Total number of directions in $F_q^2$ is $q+1$

Question

According to this paper (https://arxiv.org/pdf/0803.3525.pdf), at the beginning of page 3, the total number of directions in $F_q^2$ is $q+1$. Here I believe that direction means the number of distinct lines of the form $L_{y} = \{ty \mid y \in F_{q}^2\}$. Where did this conclusion come from?

Answer 1

The set $\mathbb F_q^2$ has $q^2$ elements, of which $q^2 - 1$ are non-zero. For each $y \in \mathbb F_q^2$ with $y \neq 0$, you get a line, or direction, $L_y = \{ty \mid t \in \mathbb F_q\}$. Such a line contains precisely $q-1$ non-zero elements, and distinct lines have no non-zero elements in common. Thus there are $$ \frac{q^2 - 1}{q-1} = q+1 $$ lines, or directions, total.

Answer 2

A line $L$ in $k^2$, where $k$ is a field, has equation $ax+by+c=0$ where $(a,b)\ne(0,0)$. Two lines ($L$ and $L'$ given by $a'x+b'y=c'$) are equal or parallel iff the vectors $(a',b')=\lambda(a,b)$ for some $\lambda\in k-\{0\})$. If $|k|=q$ there are $q^2-1$ possible $(a,b)$, and the number of "parallel classes" they fall into are $(q^2-1)/(q-1)$ as there are $q-1$ possible $\lambda$.

Alternatively, most lines are $y=mx+c$, where $m$ determines the parallel class (so accounting for $q$ parallel classes), but there's an additional parallel class of "vertical" lines $x=a$.

Answer 3

Any line in $L_y$ is uniquely identified by one of the following directions:

$$ D= \{ (1,a), (0,1) \} $$ where $a$ is any element in $\mathbb F_p$.

There are exactly $p+1$ directions.