Point
In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a point $$\mathbf p$$ is a vector having the general form
- $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ .
All points possess the geometric constraint.
The bulk of a point is given by its $$x$$, $$y$$, and $$z$$ coordinates, and the weight of a point is given by its $$w$$ coordinate. A point is unitized when $$p_w^2 = 1$$.
When used as an operator in a sandwich with the geometric antiproduct, a point is a specific kind of flector that performs an inversion through itself.
A translation operator $$\mathbf T$$ that moves a point $$\mathbf p$$ to the origin is given by
- $$\mathbf T = \underline{\mathbf p} \vee \mathbf e_{321} + 2\mathbf p \wedge \mathbf e_{321} = -p_{x\vphantom{y}} \mathbf e_{23} - p_y \mathbf e_{31} - p_{z\vphantom{y}} \mathbf e_{12} + 2p_w {\large\unicode{x1D7D9}}$$ .
Points at Infinity
If the weight of a point is zero (i.e., its $$w$$ coordinate is zero), then the point is contained in the horizon infinitely far away in the direction $$(x, y, z)$$, 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.
In the Book
- Homogeneous points are discussed in Section 2.4.1.