Difference between pages "Line" and "Rotation"

Revision as of 04:57, 14 November 2022

A rotation is a proper isometry of Euclidean space.

For a unitized line $$\boldsymbol l$$, the specific kind of motor

$$\mathbf R = \boldsymbol l\sin\phi + {\large\unicode{x1d7d9}}\cos\phi$$ ,

performs a rotation by twice the angle $$\phi$$ about the line $$\boldsymbol l$$ with the sandwich product $$\mathbf R \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \mathbf{\underset{\Large\unicode{x7E}}{R}}$$. The operator $$\mathbf R$$ differs from a general motor only in that it is always the case that $$u_w = 0$$. The line $$\boldsymbol l$$ and its weight complement $$\underline{\boldsymbol l_\smash{\unicode{x25CB}}}$$ are invariant under this operation. The line $$\underline{\boldsymbol l_\smash{\unicode{x25CB}}}$$ lies in the horizon in directions perpendicular to the direction of $$\boldsymbol l$$.

The exact rotation calculations for points, lines, and planes are shown in the following table.

Type	Transformation
Point $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$	$$\begin{split}\mathbf R \mathbin{\unicode{x27C7}} \mathbf p \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{R}}} =\, &\left[(1 - 2r_y^2 - 2r_z^2)p_x + 2(r_xr_y - r_zr_w)p_y + 2(r_zr_x + r_yr_w)p_z + 2(r_yu_z - r_zu_y + r_wu_x)p_w\right]\mathbf e_1 \\ +\, &\left[(1 - 2r_z^2 - 2r_x^2)p_y + 2(r_yr_z - r_xr_w)p_z + 2(r_xr_y + r_zr_w)p_x + 2(r_zu_x - r_xu_z + r_wu_y)p_w\right]\mathbf e_2 \\ +\, &\left[(1 - 2r_x^2 - 2r_y^2)p_z + 2(r_zr_x - r_yr_w)p_x + 2(r_yr_z + r_xr_w)p_y + 2(r_xu_y - r_yu_x + r_wu_z)p_w\right]\mathbf e_3 \\ +\, &p_w\mathbf e_4\end{split}$$
Line $$\begin{split}\boldsymbol l =\, &l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} \\ +\, &l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}\end{split}$$	$$\begin{split}\mathbf R \mathbin{\unicode{x27C7}} \boldsymbol l \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{R}}} =\, &\left[(1 - 2r_y^2 - 2r_z^2)l_{vx} + 2(r_xr_y - r_zr_w)l_{vy} + 2(r_zr_x + r_yr_w)l_{vz}\right]\mathbf e_{41} \\ +\, &\left[(1 - 2r_z^2 - 2r_x^2)l_{vy} + 2(r_yr_z - r_xr_w)l_{vz} + 2(r_xr_y + r_zr_w)l_{vx}\right]\mathbf e_{42} \\ +\, &\left[(1 - 2r_x^2 - 2r_y^2)l_{vz} + 2(r_zr_x - r_yr_w)l_{vx} + 2(r_yr_z + r_xr_w)l_{vy}\right]\mathbf e_{43} \\ +\, &\left[-4(r_yu_y + r_zu_z)l_{vx} + 2(r_yu_x + r_xu_y - r_wu_z)l_{vy} + 2(r_zu_x + r_xu_z + r_wu_y)l_{vz} + (1 - 2r_y^2 - 2r_z^2)l_{mx} + 2(r_xr_y - r_zr_w)l_{my} + 2(r_zr_x + r_yr_w)l_{mz}\right]\mathbf e_{23} \\ +\, &\left[-4(r_zu_z + r_xu_x)l_{vy} + 2(r_zu_y + r_yu_z - r_wu_x)l_{vz} + 2(r_xu_y + r_yu_x + r_wu_z)l_{vx} + (1 - 2r_z^2 - 2r_x^2)l_{my} + 2(r_yr_z - r_xr_w)l_{mz} + 2(r_xr_y + r_zr_w)l_{mx}\right]\mathbf e_{31} \\ +\, &\left[-4(r_xu_x + r_yu_y)l_{vz} + 2(r_xu_z + r_zu_x - r_wu_y)l_{vx} + 2(r_yu_z + r_zu_y + r_wu_x)l_{vy} + (1 - 2r_x^2 - 2r_y^2)l_{mz} + 2(r_zr_x - r_yr_w)l_{mx} + 2(r_yr_z + r_xr_w)l_{my}\right]\mathbf e_{12}\end{split}$$
Plane $$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$	$$\begin{split}\mathbf R \mathbin{\unicode{x27C7}} \mathbf g \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{R}}} =\, &\left[(1 - 2r_y^2 - 2r_z^2)g_x + 2(r_xr_y - r_zr_w)g_y + 2(r_zr_x + r_yr_w)g_z\right]\mathbf e_{423} \\ +\, &\left[(1 - 2r_z^2 - 2r_x^2)g_y + 2(r_yr_z - r_xr_w)g_z + 2(r_xr_y + r_zr_w)g_x\right]\mathbf e_{431} \\ +\, &\left[(1 - 2r_x^2 - 2r_y^2)g_z + 2(r_zr_x - r_yr_w)g_x + 2(r_yr_z + r_xr_w)g_y\right]\mathbf e_{412} \\ +\, &\left[2(r_yu_z - r_zu_y - r_wu_x)g_x + 2(r_zu_x - r_xu_z - r_wu_y)g_y + 2(r_xu_y - r_yu_x - r_wu_z)g_z + g_w\right]\mathbf e_{321}\end{split}$$

@@ Line 1: / Line 1: @@
-[[Image:line.svg|400px|thumb|right|'''Figure 1.''' A line is the intersection of a 4D bivector with the 3D subspace where $$w = 1$$.]]
+A ''rotation'' is a proper isometry of Euclidean space.
-In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a ''line'' $$\boldsymbol l$$ is a bivector having the general form
-:$$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$ .
+For a [[Unitization | unitized]] [[line]] $$\boldsymbol l$$, the specific kind of [[motor]]
-The components $$(l_{vx}, l_{vy}, l_{vz})$$ correspond to the line's direction, and the components $$(l_{mx}, l_{my}, l_{mz})$$ correspond to the line's moment. (These are equivalent to the six Plücker coordinates of a line.) To possess the [[geometric property]], the components of $$\boldsymbol l$$ must satisfy the equation
+:$$\mathbf R = \boldsymbol l\sin\phi + {\large\unicode{x1d7d9}}\cos\phi$$ ,
-:$$l_{vx} l_{mx} + l_{vy} l_{my} + l_{vz} l_{mz} = 0$$ ,
+performs a rotation by twice the angle $$\phi$$ about the line $$\boldsymbol l$$ with the sandwich product $$\mathbf R \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \mathbf{\underset{\Large\unicode{x7E}}{R}}$$. The operator $$\mathbf R$$ differs from a general [[motor]] only in that it is always the case that $$u_w = 0$$. The line $$\boldsymbol l$$ and its weight [[complement]] $$\underline{\boldsymbol l_\smash{\unicode{x25CB}}}$$ are invariant under this operation. The line $$\underline{\boldsymbol l_\smash{\unicode{x25CB}}}$$ lies in the [[horizon]] in directions perpendicular to the direction of $$\boldsymbol l$$.
-which means that, when regarded as vectors, the direction and moment of a line are perpendicular.
+The exact rotation calculations for points, lines, and planes are shown in the following table.
-The [[bulk]] of a line is given by its $$mx$$, $$my$$, and $$mz$$ coordinates, and the [[weight]] of a line is given by its $$vx$$, $$vy$$, and $$vz$$ coordinates. A line is [[unitized]] when $$l_{vx}^2 + l_{vy}^2 + l_{vz}^2 = 1$$. The [[attitude]] of a line is the vector $$l_{vx} \mathbf e_1 + l_{vy} \mathbf e_2 + l_{vz} \mathbf e_3$$ corresponding to its direction.
+{| class="wikitable"
+! Type || Transformation
+|-
+| style="padding: 12px;" | [[Point]]
-When used as an operator in the sandwich with the [[geometric antiproduct]], a line is a specific kind of [[motor]] that performs a 180-degree rotation about itself.
+$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$
+| style="padding: 12px;" | $$\begin{split}\mathbf R \mathbin{\unicode{x27C7}} \mathbf p \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{R}}} =\, &\left[(1 - 2r_y^2 - 2r_z^2)p_x + 2(r_xr_y - r_zr_w)p_y + 2(r_zr_x + r_yr_w)p_z + 2(r_yu_z - r_zu_y + r_wu_x)p_w\right]\mathbf e_1 \\ +\, &\left[(1 - 2r_z^2 - 2r_x^2)p_y + 2(r_yr_z - r_xr_w)p_z + 2(r_xr_y + r_zr_w)p_x + 2(r_zu_x - r_xu_z + r_wu_y)p_w\right]\mathbf e_2 \\ +\, &\left[(1 - 2r_x^2 - 2r_y^2)p_z + 2(r_zr_x - r_yr_w)p_x + 2(r_yr_z + r_xr_w)p_y + 2(r_xu_y - r_yu_x + r_wu_z)p_w\right]\mathbf e_3 \\ +\, &p_w\mathbf e_4\end{split}$$
+|-
+| style="padding: 12px;" | [[Line]]
-<br clear="right" />
+$$\begin{split}\boldsymbol l =\, &l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} \\ +\, &l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}\end{split}$$
-== Lines at Infinity ==
+| style="padding: 12px;" | $$\begin{split}\mathbf R \mathbin{\unicode{x27C7}} \boldsymbol l \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{R}}} =\, &\left[(1 - 2r_y^2 - 2r_z^2)l_{vx} + 2(r_xr_y - r_zr_w)l_{vy} + 2(r_zr_x + r_yr_w)l_{vz}\right]\mathbf e_{41} \\ +\, &\left[(1 - 2r_z^2 - 2r_x^2)l_{vy} + 2(r_yr_z - r_xr_w)l_{vz} + 2(r_xr_y + r_zr_w)l_{vx}\right]\mathbf e_{42} \\ +\, &\left[(1 - 2r_x^2 - 2r_y^2)l_{vz} + 2(r_zr_x - r_yr_w)l_{vx} + 2(r_yr_z + r_xr_w)l_{vy}\right]\mathbf e_{43} \\ +\, &\left[-4(r_yu_y + r_zu_z)l_{vx} + 2(r_yu_x + r_xu_y - r_wu_z)l_{vy} + 2(r_zu_x + r_xu_z + r_wu_y)l_{vz} + (1 - 2r_y^2 - 2r_z^2)l_{mx} + 2(r_xr_y - r_zr_w)l_{my} + 2(r_zr_x + r_yr_w)l_{mz}\right]\mathbf e_{23} \\ +\, &\left[-4(r_zu_z + r_xu_x)l_{vy} + 2(r_zu_y + r_yu_z - r_wu_x)l_{vz} + 2(r_xu_y + r_yu_x + r_wu_z)l_{vx} + (1 - 2r_z^2 - 2r_x^2)l_{my} + 2(r_yr_z - r_xr_w)l_{mz} + 2(r_xr_y + r_zr_w)l_{mx}\right]\mathbf e_{31} \\ +\, &\left[-4(r_xu_x + r_yu_y)l_{vz} + 2(r_xu_z + r_zu_x - r_wu_y)l_{vx} + 2(r_yu_z + r_zu_y + r_wu_x)l_{vy} + (1 - 2r_x^2 - 2r_y^2)l_{mz} + 2(r_zr_x - r_yr_w)l_{mx} + 2(r_yr_z + r_xr_w)l_{my}\right]\mathbf e_{12}\end{split}$$
+|-
+| style="padding: 12px;" | [[Plane]]
-[[Image:line_infinity.svg|400px|thumb|right|'''Figure 2.''' A line at infinity consists of all points at infinity in directions perpendicular to the moment $$\mathbf m$$.]]
+$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$
-If the weight of a line is zero (i.e., its $$vx$$, $$vy$$, and $$vz$$ coordinates are all zero), then the line is contained in the horizon infinitely far away in all directions perpendicular to its moment $$\mathbf m = (l_{mx}, l_{my}, l_{mz})$$, regarded as a vector, as shown in Figure 2. Such a line cannot be unitized, but it can be normalized by dividing by its [[bulk norm]].
+| style="padding: 12px;" | $$\begin{split}\mathbf R \mathbin{\unicode{x27C7}} \mathbf g \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{R}}} =\, &\left[(1 - 2r_y^2 - 2r_z^2)g_x + 2(r_xr_y - r_zr_w)g_y + 2(r_zr_x + r_yr_w)g_z\right]\mathbf e_{423} \\ +\, &\left[(1 - 2r_z^2 - 2r_x^2)g_y + 2(r_yr_z - r_xr_w)g_z + 2(r_xr_y + r_zr_w)g_x\right]\mathbf e_{431} \\ +\, &\left[(1 - 2r_x^2 - 2r_y^2)g_z + 2(r_zr_x - r_yr_w)g_x + 2(r_yr_z + r_xr_w)g_y\right]\mathbf e_{412} \\ +\, &\left[2(r_yu_z - r_zu_y - r_wu_x)g_x + 2(r_zu_x - r_xu_z - r_wu_y)g_y + 2(r_xu_y - r_yu_x - r_wu_z)g_z + g_w\right]\mathbf e_{321}\end{split}$$
+|}
-When the moment $$\mathbf m$$ is regarded as a bivector, a line at infinity can be thought of as all directions $$\mathbf v$$ parallel to the moment, which satisfy $$\mathbf m \wedge \mathbf v = 0$$.
-<br clear="right" />
-== Skew Lines ==
-[[Image:skew_lines.svg|400px|thumb|right|'''Figure 3.''' The line $$\mathbf j$$ connecting skew lines is given by a [[commutator]].]]
-Given two skew lines $$\boldsymbol l$$ and $$\mathbf k$$, as shown in Figure 3, a third line $$\mathbf j$$ that contains a point on each of the lines $$\boldsymbol l$$ and $$\mathbf k$$ is given by the [[commutator]]
-:$$\mathbf j = [\boldsymbol l, \mathbf k]^{\Large\unicode{x27C7}}_- = (l_{vy} k_{vz} - l_{vz} k_{vy})\mathbf e_{41} + (l_{vz} k_{vx} - l_{vx} k_{vz})\mathbf e_{42} + (l_{vx} k_{vy} - l_{vy} k_{vx})\mathbf e_{43} + (l_{vy} k_{mz} - l_{vz} k_{my} + l_{my} k_{vz} - l_{mz} k_{vy})\mathbf e_{23} + (l_{vz} k_{mx} - l_{vx} k_{mz} + l_{mz} k_{vx} - l_{mx} k_{vz})\mathbf e_{31} + (l_{vx} k_{my} - l_{vy} k_{mx} + l_{mx} k_{vy} - l_{my} k_{vx})\mathbf e_{12}$$ .
-The direction of $$\mathbf j$$ is perpendicular to the directions of $$\boldsymbol l$$ and $$\mathbf k$$, and it contains the closest points of approach between $$\boldsymbol l$$ and $$\mathbf k$$. The points themselves can then be found by calculating $$(\mathbf j \wedge \operatorname{att}(\boldsymbol l)) \vee \mathbf k$$ and $$(\mathbf j \wedge \operatorname{att}(\mathbf k)) \vee \boldsymbol l$$, where $$\operatorname{att}$$ is the [[attitude]] function.
-<br clear="right" />
 == See Also ==
-* [[Point]]
+* [[Dual rotation]]
-* [[Plane]]
+* [[Translation]]
+* [[Reflection]]
+* [[Inversion]]
+* [[Transflection]]

Difference between pages "Line" and "Rotation"

Revision as of 04:57, 14 November 2022

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