User contributions for Eric Lengyel
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6 August 2023
- 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
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere meet point.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere meet plane.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere meet line.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere meet dipole.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere meet circle.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere connect round.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Round.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere connect point.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere connect line.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere connect dipole.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere connect circle.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Round join round.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Point join round.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Point.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Point connect round.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Plane meet plane.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Plane meet line.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Plane meet dipole.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Plane meet circle.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Plane connect round.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Plane connect point.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Plane connect line.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Plane connect dipole.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Plane.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:GeometricProduct.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Plane connect circle.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Dipole.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Line join round.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:GeometricAntiproduct.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Line connect round.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Line.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Line connect point.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Line connect dipole.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Horosphere.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Dipole join round.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Dipole join point.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Circle.svg Uploaded with SimpleBatchUpload
