Lines in $\mathbb{R}^3$

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

$$\mathbf{x} = \lambda \mathbf{p} + (1-\lambda) \mathbf{q}$$ (1.72)

In the specific case of $\mathbb{R}^3$, these equations require six parameters to be estimated (a “bounded 3D line” in fact has six degrees of freedom).

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

$$\mathbf{x} = \mathbf{x}_0 + t \hat{\mathbf{v}}$$ (1.73)

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

$$\mathbf{x} \times \hat{\mathbf{v}} = \mathbf{x}_0 \times \hat{\mathbf{v}} = \mathbf{n}$$ (1.74)

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

In space $\mathbb{R}^3$, a line is the locus of points at the intersection of two planes (one of which may pass through the origin). Here too, we speak of at least five parameters to be estimated.

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