# Geometric norm

(Redirected from Bulk norm)

The geometric norm is a measure of the magnitude of an element that possesses the geometric property. It is the ratio of the bulk norm and the weight norm.

## Bulk Norm

The bulk norm of an element $$\mathbf a$$, denoted $$\left\Vert\mathbf a\right\Vert_\unicode{x25CF}$$, is the magnitude of its bulk components. It can be calculated by taking the square root of the geometric product of $$\mathbf a_\unicode{x25CF}$$ with its own reverse:

$$\left\Vert\mathbf a\right\Vert_\unicode{x25CF} = \sqrt{\mathbf a_\unicode{x25CF} \mathbin{\unicode{x27D1}} \mathbf{\tilde a}_\unicode{x25CF}}$$ .

If $$\mathbf a$$ possesses the geometric property, then the $$\unicode{x25CF}$$ subscripts inside the radical can be dropped because $$\mathbf a_\unicode{x25CF} \mathbin{\unicode{x27D1}} \mathbf{\tilde a}_\unicode{x25CF} = \mathbf a \mathbin{\unicode{x27D1}} \mathbf{\tilde a}$$.

An element that has a bulk norm of 1 is said to be bulk normalized.

The following table lists the bulk norms of the main types in the 4D projective geometric algebra $$\mathcal G_{3,0,1}$$.

Type Definition Bulk Norm
Point $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ $$\left\Vert\mathbf p\right\Vert_\unicode{x25CF} = \sqrt{p_x^2 + p_y^2 + p_z^2}$$
Line $$\mathbf L = v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} + m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$ $$\left\Vert\mathbf L\right\Vert_\unicode{x25CF} = \sqrt{m_x^2 + m_y^2 + m_z^2}$$
Plane $$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$ $$\left\Vert\mathbf f\right\Vert_\unicode{x25CF} = |f_w|$$
Motor $$\mathbf Q = r_x \mathbf e_{41} + r_y \mathbf e_{42} + r_z \mathbf e_{43} + r_w {\large\unicode{x1d7d9}} + u_x \mathbf e_{23} + u_y \mathbf e_{31} + u_z \mathbf e_{12} + u_w$$ $$\left\Vert\mathbf Q\right\Vert_\unicode{x25CF} = \sqrt{u_x^2 + u_y^2 + u_z^2 + u_w^2}$$
Flector $$\mathbf G = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 + s_w \mathbf e_4 + h_x \mathbf e_{234} + h_y \mathbf e_{314} + h_z \mathbf e_{124} + h_w \mathbf e_{321}$$ $$\left\Vert\mathbf G\right\Vert_\unicode{x25CF} = \sqrt{s_x^2 + s_y^2 + s_z^2 + h_w^2}$$

## Weight Norm

The weight norm of an element $$\mathbf a$$, denoted $$\left\Vert\mathbf a\right\Vert_\unicode{x25CB}$$, is the magnitude of its weight components. It can be calculated by taking the square root of the geometric antiproduct of $$\mathbf a_\unicode{x25CB}$$ with its own antireverse:

$$\left\Vert\mathbf a\right\Vert_\unicode{x25CB} = \sqrt{\mathbf a_\unicode{x25CB} \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{a}}}_\unicode{x25CB}\vphantom{\mathbf{\tilde a}}}$$ .

Note that the square root in this case is taken with respect to the geometric antiproduct. If $$\mathbf a$$ possesses the geometric property, then the $$\unicode{x25CB}$$ subscripts inside the radical can be dropped because $$\mathbf a_\unicode{x25CB} \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{a}}}_\unicode{x25CB} = \mathbf a \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{a}}}$$.

An element that has a weight norm of $$\large\unicode{x1D7D9}$$ is said to be weight normalized or unitized.

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

Type Definition Weight Norm
Point $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ $$\left\Vert\mathbf p\right\Vert_\unicode{x25CB} = |p_w|{\large\unicode{x1D7D9}}$$
Line $$\mathbf L = v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} + m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$ $$\left\Vert\mathbf L\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{v_x^2 + v_y^2 + v_z^2}$$
Plane $$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$ $$\left\Vert\mathbf f\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{f_x^2 + f_y^2 + f_z^2}$$
Motor $$\mathbf Q = r_x \mathbf e_{41} + r_y \mathbf e_{42} + r_z \mathbf e_{43} + r_w {\large\unicode{x1d7d9}} + u_x \mathbf e_{23} + u_y \mathbf e_{31} + u_z \mathbf e_{12} + u_w$$ $$\left\Vert\mathbf Q\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{r_x^2 + r_y^2 + r_z^2 + r_w^2}$$
Flector $$\mathbf G = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 + s_w \mathbf e_4 + h_x \mathbf e_{234} + h_y \mathbf e_{314} + h_z \mathbf e_{124} + h_w \mathbf e_{321}$$ $$\left\Vert\mathbf G\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{h_x^2 + h_y^2 + h_z^2 + s_w^2}$$

## Geometric Norm

The geometric norm is a scalar value representing the magnitude of an element, which corresponds to a distance in Euclidean space.

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

Type Definition Geometric Norm Interpretation
Point $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ $$\left\Vert\mathbf p\right\Vert = \dfrac{\sqrt{p_x^2 + p_y^2 + p_z^2}}{|p_w|}$$ Distance from the origin to the point $$\mathbf p$$.

Half the distance that the origin is moved by the flector $$\mathbf p$$.

Line $$\mathbf L = v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} + m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$ $$\left\Vert\mathbf L\right\Vert = \sqrt{\dfrac{m_x^2 + m_y^2 + m_z^2}{v_x^2 + v_y^2 + v_z^2}}$$ Perpendicular distance from the origin to the line $$\mathbf L$$.

Half the distance that the origin is moved by the motor $$\mathbf L$$.

Plane $$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$ $$\left\Vert\mathbf f\right\Vert = \dfrac{|f_w|}{\sqrt{f_x^2 + f_y^2 + f_z^2}}$$ Perpendicular distance from the origin to the plane $$\mathbf f$$.

Half the distance that the origin is moved by the flector $$\mathbf f$$.

Motor $$\mathbf Q = r_x \mathbf e_{41} + r_y \mathbf e_{42} + r_z \mathbf e_{43} + r_w {\large\unicode{x1d7d9}} + u_x \mathbf e_{23} + u_y \mathbf e_{31} + u_z \mathbf e_{12} + u_w$$ $$\left\Vert\mathbf Q\right\Vert = \sqrt{\dfrac{u_x^2 + u_y^2 + u_z^2 + u_w^2}{r_x^2 + r_y^2 + r_z^2 + r_w^2}}$$ Half the distance that the origin is moved by the motor $$\mathbf Q$$.
Flector $$\mathbf G = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 + s_w \mathbf e_4 + h_x \mathbf e_{234} + h_y \mathbf e_{314} + h_z \mathbf e_{124} + h_w \mathbf e_{321}$$ $$\left\Vert\mathbf G\right\Vert = \sqrt{\dfrac{s_x^2 + s_y^2 + s_z^2 + h_w^2}{h_x^2 + h_y^2 + h_z^2 + s_w^2}}$$ Half the distance that the origin is moved by the flector $$\mathbf G$$.