Attitude - Revision history

Eric Lengyel at 05:13, 13 April 2024

2024-04-13T05:13:32Z

← Older revision		Revision as of 05:13, 13 April 2024
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	\| style="padding: 12px;" \| $$\operatorname{att}(\mathbf g) = g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}$$		\| style="padding: 12px;" \| $$\operatorname{att}(\mathbf g) = g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}$$
	\|}		\|}

			== In the Book ==

			* The attitude is discussed in Section 2.8.4.

Eric Lengyel at 07:20, 23 October 2023

2023-10-23T07:20:09Z

← Older revision		Revision as of 07:20, 23 October 2023
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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		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

	:$$\operatorname{att}(\mathbf x) = \mathbf x \vee \overline{\mathbf e_4}$$ .		:$$\operatorname{att}(\mathbf u) = \mathbf u \vee \overline{\mathbf e_4}$$ .

	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.		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.

Eric Lengyel: Created 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 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. The following table lists the attitude for the main types in the 4D rigid geometric algebra..."

2023-07-15T05:29:38Z

Created 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 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. The following table lists the attitude for the main types in the 4D rigid geometric algebra..."

New page

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 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.

The following table lists the attitude for the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.

{| class="wikitable"
! Type !! Definition !! Attitude
|-
| style="padding: 12px;" | [[Magnitude]]
| style="padding: 12px;" | $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$
| style="padding: 12px;" | $$\operatorname{att}(\mathbf z) = y \mathbf e_{321}$$
|-
| style="padding: 12px;" | [[Point]]
| 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;" | $$\operatorname{att}(\mathbf p) = p_w \mathbf 1$$
|-
| style="padding: 12px;" | [[Line]]
| style="padding: 12px;" | $$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$
| style="padding: 12px;" | $$\operatorname{att}(\boldsymbol l) = l_{vx} \mathbf e_1 + l_{vy} \mathbf e_2 + l_{vz} \mathbf e_3$$
|-
| style="padding: 12px;" | [[Plane]]
| style="padding: 12px;" | $$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$
| style="padding: 12px;" | $$\operatorname{att}(\mathbf g) = g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}$$
|}