Geometric norm

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The geometric norm is a measure of the magnitude of an element. It has two components called the bulk norm and the weight norm.

For points, lines, and planes, the geometric norm is equal to the shortest Euclidean distance between the geometry and the origin. For motors and flectors, the geometric norm is equal to half the distance that the origin is moved by the isometry operator.

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 dot product of $$\mathbf a$$ with its own reverse:

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

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 antidot product of $$\mathbf a$$ with its own antireverse:

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

(Note that the square root in this case is taken with respect to the geometric antiproduct.)

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 bulk norm and weight norm are summed to construct a geometric norm given by

$$\left\Vert\mathbf a\right\Vert = \left\Vert\mathbf a\right\Vert_\unicode{x25CF} + \left\Vert\mathbf a\right\Vert_\unicode{x25CB} = \sqrt{\mathbf a \mathbin{\unicode{x25CF}} \mathbf{\tilde a}} + \sqrt{\mathbf a \mathbin{\unicode{x25CB}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{a}}}\vphantom{\mathbf{\tilde a}}}$$ .

This quantity is the sum of a scalar $$x\mathbf 1$$ and antiscalar $$y{\large\unicode{x1D7D9}}$$ representing a homogeneous magnitude that itself has a bulk and a weight. Its bulk norm is simply the magnitude of its scalar part, and its weight norm is simply the magnitude of its antiscalar part. The geometric norm is idempotent because

$$\left\Vert x\mathbf 1 + y{\large\unicode{x1D7D9}}\right\Vert = |x|\mathbf 1 + |y|{\large\unicode{x1D7D9}}$$ .

Like all other homogeneous quantities, the magnitude given by the geometric norm is unitized by dividing by its weight norm. The unitized magnitude of an element $$\mathbf a$$ is given by

$$\widehat{\left\Vert\mathbf a\right\Vert} = \dfrac{\left\Vert\mathbf a\right\Vert}{\left\Vert\mathbf a\right\Vert_\unicode{x25CB}} = \dfrac{\left\Vert\mathbf a\right\Vert_\unicode{x25CF}}{\left\Vert\mathbf a\right\Vert_\unicode{x25CB}} + {\large\unicode{x1D7D9}} = \dfrac{\sqrt{\mathbf a \mathbin{\unicode{x25CF}} \mathbf{\tilde a}}}{\sqrt{\mathbf a \mathbin{\unicode{x25CB}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{a}}}\vphantom{\mathbf{\tilde a}}}} + {\large\unicode{x1D7D9}}$$ .

The following table lists the unitized geometric norms of the main types in the 4D projective geometric algebra $$\mathcal G_{3,0,1}$$ after dropping the constant $${\large\unicode{x1D7D9}}$$ term.

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

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

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

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

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

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

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

See Also