Complements

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Complements are unary operations in geometric algebra that perform a specific type of dualization.

Every basis element $$\mathbf a$$ has a right complement, which we denote by $$\overline{\mathbf a}$$, that satisfies the equation

$$\mathbf a \wedge \overline{\mathbf a} = {\large\unicode{x1D7D9}}$$ .

There is also a left complement, which we denote by $$\underline{\mathbf a}$$, that satisfies the equation

$$\underline{\mathbf a} \wedge \mathbf a = {\large\unicode{x1D7D9}}$$ .

Complements exchange full and empty dimensions, and the left and right complements can differ only by sign according to the relationship

$$\underline{\mathbf a} = (-1)^{\operatorname{gr}(\mathbf a)\operatorname{ag}(\mathbf a)}\,\overline{\mathbf a}$$ .

This shows that the left and right complements of an element $$\mathbf a$$ are always the same if either its grade $$\operatorname{gr}(\mathbf a)$$ or its antigrade $$\operatorname{ag}(\mathbf a)$$ is even. If the number of dimensions is odd, then it is always true that one of these is even, so left and right complements are the same for all elements in an odd-dimensional algebra. As shown in the table below, applying the right or left complement twice can negate the operand in even numbers of dimensions. However, the right and left complements are inverse operations, so we always have $$\overline{\underline{\mathbf a}} = \mathbf a$$.

The right and left complements under the wedge product are also the right and left complements under the antiwedge product, so we can write

$$\mathbf a \vee \overline{\mathbf a} = \mathbf 1$$
$$\underline{\mathbf a} \vee\mathbf a = \mathbf 1$$ .

To extend the complements to all elements of an algebra, we simply require that they are linear operations. For any basis elements $$\mathbf a$$ and $$\mathbf b$$, and for any scalars $$x$$ and $$y$$, we must have, for the right complement,

$$\overline{(x\mathbf a + y\mathbf b)} = x\overline{\mathbf{a\vphantom b}} + y\overline{\mathbf b}$$ ,

and similarly for the left complement.

The following table lists the left and right complements for all of the basis elements in the 4D geometric algebra $$\mathcal G_{3,0,1}$$.

Complements.svg

Explicit Calculation

Left and right complements of the bulk and weight of an element $$\mathbf a$$ can be calculated by taking the geometric product and antiproduct with the antiscalar $$\large\unicode{x1D7D9}$$ and the scalar $$\mathbf 1$$, respectively, as follows.

$$\overline{\mathbf a_\smash{\unicode{x25CF}}} = \mathbf{\tilde a} \mathbin{\unicode{x27D1}} {\large\unicode{x1D7D9}}$$
$$\underline{\mathbf a_\smash{\unicode{x25CF}}} = \smash{\mathbf{\underset{\Large\unicode{x7E}}{a}}} \mathbin{\unicode{x27D1}} {\large\unicode{x1D7D9}}$$
$$\overline{\mathbf a_\smash{\unicode{x25CB}}} = \mathbf 1 \mathbin{\unicode{x27C7}} \mathbf{\tilde a}$$
$$\underline{\mathbf a_\smash{\unicode{x25CB}}} = \mathbf 1 \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{a}}}$$

The full complement of $$\mathbf a$$ is the sum of the complements of its bulk and weight, so we can write

$$\overline{\mathbf a} = \mathbf{\tilde a} \mathbin{\unicode{x27D1}} {\large\unicode{x1D7D9}} + \mathbf 1 \mathbin{\unicode{x27C7}} \mathbf{\tilde a}$$
$$\underline{\mathbf a} = \smash{\mathbf{\underset{\Large\unicode{x7E}}{a}}} \mathbin{\unicode{x27D1}} {\large\unicode{x1D7D9}} + \mathbf 1 \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{a}}}$$ .

Weight Complements

The complement of an element's weight is particularly useful because it extracts the attitude of a geometric object as an element expressed on an orthogonal basis. These arise naturally in projections, which make use of the interior product.

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

Type Definition Weight Left Complement Description
Point $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ $$\underline{\mathbf p_\smash{\unicode{x25CB}}} = -p_w \mathbf e_{321}$$ Plane at infinity.
Line $$\mathbf L = v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} + m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$ $$\underline{\mathbf L_\smash{\unicode{x25CB}}} = -v_x \mathbf e_{23} - v_y \mathbf e_{31} - v_z \mathbf e_{12}$$ Line at infinity perpendicular to line $$\mathbf L$$.
Plane $$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$ $$\underline{\mathbf f_\smash{\unicode{x25CB}}} = f_x \mathbf e_1 + f_y \mathbf e_2 + f_z \mathbf e_3$$ Normal vector, or point at infinity in direction perpendicular to plane $$\mathbf f$$.
Motor $$\mathbf Q = r_x \mathbf e_{41} + r_y \mathbf e_{42} + r_z \mathbf e_{43} + r_w {\large\unicode{x1d7d9}} + u_x \mathbf e_{23} + u_y \mathbf e_{31} + u_z \mathbf e_{12} + u_w$$ $$\underline{\mathbf Q_\smash{\unicode{x25CB}}} = -r_x \mathbf e_{23} - r_y \mathbf e_{31} - r_z \mathbf e_{12} + r_w$$ Conventional quaternion $$\mathbf q = r_x \mathbf i + r_y \mathbf j + r_z \mathbf k + r_w = (a_x \mathbf i + a_y \mathbf j + a_z \mathbf k)\sin\phi + \cos\phi$$, which is the 3D position-free counterpart of a motor.

The sandwich product $$\mathbf q \mathbin{\unicode{x27D1}} \mathbf v \mathbin{\unicode{x27D1}} \mathbf{\tilde q}$$ rotates the vector $$\mathbf v$$ through the angle $$2\phi$$ about the axis $$\mathbf a$$.

Flector $$\mathbf G = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 + s_w \mathbf e_4 + h_x \mathbf e_{234} + h_y \mathbf e_{314} + h_z \mathbf e_{124} + h_w \mathbf e_{321}$$ $$\underline{\mathbf G_\smash{\unicode{x25CB}}} = h_x \mathbf e_1 + h_y \mathbf e_2 + h_z \mathbf e_3 - s_w \mathbf e_{321}$$ 3D position-free counterpart of a flector having the form $$\mathbf g = (a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3)\cos\phi + \mathbf e_{123}\sin\phi$$.

The sandwich product $$-\mathbf g \mathbin{\unicode{x27D1}} \mathbf v \mathbin{\unicode{x27D1}} \mathbf{\tilde g}$$ rotates the vector $$\mathbf v$$ through the angle $$2\phi$$ about the axis $$\mathbf a$$ and reflects it along the direction of $$\mathbf a$$.

See Also