Geometric norm

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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 $$\mathbf G$$.

See Also