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Figure 1. A point is the intersection of a 4D vector with the 3D subspace where $$w = 1$$.

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.

See Also