From Rigid Geometric Algebra
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| 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
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| :$$\operatorname{att}(\mathbf x) = \mathbf x \vee \overline{\mathbf e_4}$$ .
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| The attitude of a [[line]] is the line's direction as a vector, and the attitude of a [[plane]] is the plane's normal as a bivector.
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| The following table lists the attitude for the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.
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| {| class="wikitable"
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| ! Type !! Definition !! Attitude
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| |-
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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;" | $$\operatorname{att}(\mathbf z) = y \mathbf e_{321}$$
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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;" | $$\operatorname{att}(\mathbf p) = p_w \mathbf 1$$
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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;" | $$\operatorname{att}(\boldsymbol l) = l_{vx} \mathbf e_1 + l_{vy} \mathbf e_2 + l_{vz} \mathbf e_3$$
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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;" | $$\operatorname{att}(\mathbf g) = g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}$$
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| |}
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Latest revision as of 01:31, 12 May 2024
