Numerical Calculations in Clifford Algebra

18Lines Not Parallel

Two lines which are not parallel intersect if they lie in a common plane, but fail to intersect if they have no plane in common. In the first case, as in Figure 18.1a, the shortest line segment which connects the two lines is of length zero and identifies a single point in space. In the second case, as in Figure 18.1b, the shortest line segment connecting the two lines is non‐zero in length and identifies two points in space, one on each line.

Identifying the shortest line segment between two non‐parallel lines in space is taken here as a simple geometrical problem on which to demonstrate the use of Grassmann's inner and outer products. Since both of these products operate consistently in any number of dimensions¹, the solutions given here can be used in all dimensions . Although the code used to demonstrate these solutions numerically is applied only in three dimensions, it also applies directly to higher dimensions simply by declaring a greater number of dimensions at the outset and providing higher dimensional data.

18.1 Theory

18.1.1 Common Plane

The intersection of two lines can be calculated by solving a system of linear equations involving Cartesian (or other) components of vectors. A linear system of two interdependent equations arises when Grassmann's inner product is used. When using Grassmann's outer product, the linear system reduces to two ...