Why is the line connecting the identity point and another point vertical in an elliptic curve? I'm reading through Elliptic Tales.
Addition of 2 points on an elliptic curve is described as follows:

$L$ is the line between $P$ and $Q$ and $R$
$L'$ is the line between $O$ and $P + Q$ and $R$
The book describes the algebraic process of adding together 2 points on an elliptic curve.
First: It describes adding together $P$ and $Q$ to get $R$. It then says we need to connect $O$ and $R$ with a line, and where that line intersects $E$ will be the point $P + Q$. So far so good.
It then says the line connecting $O$ and $R$ is vertical and is easy to describe in projective coordinates as $x = x_3z$ where $R$ is $(x_3, y_3)$.
The line connecting $O$ and $R$ is $L'$ doesn't seem to be vertical. Clearly, in the picture it's slanted downwards.
Does anyone know what's going on?

The construction and diagrams you're quoting are in Section 8.1 of the book.
In Section 8.2, the author changes the definition of $\mathcal O$ to $(0:1:0)$, i.e. the point at infinity in the vertical direction.  In this new context, $L'$ becomes a vertical line.
This group theoretic construction in 8.1 works for nonsingular cubic curves in general where $\mathcal O$ can be any point on the curve.  Elliptic curves are a special case of these curves, and always pass through (0:1:0).  This simplifies the construction somewhat, in that the final stage is a simple reflection across the x-axis, rather than the intersection of the curve and a line.
