Lines in $\mathbb{R}^3$

One can view a generic line in a space $\mathbb{R}^n$ as the interpolation of two points in the same space:

$$\mathbf{x} = \lambda \mathbf{p} + (1-\lambda) \mathbf{q}$$ (1.36)

In the specific case of $\mathbb{R}^3$, these equations require 6 parameters to estimate (a "bounded 3D line" indeed has 6 degrees of freedom).

A line in the space $\mathbb{R}^n$ can be seen as a point plus a direction vector:

$$\mathbf{x} = \mathbf{x}_0 + t \hat{\mathbf{v}}$$ (1.37)

In the specific case of $\mathbb{R}^3$, these equations require 5 parameters (since a direction vector can be described by only 2 variables). In this case, the locus of points can be derived by multiplying by $\times \hat{\mathbf{v}}$:

$$\mathbf{x} \times \hat{\mathbf{v}} = \mathbf{x}_0 \times \hat{\mathbf{v}} = \mathbf{n}$$ (1.38)

The vector $\mathbf{x}_0 \times \hat{\mathbf{v}}$ obviously describes a vector orthogonal to the other two, but whose length is important. This representation is identical to that obtained using the PlÃ14cker coordinate system.

In the space $\mathbb{R}^3$, the line is the locus of points at the intersection of 2 planes (one of which may potentially pass through the origin). Again, we are discussing at least 5 parameters to estimate.

However, in $\mathbb{R}^3$, lines have only 4 degrees of freedom: we can see that every line is tangent to a sphere of radius $r$, intersecting at point $m=(r, \theta, \phi)$ in spherical coordinates. The last parameter is a rotation angle $\gamma$ around vector $m$ to indicate the direction of the line (this part requires a couple of additional conditions to avoid singularities).