|
|
| Line 1: |
Line 1: |
| The degenerate metric of rigid geometric algebra naturally divides the components of every quantity into two groups called the ''bulk'' and the ''weight''. | | The ''metric'' used in the 4D rigid geometric algebra over 3D Euclidean space is the $$4 \times 4$$ matrix $$\mathfrak g$$ given by |
|
| |
|
| The bulk of an element $$\mathbf u$$ is denoted by $$\mathbf u_\unicode{x25CF}$$, and it is defined as
| | :$$\mathfrak g = \begin{bmatrix} |
| | 1 & 0 & 0 & 0 \\ |
| | 0 & 1 & 0 & 0 \\ |
| | 0 & 0 & 1 & 0 \\ |
| | 0 & 0 & 0 & 0 \\\end{bmatrix}$$ . |
|
| |
|
| :$$\mathbf u_\unicode{x25CF} = \mathbf G \mathbf u$$,
| | The ''metric exomorphism matrix'' $$\mathbf G$$, often just called the "metric" itself, corresponding to the metric $$\mathfrak g$$ is the $$16 \times 16$$ matrix shown below. |
|
| |
|
| where $$\mathbf G$$ is the metric exomorphism matrix. The bulk consists of the components of $$\mathbf u$$ that do not have the projective basis vector $$\mathbf e_4$$ as a factor.
| | [[Image:metric-rga-3d.svg|420px]] |
|
| |
|
| The weight is denoted by $$\mathbf u_\unicode{x25CB}$$, and it is defined as | | The ''metric antiexomorphism matrix'' $$\mathbb G$$, often called the "antimetric", corresponding to the metric $$\mathfrak g$$ is the $$16 \times 16$$ matrix shown below. |
|
| |
|
| :$$\mathbf u_\unicode{x25CB} = \mathbb G \mathbf u$$, | | [[Image:antimetric-rga-3d.svg|420px]] |
|
| |
|
| where $$\mathbb G$$ is the metric anti-exomorphism matrix. The weight consists of the components of $$\mathbf u$$ that do have the projective basis vector $$\mathbf e_4$$ as a factor.
| | The product of the metric exomorphism matrix $$\mathbf G$$ and metric antiexomorphism matrix $$\mathbb G$$ for any metric $$\mathfrak g$$ is always equal to the $$16 \times 16$$ identity matrix times the determinant of $$\mathfrak g$$. That is, $$\mathbf G \mathbb G = \det(\mathfrak g) \mathbf I$$. |
|
| |
|
| 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. | | The metric and antimetric determine [[bulk and weight]], [[duals]], [[dot products]], and [[geometric products]]. |
|
| |
|
| An element is [[unitized]] when the magnitude of its weight is one.
| | == See Also == |
|
| |
|
| The following table lists the bulk and weight for the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.
| | * [[Bulk and weight]] |
| | * [[Duals]] |
| | * [[Dot products]] |
|
| |
|
| {| class="wikitable"
| | == In the Book == |
| ! Type !! Definition !! Bulk !! Weight
| |
| |-
| |
| | style="padding: 12px;" | [[Magnitude]]
| |
| | style="padding: 12px;" | $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$
| |
| | style="padding: 12px;" | $$\mathbf z_\unicode{x25CF} = x \mathbf 1$$
| |
| | style="padding: 12px;" | $$\mathbf z_\unicode{x25CB} = y {\large\unicode{x1d7d9}}$$
| |
| |-
| |
| | 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;" | $$\mathbf p_\unicode{x25CF} = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3$$
| |
| | style="padding: 12px;" | $$\mathbf p_\unicode{x25CB} = p_w \mathbf e_4$$
| |
| |-
| |
| | 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;" | $$\boldsymbol l_\unicode{x25CF} = l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$
| |
| | style="padding: 12px;" | $$\boldsymbol l_\unicode{x25CB} = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43}$$
| |
| |-
| |
| | 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;" | $$\mathbf g_\unicode{x25CF} = g_w \mathbf e_{321}$$
| |
| | style="padding: 12px;" | $$\mathbf g_\unicode{x25CB} = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412}$$
| |
| |-
| |
| | style="padding: 12px;" | [[Motor]]
| |
| | style="padding: 12px;" | $$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$
| |
| | style="padding: 12px;" | $$\mathbf Q_\unicode{x25CF} = Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$
| |
| | style="padding: 12px;" | $$\mathbf Q_\unicode{x25CB} = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}}$$
| |
| |-
| |
| | style="padding: 12px;" | [[Flector]]
| |
| | style="padding: 12px;" | $$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$
| |
| | style="padding: 12px;" | $$\mathbf F_\unicode{x25CF} = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{gw} \mathbf e_{321}$$
| |
| | style="padding: 12px;" | $$\mathbf F_\unicode{x25CB} = F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412}$$
| |
| |}
| |
|
| |
|
| == See Also ==
| | * The metric and antimetric are introduced in Section 2.8. |
| | |
| * [[Attitude]] | |
| * [[Geometric norm]]
| |
| * [[Unitization]]
| |
| * [[Complements]]
| |
| * [[Duals]]
| |
The metric used in the 4D rigid geometric algebra over 3D Euclidean space is the $$4 \times 4$$ matrix $$\mathfrak g$$ given by
- $$\mathfrak g = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\\end{bmatrix}$$ .
The metric exomorphism matrix $$\mathbf G$$, often just called the "metric" itself, corresponding to the metric $$\mathfrak g$$ is the $$16 \times 16$$ matrix shown below.
The metric antiexomorphism matrix $$\mathbb G$$, often called the "antimetric", corresponding to the metric $$\mathfrak g$$ is the $$16 \times 16$$ matrix shown below.
The product of the metric exomorphism matrix $$\mathbf G$$ and metric antiexomorphism matrix $$\mathbb G$$ for any metric $$\mathfrak g$$ is always equal to the $$16 \times 16$$ identity matrix times the determinant of $$\mathfrak g$$. That is, $$\mathbf G \mathbb G = \det(\mathfrak g) \mathbf I$$.
The metric and antimetric determine bulk and weight, duals, dot products, and geometric products.
See Also
In the Book
- The metric and antimetric are introduced in Section 2.8.
