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| == Introduction ==
| | The components of an element of a rigid geometric algebra can be divided into two groups called the ''bulk'' and the ''weight'' of the element. The bulk of an element $$\mathbf a$$ is denoted by $$\mathbf a_\unicode{x25CF}$$, and it consists of the components of $$\mathbf a$$ that do not have the projective basis vector as a factor. The weight is denoted by $$\mathbf a_\unicode{x25CB}$$, and it consists of the components that do have the projective basis vector as a factor. In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, the bulk is thus all components that do not contain the factor $$\mathbf e_4$$, and the weight is all components that do contain the factor $$\mathbf e_4$$. |
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| [[Image:Basis201.svg|thumb|right|400px|'''Table 1.''' The 8 basis elements of the 3D rigid geometric algebra.]]
| | 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. |
| In the three-dimensional rigid geometric algebra, there are 8 graded basis elements. These are listed in Table 1.
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| There is a single ''scalar'' basis element $$\mathbf 1$$, and its multiples correspond to the real numbers, which are values that have no dimensions.
| | An element is [[unitized]] when the magnitude of its weight is one. |
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| There are three ''vector'' basis elements named $$\mathbf e_1$$, $$\mathbf e_2$$, and $$\mathbf e_3$$ that have one-dimensional extents. A general vector $$\mathbf v = (v_x, v_y, v_z)$$ has the form
| | The following table lists the bulk and weight for the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. |
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| :$$\mathbf v = v_x \mathbf e_1 + v_y \mathbf e_2 + v_z \mathbf e_3$$ .
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| There are three ''bivector'' basis elements named $$\mathbf e_{23}$$, $$\mathbf e_{31}$$, and $$\mathbf e_{12}$$ having two-dimensional extents.
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| Finally, there is a single ''trivector'' basis element $${\large\unicode{x1D7D9}} = \mathbf e_3 \wedge \mathbf e_2 \wedge \mathbf e_1$$ having three-dimensional extents.
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| __TOC__
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| <br clear="right" />
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| == Unary Operations ==
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| The 3D rigid geometric algebra has a single [[complement]] operation, a [[reverse]] operation, and an [[antireverse]] operation. (In three dimensions, the left and right [[complements]] are identical.) These are listed for all basis elements in the following table.
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| [[Image:Unary201.svg|480px]]
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| == Geometric Products ==
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| The geometric product is characterized by a metric that defines the products of the basis vectors with themselves. The subscript in $$\mathcal G_{2,0,1}$$ means that two basis vectors square to +1, zero basis vectors square to −1, and one basis vector squares to 0. The geometric product between two different basis vectors is given by the [[wedge product]]. We can write these rules as follows.
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| :$$\mathbf e_1 \mathbin{\unicode{x27D1}} \mathbf e_1 = 1$$
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| :$$\mathbf e_2 \mathbin{\unicode{x27D1}} \mathbf e_2 = 1$$
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| :$$\mathbf e_3 \mathbin{\unicode{x27D1}} \mathbf e_3 = 0$$
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| :$$\mathbf e_i \mathbin{\unicode{x27D1}} \mathbf e_j = \mathbf e_i \wedge \mathbf e_j$$, for $$i \neq j$$.
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| The following Cayley table shows the geometric products between all pairs of basis elements in the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$. The identity of the geometric product is the [[scalar]] basis element $$\mathbf 1$$.
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| [[Image:GeometricProduct201.svg|360px]]
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| The following Cayley table shows the geometric antiproducts between all pairs of basis elements in the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$. The identity of the geometric antiproduct is the [[antiscalar]] basis element $$\large\unicode{x1D7D9}$$.
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| [[Image:GeometricAntiproduct201.svg|360px]]
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| == Points ==
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| In the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$, a ''point'' $$\mathbf p$$ is a vector having the general form
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| :$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3$$ .
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| The [[bulk]] of a point is given by its $$x$$ and $$y$$ coordinates, and the [[weight]] of a point is given by its $$z$$ coordinate. A point is [[unitized]] when $$p_z^2 = 1$$.
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| When used as an operator in the sandwich product, a point is a specific kind of [[motor]] that performs a [[rotation]] about itself.
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| If the weight of a point is zero (i.e., its $$z$$ coordinate is zero), then the point lies at infinity in the direction $$(x, y)$$, and it cannot be unitized. A point with zero weight can also be interpreted as a direction vector, and it is normalized to unit length by dividing by its [[bulk norm]].
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| == Lines ==
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| In the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$, a ''line'' $$\mathbf L$$ is a bivector having the general form
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| :$$\mathbf L = L_x \mathbf e_{23} + L_y \mathbf e_{31} + L_z \mathbf e_{12}$$ .
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| The [[bulk]] of a line is given by its $$z$$ coordinate, and the [[weight]] of a line is given by its $$x$$ and $$y$$ coordinates. A line is [[unitized]] when $$L_x^2 + L_y^2 = 1$$.
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| When used as an operator in the sandwich product, a line is a specific kind of [[flector]] that performs a [[reflection]] through itself.
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| If the weight of a line is zero (i.e., its $$x$$ and $$y$$ coordinates are both zero), then the line lies at infinity in all directions. Such a line is normalized when $$L_z = \pm 1$$. This is the ''horizon'' of two-dimensional space.
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| == Bulk and Weight ==
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| The following table lists the [[bulk]] and [[weight]] for the main types in the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$. | |
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| {| class="wikitable" | | {| class="wikitable" |
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| | style="padding: 12px;" | [[Point]] | | | style="padding: 12px;" | [[Point]] |
| | style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3$$ | | | style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ |
| | style="padding: 12px;" | $$\mathbf p_\unicode{x25CF} = p_x \mathbf e_1 + p_y \mathbf e_2$$ | | | style="padding: 12px;" | $$\mathbf p_\unicode{x25CF} = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3$$ |
| | style="padding: 12px;" | $$\mathbf p_\unicode{x25CB} = p_z \mathbf e_3$$ | | | style="padding: 12px;" | $$\mathbf p_\unicode{x25CB} = p_w \mathbf e_4$$ |
| |- | | |- |
| | style="padding: 12px;" | [[Line]] | | | style="padding: 12px;" | [[Line]] |
| | style="padding: 12px;" | $$\mathbf L = L_x \mathbf e_{23} + L_y \mathbf e_{31} + L_z \mathbf e_{12}$$ | | | style="padding: 12px;" | $$\mathbf L = v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} + m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$ |
| | style="padding: 12px;" | $$\mathbf L_\unicode{x25CF} = L_z \mathbf e_{12}$$ | | | style="padding: 12px;" | $$\mathbf L_\unicode{x25CF} = m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$ |
| | style="padding: 12px;" | $$\mathbf L_\unicode{x25CB} = L_x \mathbf e_{23} + L_y \mathbf e_{31}$$ | | | style="padding: 12px;" | $$\mathbf L_\unicode{x25CB} = v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43}$$ |
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| | style="padding: 12px;" | [[Motor]] | | | style="padding: 12px;" | [[Plane]] |
| | style="padding: 12px;" | $$\mathbf Q = q_x \mathbf e_{1} + q_y \mathbf e_{2} + q_z \mathbf e_{3} + q_w {\large\unicode{x1d7d9}}$$ | | | style="padding: 12px;" | $$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$ |
| | style="padding: 12px;" | $$\mathbf Q_\unicode{x25CF} = q_x \mathbf e_{1} + q_y \mathbf e_{2}$$
| | | style="padding: 12px;" | $$\mathbf f_\unicode{x25CF} = f_w \mathbf e_{321}$$ |
| | style="padding: 12px;" | $$\mathbf Q_\unicode{x25CB} = q_z \mathbf e_{3} + q_w {\large\unicode{x1d7d9}}$$
| | | style="padding: 12px;" | $$\mathbf f_\unicode{x25CB} = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124}$$ |
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| | style="padding: 12px;" | [[Flector]]
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| | style="padding: 12px;" | $$\mathbf G = g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12} + g_w$$
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| | style="padding: 12px;" | $$\mathbf G_\unicode{x25CF} = g_z \mathbf e_{12} + g_w$$ | |
| | style="padding: 12px;" | $$\mathbf G_\unicode{x25CB} = g_x \mathbf e_{23} + g_y \mathbf e_{31}$$ | |
| |}
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| == Unitization ==
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| The following table lists the [[unitization]] conditions for the main types in the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$.
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| {| class="wikitable"
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| ! Type !! Definition !! Unitization
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| | style="padding: 12px;" | [[Magnitude]]
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| | style="padding: 12px;" | $$\mathbf z = x + y {\large\unicode{x1d7d9}}$$
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| | style="padding: 12px;" | $$y^2 = 1$$
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| |-
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| | style="padding: 12px;" | [[Point]]
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| | style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3$$
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| | style="padding: 12px;" | $$p_z^2 = 1$$
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| |-
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| | style="padding: 12px;" | [[Line]]
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| | style="padding: 12px;" | $$\mathbf L = L_x \mathbf e_{23} + L_y \mathbf e_{31} + L_z \mathbf e_{12}$$
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| | style="padding: 12px;" | $$L_x^2 + L_y^2 = 1$$
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| | style="padding: 12px;" | [[Motor]] | | | style="padding: 12px;" | [[Motor]] |
| | style="padding: 12px;" | $$\mathbf Q = q_x \mathbf e_{1} + q_y \mathbf e_{2} + q_z \mathbf e_{3} + q_w {\large\unicode{x1d7d9}}$$ | | | style="padding: 12px;" | $$\mathbf Q = r_x \mathbf e_{41} + r_y \mathbf e_{42} + r_z \mathbf e_{43} + r_w {\large\unicode{x1d7d9}} + u_x \mathbf e_{23} + u_y \mathbf e_{31} + u_z \mathbf e_{12} + u_w$$ |
| | style="padding: 12px;" | $$q_z^2 + q_w^2 = 1$$ | | | style="padding: 12px;" | $$\mathbf Q_\unicode{x25CF} = u_x \mathbf e_{23} + u_y \mathbf e_{31} + u_z \mathbf e_{12} + u_w$$ |
| | | style="padding: 12px;" | $$\mathbf Q_\unicode{x25CB} = r_x \mathbf e_{41} + r_y \mathbf e_{42} + r_z \mathbf e_{43} + r_w {\large\unicode{x1d7d9}}$$ |
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| | style="padding: 12px;" | [[Flector]] | | | style="padding: 12px;" | [[Flector]] |
| | style="padding: 12px;" | $$\mathbf G = g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12} + g_w$$ | | | style="padding: 12px;" | $$\mathbf G = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 + s_w \mathbf e_4 + h_x \mathbf e_{234} + h_y \mathbf e_{314} + h_z \mathbf e_{124} + h_w \mathbf e_{321}$$ |
| | style="padding: 12px;" | $$g_x^2 + g_y^2 = 1$$ | | | style="padding: 12px;" | $$\mathbf G_\unicode{x25CF} = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 + h_w \mathbf e_{321}$$ |
| | | style="padding: 12px;" | $$\mathbf G_\unicode{x25CB} = s_w \mathbf e_4 + h_x \mathbf e_{234} + h_y \mathbf e_{314} + h_z \mathbf e_{124}$$ |
| |} | | |} |
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| == Geometric Norm == | | == See Also == |
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| The following table lists the [[bulk norms]] of the main types in the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$.
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| {| class="wikitable"
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| ! Type !! Definition !! Bulk Norm
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| | style="padding: 12px;" | [[Magnitude]]
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| | style="padding: 12px;" | $$\mathbf z = x + y {\large\unicode{x1d7d9}}$$
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| | style="padding: 12px;" | $$\left\Vert\mathbf z\right\Vert_\unicode{x25CF} = |x|$$
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| | style="padding: 12px;" | [[Point]]
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| | style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3$$
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| | style="padding: 12px;" | $$\left\Vert\mathbf p\right\Vert_\unicode{x25CF} = \sqrt{p_x^2 + p_y^2}$$
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| | style="padding: 12px;" | [[Line]]
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| | style="padding: 12px;" | $$\mathbf L = L_x \mathbf e_{23} + L_y \mathbf e_{31} + L_z \mathbf e_{12}$$
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| | style="padding: 12px;" | $$\left\Vert\mathbf L\right\Vert_\unicode{x25CF} = |L_z|$$
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| | style="padding: 12px;" | [[Motor]]
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| | style="padding: 12px;" | $$\mathbf Q = q_x \mathbf e_{1} + q_y \mathbf e_{2} + q_z \mathbf e_{3} + q_w {\large\unicode{x1d7d9}}$$
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| | style="padding: 12px;" | $$\left\Vert\mathbf Q\right\Vert_\unicode{x25CF} = \sqrt{q_x^2 + q_y^2}$$
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| | style="padding: 12px;" | [[Flector]]
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| | style="padding: 12px;" | $$\mathbf G = g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12} + g_w$$
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| | style="padding: 12px;" | $$\left\Vert\mathbf G\right\Vert_\unicode{x25CF} = \sqrt{g_z^2 + g_w^2}$$
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| |}
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| The following table lists the [[weight norms]] of the main types in the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$.
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| {| class="wikitable"
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| ! Type !! Definition !! Weight Norm
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| | style="padding: 12px;" | [[Magnitude]]
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| | style="padding: 12px;" | $$\mathbf z = x + y {\large\unicode{x1d7d9}}$$
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| | style="padding: 12px;" | $$\left\Vert\mathbf z\right\Vert_\unicode{x25CB} = |y|{\large\unicode{x1D7D9}}$$
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| | style="padding: 12px;" | [[Point]]
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| | style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3$$
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| | style="padding: 12px;" | $$\left\Vert\mathbf p\right\Vert_\unicode{x25CB} = |p_z|{\large\unicode{x1D7D9}}$$
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| | style="padding: 12px;" | [[Line]]
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| | style="padding: 12px;" | $$\mathbf L = L_x \mathbf e_{23} + L_y \mathbf e_{31} + L_z \mathbf e_{12}$$
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| | style="padding: 12px;" | $$\left\Vert\mathbf L\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{L_x^2 + L_y^2}$$
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| | style="padding: 12px;" | [[Motor]]
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| | style="padding: 12px;" | $$\mathbf Q = q_x \mathbf e_{1} + q_y \mathbf e_{2} + q_z \mathbf e_{3} + q_w {\large\unicode{x1d7d9}}$$
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| | style="padding: 12px;" | $$\left\Vert\mathbf Q\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{q_z^2 + q_w^2}$$
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| | style="padding: 12px;" | [[Flector]]
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| | style="padding: 12px;" | $$\mathbf G = g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12} + g_w$$
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| | style="padding: 12px;" | $$\left\Vert\mathbf G\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{g_x^2 + g_y^2}$$
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| |}
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| The following table lists the unitized [[geometric norms]] of the main types in the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$ after dropping the constant $${\large\unicode{x1D7D9}}$$ term.
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| {| class="wikitable"
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| ! Type !! Definition !! Geometric Norm !! Interpretation
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| | style="padding: 12px;" | [[Magnitude]]
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| | style="padding: 12px;" | $$\mathbf z = x + y {\large\unicode{x1d7d9}}$$
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| | style="padding: 12px;" | $$\widehat{\left\Vert\mathbf z\right\Vert} = \dfrac{|x|}{|y|}$$
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| | style="padding: 12px;" | A Euclidean distance.
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| | style="padding: 12px;" | [[Point]]
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| | style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3$$
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| | style="padding: 12px;" | $$\widehat{\left\Vert\mathbf p\right\Vert} = \dfrac{\sqrt{p_x^2 + p_y^2}}{|p_z|}$$
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| | style="padding: 12px;" | Distance from the origin to the point $$\mathbf p$$.
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| Half the distance that the origin is moved by the [[motor]] $$\mathbf p$$.
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| | style="padding: 12px;" | [[Line]]
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| | style="padding: 12px;" | $$\mathbf L = L_x \mathbf e_{23} + L_y \mathbf e_{31} + L_z \mathbf e_{12}$$
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| | style="padding: 12px;" | $$\widehat{\left\Vert\mathbf L\right\Vert} = \dfrac{|L_z|}{\sqrt{L_x^2 + L_y^2}}$$
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| | style="padding: 12px;" | Perpendicular distance from the origin to the line $$\mathbf L$$.
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| Half the distance that the origin is moved by the [[flector]] $$\mathbf L$$.
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| | style="padding: 12px;" | [[Motor]]
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| | style="padding: 12px;" | $$\mathbf Q = q_x \mathbf e_{1} + q_y \mathbf e_{2} + q_z \mathbf e_{3} + q_w {\large\unicode{x1d7d9}}$$
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| | style="padding: 12px;" | $$\widehat{\left\Vert\mathbf Q\right\Vert} = \sqrt{\dfrac{q_x^2 + q_y^2}{q_z^2 + q_w^2}}$$
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| | style="padding: 12px;" | Half the distance that the origin is moved by the [[motor]] $$\mathbf Q$$.
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| | style="padding: 12px;" | [[Flector]]
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| | style="padding: 12px;" | $$\mathbf G = g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12} + g_w$$
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| | style="padding: 12px;" | $$\widehat{\left\Vert\mathbf G\right\Vert} = \sqrt{\dfrac{g_z^2 + g_w^2}{g_x^2 + g_y^2}}$$
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| | style="padding: 12px;" | Half the distance that the origin is moved by the [[flector]] $$\mathbf G$$.
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| |}
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| == Join and Meet ==
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| 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.
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| The points and lines appearing in the following tables are defined as follows:
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| :$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3$$
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| :$$\mathbf q = q_x \mathbf e_1 + q_y \mathbf e_2 + q_z \mathbf e_3$$
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| :$$\mathbf L = L_x \mathbf e_{23} + L_y \mathbf e_{31} + L_z \mathbf e_{12}$$
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| :$$\mathbf K = K_x \mathbf e_{23} + K_y \mathbf e_{31} + K_z \mathbf e_{12}$$
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| The join operation is performed by taking the [[wedge product]] between two geometric objects. The meet operation is performed by taking the [[antiwedge product]] between two geometric objects.
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| {| class="wikitable"
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| ! Formula || Commutator || Description
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| | style="padding: 12px;" | $$\mathbf p \wedge \mathbf q = (p_yq_z - q_yp_z)\mathbf e_{23} + (q_xp_z - p_xq_z)\mathbf e_{31} + (p_xq_y - p_yq_x)\mathbf e_{12}$$
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| | style="padding: 12px;" | $$[\mathbf p, \mathbf q]^{\Large\unicode{x27D1}}_-$$
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| | style="padding: 12px;" | Line containing points $$\mathbf p$$ and $$\mathbf q$$.
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| Zero if $$\mathbf p$$ and $$\mathbf q$$ are coincident.
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| | style="padding: 12px;" | $$\mathbf L \vee \mathbf K = (L_yK_z - K_yL_z)\mathbf e_1 + (L_zK_x - L_xK_z)\mathbf e_2 + (L_xK_y - L_yK_x)\mathbf e_3$$
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| | style="padding: 12px;" | $$[\mathbf K, \mathbf L]^{\Large\unicode{x27C7}}_-$$
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| | style="padding: 12px;" | Point where lines $$\mathbf L$$ and $$\mathbf K$$ intersect.
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| Point at infinity if $$\mathbf L$$ and $$\mathbf K$$ are parallel.
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| | style="padding: 12px;" | $$\underline{\mathbf L_\smash{\unicode{x25CB}}} \wedge \mathbf p = L_yp_z\mathbf e_{23} - L_xp_z\mathbf e_{31} + (L_xp_y - L_yp_x)\mathbf e_{12}$$
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| | style="padding: 12px;" | $$-[\mathbf p, \mathbf L]^{\Large\unicode{x27C7}}_+$$
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| | style="padding: 12px;" | Line perpendicular to line $$\mathbf L$$ passing through point $$\mathbf p$$.
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| |}
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| == Projections ==
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| The only nontrivial [[projections]] in 2D space are the projection of a point onto a line and its corresponding antiprojection. These are given by the following formulas.
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| {| class="wikitable"
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| ! Formula !! Description
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| | style="padding: 12px;" | $$\left(\underline{\mathbf L_\smash{\unicode{x25CB}}} \wedge \mathbf p\right) \vee \mathbf L = (L_x^2 + L_y^2)\mathbf p - (L_xp_x + L_yp_y + L_zp_z)(L_x \mathbf e_1 + L_y \mathbf e_2)$$
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| | style="padding: 12px;" | Projection of point $$\mathbf p$$ onto line $$\mathbf L$$.
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| | style="padding: 12px;" | $$\left(\underline{\mathbf p_\smash{\unicode{x25CB}}} \vee \mathbf L\right) \wedge \mathbf p = L_xp_z^2 \mathbf e_{23} + L_yp_z^2 \mathbf e_{31} - (L_xp_x + L_yp_y)p_z \mathbf e_{12}$$
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| | style="padding: 12px;" | Antiprojection of line $$\mathbf L$$ onto point $$\mathbf p$$.
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| |}
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| Projecting the origin onto a line gives us the following formula for the point on a line $$\mathbf L$$ closest to the origin.
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| :$$\left(\underline{\mathbf L_\smash{\unicode{x25CB}}} \wedge \mathbf e_3\right) \vee \mathbf L = -L_xL_z \mathbf e_1 - L_yL_z \mathbf e_2 + (L_x^2 + L_y^2)\mathbf e_3$$
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| Symmetrically, antiprojecting the line at infinity onto a point gives us the following formula for the line farthest from the origin containing a point $$\mathbf p$$.
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| :$$\left(\underline{\mathbf p_\smash{\unicode{x25CF}}} \vee \mathbf e_{12}\right) \wedge \mathbf p = -p_xp_z \mathbf e_{23} - p_yp_z \mathbf e_{31} + (p_x^2 + p_y^2)\mathbf e_{12}$$
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| == Motors ==
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| The set of all motors corresponds to the set of all proper Euclidean isometries in two-dimensional space. In the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$, a [[motor]] $$\mathbf Q$$ has the general form
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| :$$\mathbf Q = q_x \mathbf e_{1} + q_y \mathbf e_{2} + q_z \mathbf e_{3} + q_w {\large\unicode{x1d7d9}}$$ .
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| A motor represents a rotation about the center $$q_x \mathbf e_{1} + q_y \mathbf e_{2} + q_z \mathbf e_{3}$$.
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| A motor $$\mathbf Q$$ can be expressed as the exponential of a unitized point $$\mathbf c$$ multiplied by $$\phi{\large\unicode{x1D7D9}}$$, where $$\phi$$ is half the angle of rotation about the point $$\mathbf c$$. The exponential form can be written as
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| :$$\mathbf Q = \exp_\unicode{x27C7}(\phi{\large\unicode{x1D7D9}} \mathbin{\unicode{x27C7}} \mathbf c) = {\large\unicode{x1D7D9}}\cos\phi + \mathbf c\sin\phi$$ .
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| == Flectors ==
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| The set of all flectors corresponds to the set of all improper Euclidean isometries in two-dimensional space. In the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$, a [[flector]] $$\mathbf G$$ has the general form
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| :$$\mathbf G = g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12} + g_w$$ .
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| A flector represents a transflection with respect to the line $$g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}$$. When the line is unitized, $$g_w$$ is half the translation distance parallel to the line.
| | * [[Geometric norm]] |
| | * [[Unitization]] |
| | * [[Complements]] |