Parametric equations of projective line

Question

The parametric equations of a projective line passing through $P = \left[X_1:Y_1:Z_1\right]$ and $Q = \left[X_2:Y_2:Z_2\right]$ are :

$\left[X:Y:Z\right] = \alpha\left[X_1:Y_1:Z_1\right] + \beta\left[X_2:Y_2:Z_2\right]$, $\alpha,\beta \in \mathbb{R}$?.

Is it an abuse of notation?

Answer 1

Your equation is perfectly valid (with the caveat spotted by @Semiclassical).

The parametric equation of a line defined by 2 points in projective space is usually derived from a reasoning on the rank of matrix:

$$\left(\begin{array}{l l l} X_1 & X_2 & X \\ Y_1 & Y_2 & Y \\ Z_1 & Z_2 & Z \end{array}\right) $$

By definition of a projective line, this rank has to be 2; which is equivalent to the fact that the last column can be expressed as linear combination of the first two columns:

$$\tag{1}M:=\left(\begin{array}{l l l} X \\ Y \\ Z \end{array}\right)=\alpha_1\left(\begin{array}{l l l} X_1 \\ Y_1 \\ Z_1 \end{array}\right)+\alpha_2\left(\begin{array}{l l l} X_2 \\ Y_2 \\ Z_2 \end{array}\right)$$

Edit (following a remark by @MvG pointing an error of mine in a first draft):

Two cases can occur:

• either $\alpha_1+\alpha_2 \neq 0$, and point $M$ is said "an ordinary point". In this case, we can further normalize this linear combination (considering that we can take taking $\alpha_1+\alpha_2=1$), due do the fact that projective coordinates are defined up to a multiplicative factor. Thus (1) implies:

$$M:=\left(\begin{array}{l l l} X \\ Y \\ Z \end{array}\right) \sim k\alpha_1\left(\begin{array}{l l l} X_1 \\ Y_1 \\ Z_1 \end{array}\right)+k\alpha_2\left(\begin{array}{l l l} X_2 \\ Y_2 \\ Z_2 \end{array}\right)$$

where $k$ is any nonzero real number, for example $1/(\alpha_1+\alpha_2)$, and symbol $\sim$ means "proportional to" (= with equivalence relationship "defined up to a non zero multiplicative factor"). We check immediately that the new coefficients $\alpha'_1:=\alpha_1/(\alpha_1+\alpha_2),\alpha'_2:=\alpha_2/(\alpha_1+\alpha_2)$ sum up to $1$.

Remark: this way of writing things is important because, in this way, as we are in affine geometry, the bridge is made with the barycentrical approach.

• or $\alpha_1+\alpha_2=0$. In this case, (1) becomes:

$$M:=\left(\begin{array}{l l l} X \\ Y \\ Z \end{array}\right) \sim \left(\begin{array}{l l l} X_1 \\ Y_1 \\ Z_1 \end{array}\right)-\left(\begin{array}{l l l} X_2 \\ Y_2 \\ Z_2 \end{array}\right)$$

i.e., is equivalent to a vector. Point $M$ is said "at infinity".

Remark: A key point in the understanding of projective geometry is the bijective correspondence between vectors (up to a sign change) and points at infinity.