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| An element $$\mathbf x$$ of a geometric algebra possesses the ''geometric constraint'' if and only if the [[geometric product]] between $$\mathbf u$$ and its own reverse is a scalar, which is given by the [[dot product]], and the [[geometric antiproduct]] between $$\mathbf u$$ and its own antireverse is an antiscalar, which is given by the [[antidot product]]. That is,
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| :$$\mathbf u \mathbin{\unicode{x27D1}} \mathbf{\tilde u} = \mathbf u \mathbin{\unicode{x25CF}} \mathbf u$$
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| and
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| :$$\mathbf u \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} = \mathbf u \mathbin{\unicode{x25CB}} \mathbf u$$ .
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| The set of all elements satisfying the geometric constraint is closed under both the [[geometric product]] and [[geometric antiproduct]].
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| The following table lists the requirements that must be satisfied for the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ due to the geometric constraint. Points and planes do not have any requirements—they have no constraints.
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| {| class="wikitable"
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| ! Type !! Definition !! Requirement
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| | style="padding: 12px;" | [[Magnitude]]
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| | style="padding: 12px;" | $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$
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| | style="padding: 12px;" | $$xy = 0$$
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| |-
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| | style="padding: 12px;" | [[Point]]
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| | style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$
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| | style="padding: 12px;" | —
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| | style="padding: 12px;" | [[Line]]
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| | style="padding: 12px;" | $$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$
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| | style="padding: 12px;" | $$l_{vx} l_{mx} + l_{vy} l_{my} + l_{vz} l_{mz} = 0$$
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| |-
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| | style="padding: 12px;" | [[Plane]]
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| | style="padding: 12px;" | $$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$
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| | style="padding: 12px;" | —
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| | style="padding: 12px;" | [[Motor]]
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| | style="padding: 12px;" | $$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$
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| | style="padding: 12px;" | $$Q_{vx} Q_{mx} + Q_{vy} Q_{my} + Q_{vz} Q_{mz} + Q_{vw} Q_{mw} = 0$$
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| | style="padding: 12px;" | [[Flector]]
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| | style="padding: 12px;" | $$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$
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| | style="padding: 12px;" | $$F_{px} F_{gx} + F_{py} F_{gy} + F_{pz} F_{gz} + F_{pw} F_{gw} = 0$$
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| |}
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| == See Also ==
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| * [[Geometric norm]]
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