist +3,293 N DualsCreated page with "The ''dual'' of an object $$\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 AttitudeCreated 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 PartnersCreated 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 $$\mathbf x^*$$: :$$\operatorname{par}(\mathbf x) = \operatorna..."
  • 03:1603:16, 6 August 2023 diff hist +4,839 N CarriersCreated page with "== Carrier == The ''carrier'' of a round object (a round point, dipole, circle, or sphere) is the lowest dimensional flat object (a flat point, line, or plane) that contains it. The carrier of an object $$\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 CentersCreated 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..."
(newest | oldest) View ( | ) (20 | 50 | 100 | 250 | 500)