View source for Rigid Geometric Algebra for 2D Space

== Introduction ==

[[Image:Basis201.svg|thumb|right|400px|'''Table 1.''' The 8 basis elements of the 3D rigid geometric algebra.]]
In the three-dimensional rigid geometric algebra, there are 8 graded basis elements. These are listed in Table 1.

There is a single ''scalar'' basis element $$\mathbf 1$$, and its multiples correspond to the real numbers, which are values that have no dimensions.

There are three ''vector'' basis elements named $$\mathbf e_1$$, $$\mathbf e_2$$, and $$\mathbf e_3$$ that have one-dimensional extents. A general vector $$\mathbf v = (v_x, v_y, v_z)$$ has the form

:$$\mathbf v = v_x \mathbf e_1 + v_y \mathbf e_2 + v_z \mathbf e_3$$ .

There are three ''bivector'' basis elements named $$\mathbf e_{23}$$, $$\mathbf e_{31}$$, and $$\mathbf e_{12}$$ having two-dimensional extents.

Finally, there is a single ''trivector'' basis element $${\large\unicode{x1D7D9}} = \mathbf e_3 \wedge \mathbf e_2 \wedge \mathbf e_1$$ having three-dimensional extents.

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== Unary Operations ==

The 3D rigid geometric algebra has a single [[complement]] operation, a [[reverse]] operation, and an [[antireverse]] operation. (In three dimensions, the left and right [[complements]] are identical.) These are listed for all basis elements in the following table.

[[Image:Unary201.svg|480px]]

== Geometric Products ==

The geometric product is characterized by a metric that defines the products of the basis vectors with themselves. The subscript in $$\mathcal G_{2,0,1}$$ means that two basis vectors square to +1, zero basis vectors square to &minus;1, and one basis vector squares to 0. The geometric product between two different basis vectors is given by the [[wedge product]]. We can write these rules as follows.

:$$\mathbf e_1 \mathbin{\unicode{x27D1}} \mathbf e_1 = 1$$

:$$\mathbf e_2 \mathbin{\unicode{x27D1}} \mathbf e_2 = 1$$

:$$\mathbf e_3 \mathbin{\unicode{x27D1}} \mathbf e_3 = 0$$

:$$\mathbf e_i \mathbin{\unicode{x27D1}} \mathbf e_j = \mathbf e_i \wedge \mathbf e_j$$, for $$i \neq j$$.

The following Cayley table shows the geometric products between all pairs of basis elements in the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$. The identity of the geometric product is the [[scalar]] basis element $$\mathbf 1$$.

[[Image:GeometricProduct201.svg|360px]]

The following Cayley table shows the geometric antiproducts between all pairs of basis elements in the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$. The identity of the geometric antiproduct is the [[antiscalar]] basis element $$\large\unicode{x1D7D9}$$.

[[Image:GeometricAntiproduct201.svg|360px]]

== Points ==

In the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$, a ''point'' $$\mathbf p$$ is a vector having the general form

:$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3$$ .

The [[bulk]] of a point is given by its $$x$$ and $$y$$ coordinates, and the [[weight]] of a point is given by its $$z$$ coordinate. A point is [[unitized]] when $$p_z^2 = 1$$.

When used as an operator in the sandwich product, a point is a specific kind of [[motor]] that performs a [[rotation]] about itself.

If the weight of a point is zero (i.e., its $$z$$ coordinate is zero), then the point lies at infinity in the direction $$(x, y)$$, and it cannot be unitized. A point with zero weight can also be interpreted as a direction vector, and it is normalized to unit length by dividing by its [[bulk norm]].

== Lines ==

In the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$, a ''line'' $$\mathbf g$$ is a bivector having the general form

:$$\mathbf g = g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}$$ .

The [[bulk]] of a line is given by its $$z$$ coordinate, and the [[weight]] of a line is given by its $$x$$ and $$y$$ coordinates. A line is [[unitized]] when $$g_x^2 + g_y^2 = 1$$.

When used as an operator in the sandwich product, a line is a specific kind of [[flector]] that performs a [[reflection]] through itself.

If the weight of a line is zero (i.e., its $$x$$ and $$y$$ coordinates are both zero), then the line lies at infinity in all directions. Such a line is normalized when $$g_z = \pm 1$$. This is the ''horizon'' of two-dimensional space.

== Bulk and Weight ==

The following table lists the [[bulk]] and [[weight]] for the main types in the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$.

{| class="wikitable"
! 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;" | $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3$$
| style="padding: 12px;" | $$\mathbf p_\unicode{x25CF} = p_x \mathbf e_1 + p_y \mathbf e_2$$
| style="padding: 12px;" | $$\mathbf p_\unicode{x25CB} = p_z \mathbf e_3$$
|-
| style="padding: 12px;" | [[Line]]
| style="padding: 12px;" | $$\mathbf g = g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}$$
| style="padding: 12px;" | $$\mathbf g_\unicode{x25CF} = g_z \mathbf e_{12}$$
| style="padding: 12px;" | $$\mathbf g_\unicode{x25CB} = g_x \mathbf e_{23} + g_y \mathbf e_{31}$$
|-
| style="padding: 12px;" | [[Motor]]
| style="padding: 12px;" | $$\mathbf Q = q_x \mathbf e_{1} + q_y \mathbf e_{2} + q_z \mathbf e_{3} + q_w {\large\unicode{x1d7d9}}$$
| style="padding: 12px;" | $$\mathbf Q_\unicode{x25CF} = q_x \mathbf e_{1} + q_y \mathbf e_{2}$$
| style="padding: 12px;" | $$\mathbf Q_\unicode{x25CB} = q_z \mathbf e_{3} + q_w {\large\unicode{x1d7d9}}$$
|-
| style="padding: 12px;" | [[Flector]]
| style="padding: 12px;" | $$\mathbf F = f_x \mathbf e_{23} + f_y \mathbf e_{31} + f_z \mathbf e_{12} + f_w \mathbf 1$$
| style="padding: 12px;" | $$\mathbf F_\unicode{x25CF} = f_z \mathbf e_{12} + f_w \mathbf 1$$
| style="padding: 12px;" | $$\mathbf F_\unicode{x25CB} = f_x \mathbf e_{23} + f_y \mathbf e_{31}$$
|}

== Unitization ==

The following table lists the [[unitization]] conditions for the main types in the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$.

{| class="wikitable"
! Type !! Definition !! Unitization
|-
| style="padding: 12px;" | [[Magnitude]]
| style="padding: 12px;" | $$\mathbf z = x\mathbf 1 + y {\large\unicode{x1d7d9}}$$
| style="padding: 12px;" | $$y^2 = 1$$
|-
| style="padding: 12px;" | [[Point]]
| style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3$$
| style="padding: 12px;" | $$p_z^2 = 1$$
|-
| style="padding: 12px;" | [[Line]]
| style="padding: 12px;" | $$\mathbf g = g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}$$
| style="padding: 12px;" | $$g_x^2 + g_y^2 = 1$$
|-
| style="padding: 12px;" | [[Motor]]
| style="padding: 12px;" | $$\mathbf Q = q_x \mathbf e_{1} + q_y \mathbf e_{2} + q_z \mathbf e_{3} + q_w {\large\unicode{x1d7d9}}$$
| style="padding: 12px;" | $$q_z^2 + q_w^2 = 1$$
|-
| style="padding: 12px;" | [[Flector]]
| style="padding: 12px;" | $$\mathbf F = f_x \mathbf e_{23} + f_y \mathbf e_{31} + f_z \mathbf e_{12} + f_w \mathbf 1$$
| style="padding: 12px;" | $$f_x^2 + f_y^2 = 1$$
|}

== Geometric Norm ==

The following table lists the [[bulk norms]] of the main types in the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$.

{| class="wikitable"
! Type !! Definition !! Bulk Norm
|-
| style="padding: 12px;" | [[Magnitude]]
| style="padding: 12px;" | $$\mathbf z = x\mathbf 1 + y {\large\unicode{x1d7d9}}$$
| style="padding: 12px;" | $$\left\Vert\mathbf z\right\Vert_\unicode{x25CF} = |x|$$
|-
| style="padding: 12px;" | [[Point]]
| style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3$$
| style="padding: 12px;" | $$\left\Vert\mathbf p\right\Vert_\unicode{x25CF} = \sqrt{p_x^2 + p_y^2}$$
|-
| style="padding: 12px;" | [[Line]]
| style="padding: 12px;" | $$\mathbf g = g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}$$
| style="padding: 12px;" | $$\left\Vert\mathbf g\right\Vert_\unicode{x25CF} = |g_z|$$
|-
| style="padding: 12px;" | [[Motor]]
| style="padding: 12px;" | $$\mathbf Q = q_x \mathbf e_{1} + q_y \mathbf e_{2} + q_z \mathbf e_{3} + q_w {\large\unicode{x1d7d9}}$$
| style="padding: 12px;" | $$\left\Vert\mathbf Q\right\Vert_\unicode{x25CF} = \sqrt{q_x^2 + q_y^2}$$
|-
| style="padding: 12px;" | [[Flector]]
| style="padding: 12px;" | $$\mathbf F = f_x \mathbf e_{23} + f_y \mathbf e_{31} + f_z \mathbf e_{12} + f_w \mathbf 1$$
| style="padding: 12px;" | $$\left\Vert\mathbf F\right\Vert_\unicode{x25CF} = \sqrt{f_z^2 + f_w^2}$$
|}

The following table lists the [[weight norms]] of the main types in the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$.

{| class="wikitable"
! Type !! Definition !! Weight Norm
|-
| style="padding: 12px;" | [[Magnitude]]
| style="padding: 12px;" | $$\mathbf z = x\mathbf 1 + y {\large\unicode{x1d7d9}}$$
| style="padding: 12px;" | $$\left\Vert\mathbf z\right\Vert_\unicode{x25CB} = |y|{\large\unicode{x1D7D9}}$$
|-
| style="padding: 12px;" | [[Point]]
| style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3$$
| style="padding: 12px;" | $$\left\Vert\mathbf p\right\Vert_\unicode{x25CB} = |p_z|{\large\unicode{x1D7D9}}$$
|-
| style="padding: 12px;" | [[Line]]
| style="padding: 12px;" | $$\mathbf g = g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}$$
| style="padding: 12px;" | $$\left\Vert\mathbf g\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{g_x^2 + g_y^2}$$
|-
| style="padding: 12px;" | [[Motor]]
| style="padding: 12px;" | $$\mathbf Q = q_x \mathbf e_{1} + q_y \mathbf e_{2} + q_z \mathbf e_{3} + q_w {\large\unicode{x1d7d9}}$$
| style="padding: 12px;" | $$\left\Vert\mathbf Q\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{q_z^2 + q_w^2}$$
|-
| style="padding: 12px;" | [[Flector]]
| style="padding: 12px;" | $$\mathbf F = f_x \mathbf e_{23} + f_y \mathbf e_{31} + f_z \mathbf e_{12} + f_w \mathbf 1$$
| style="padding: 12px;" | $$\left\Vert\mathbf F\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{f_x^2 + f_y^2}$$
|}

The following table lists the unitized [[geometric norms]] of the main types in the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$ after dropping the constant $${\large\unicode{x1D7D9}}$$ term.

{| class="wikitable"
! Type !! Definition !! Geometric Norm !! Interpretation
|-
| style="padding: 12px;" | [[Magnitude]]
| style="padding: 12px;" | $$\mathbf z = x\mathbf 1 + y {\large\unicode{x1d7d9}}$$
| style="padding: 12px;" | $$\widehat{\left\Vert\mathbf z\right\Vert} = \dfrac{|x|}{|y|}$$
| style="padding: 12px;" | A Euclidean distance.
|-
| style="padding: 12px;" | [[Point]]
| style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3$$
| style="padding: 12px;" | $$\widehat{\left\Vert\mathbf p\right\Vert} = \dfrac{\sqrt{p_x^2 + p_y^2}}{|p_z|}$$
| style="padding: 12px;" | Distance from the origin to the point $$\mathbf p$$.

Half the distance that the origin is moved by the [[motor]] $$\mathbf p$$.
|-
| style="padding: 12px;" | [[Line]]
| style="padding: 12px;" | $$\mathbf g = g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}$$
| style="padding: 12px;" | $$\widehat{\left\Vert\mathbf g\right\Vert} = \dfrac{|g_z|}{\sqrt{g_x^2 + g_y^2}}$$
| style="padding: 12px;" | Perpendicular distance from the origin to the line $$\mathbf g$$.

Half the distance that the origin is moved by the [[flector]] $$\mathbf g$$.
|-
| style="padding: 12px;" | [[Motor]]
| style="padding: 12px;" | $$\mathbf Q = q_x \mathbf e_{1} + q_y \mathbf e_{2} + q_z \mathbf e_{3} + q_w {\large\unicode{x1d7d9}}$$
| style="padding: 12px;" | $$\widehat{\left\Vert\mathbf Q\right\Vert} = \sqrt{\dfrac{q_x^2 + q_y^2}{q_z^2 + q_w^2}}$$
| style="padding: 12px;" | Half the distance that the origin is moved by the [[motor]] $$\mathbf Q$$.
|-
| style="padding: 12px;" | [[Flector]]
| style="padding: 12px;" | $$\mathbf F = f_x \mathbf e_{23} + f_y \mathbf e_{31} + f_z \mathbf e_{12} + f_w \mathbf 1$$
| style="padding: 12px;" | $$\widehat{\left\Vert\mathbf F\right\Vert} = \sqrt{\dfrac{f_z^2 + f_w^2}{f_x^2 + f_y^2}}$$
| style="padding: 12px;" | Half the distance that the origin is moved by the [[flector]] $$\mathbf F$$.
|}

== Join and Meet ==

The ''join'' is a binary operation that calculates the higher-dimensional geometry containing its two operands, similar to a union. The ''meet'' is another binary operation that calculates the lower-dimensional geometry shared by its two operands, similar to an intersection.

The points and lines appearing in the following tables are defined as follows:

:$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3$$
:$$\mathbf q = q_x \mathbf e_1 + q_y \mathbf e_2 + q_z \mathbf e_3$$
:$$\mathbf g = g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}$$
:$$\mathbf h = h_x \mathbf e_{23} + h_y \mathbf e_{31} + h_z \mathbf e_{12}$$

The join operation is performed by taking the [[wedge product]] between two geometric objects. The meet operation is performed by taking the [[antiwedge product]] between two geometric objects.

{| class="wikitable"
! Formula || Description
|-
| style="padding: 12px;" | $$\mathbf p \wedge \mathbf q = (p_yq_z - q_yp_z)\mathbf e_{23} + (q_xp_z - p_xq_z)\mathbf e_{31} + (p_xq_y - p_yq_x)\mathbf e_{12}$$
| style="padding: 12px;" | Line containing points $$\mathbf p$$ and $$\mathbf q$$.

Zero if $$\mathbf p$$ and $$\mathbf q$$ are coincident.
|-
| style="padding: 12px;" | $$\mathbf g \vee \mathbf h = (g_yh_z - h_yg_z)\mathbf e_1 + (g_zh_x - g_xh_z)\mathbf e_2 + (g_xh_y - g_yh_x)\mathbf e_3$$
| style="padding: 12px;" | Point where lines $$\mathbf g$$ and $$\mathbf h$$ intersect.

Point at infinity if $$\mathbf g$$ and $$\mathbf h$$ are parallel.
|-
| style="padding: 12px;" | $$\mathbf g^\unicode["segoe ui symbol"]{x2605} \wedge \mathbf p = -g_yp_z\mathbf e_{23} + g_xp_z\mathbf e_{31} + (g_yp_x - g_xp_y)\mathbf e_{12}$$
| style="padding: 12px;" | Line perpendicular to line $$\mathbf g$$ passing through point $$\mathbf p$$.
|}

== Projections ==

The only nontrivial [[projections]] in 2D space are the projection of a point onto a line and its corresponding antiprojection. These are given by the following formulas.

{| class="wikitable"
! Formula !! Description
|-
| style="padding: 12px;" | $$\left(\mathbf g^\unicode["segoe ui symbol"]{x2605} \wedge \mathbf p\right) \vee \mathbf g = (g_x^2 + g_y^2)\mathbf p - (g_xp_x + g_yp_y + g_zp_z)(g_x \mathbf e_1 + g_y \mathbf e_2)$$
| style="padding: 12px;" | Projection of point $$\mathbf p$$ onto line $$\mathbf g$$.
|-
| style="padding: 12px;" | $$\left(\mathbf p^\unicode["segoe ui symbol"]{x2605} \vee \mathbf g\right) \wedge \mathbf p = g_xp_z^2 \mathbf e_{23} + g_yp_z^2 \mathbf e_{31} - (g_xp_x + g_yp_y)p_z \mathbf e_{12}$$
| style="padding: 12px;" | Antiprojection of line $$\mathbf g$$ onto point $$\mathbf p$$.
|}

Projecting the origin onto a line gives us the following formula for the point on a line $$\mathbf g$$ closest to the origin.

:$$\left(\mathbf g^\unicode["segoe ui symbol"]{x2605} \wedge \mathbf e_3\right) \vee \mathbf g = -g_xg_z \mathbf e_1 - g_yg_z \mathbf e_2 + (g_x^2 + g_y^2)\mathbf e_3$$

Symmetrically, antiprojecting the line at infinity onto a point gives us the following formula for the line farthest from the origin containing a point $$\mathbf p$$.

:$$\left(\mathbf p^\unicode["segoe ui symbol"]{x2606} \vee \mathbf e_{12}\right) \wedge \mathbf p = -p_xp_z \mathbf e_{23} - p_yp_z \mathbf e_{31} + (p_x^2 + p_y^2)\mathbf e_{12}$$

== Motors ==

The set of all motors corresponds to the set of all proper Euclidean isometries in two-dimensional space. In the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$, a [[motor]] $$\mathbf Q$$ has the general form

:$$\mathbf Q = q_x \mathbf e_{1} + q_y \mathbf e_{2} + q_z \mathbf e_{3} + q_w {\large\unicode{x1d7d9}}$$ .

A motor represents a rotation about the center $$q_x \mathbf e_{1} + q_y \mathbf e_{2} + q_z \mathbf e_{3}$$.

A motor $$\mathbf Q$$ can be expressed as the exponential of a unitized point $$\mathbf p$$ multiplied by $$\phi{\large\unicode{x1D7D9}}$$, where $$\phi$$ is half the angle of rotation about the point $$\mathbf p$$. The exponential form can be written as

:$$\mathbf Q = \exp_\unicode{x27C7}(\phi{\large\unicode{x1D7D9}} \mathbin{\unicode{x27C7}} \mathbf p) = {\large\unicode{x1D7D9}}\cos\phi + \mathbf p\sin\phi$$ .

== Flectors ==

The set of all flectors corresponds to the set of all improper Euclidean isometries in two-dimensional space. In the 3D rigid geometric algebra $$\mathcal G_{2,0,1}$$, a [[flector]] $$\mathbf F$$ has the general form

:$$\mathbf F = g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12} + g_w \mathbf 1$$ .

A flector represents a transflection with respect to the line $$g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}$$. When the line is unitized, $$g_w$$ is half the translation distance parallel to the line.