用参数来定义一条直线/射线、缩放和有向直线上方/下方的半平面

A line is a set of points of constant direction, up to sign. More precisely, a line is a set of points such that there exists a nonzero vector $v$ such that for any two distinct points $P,Q$ on the line, $\overrightarrow{PQ} = Q-P$ is a vector either in the direction of $v$ or $-v$. Algebraically, a line is just a set of the form $\{(a+tx,b+ty):t\in\mathbb{R}\}$, where $(a,b)$ is any point on the line and $v$ is in the direction of $\pm(x,y)$.

The ray starting from $(a,b)$ (in the direction $(x,y)$) is the set of points $\{(a+tx,b+ty) : t \ge 0\}$.

The dilation/scaling/homothety centered at $(a,b)$ with nonzero ratio/factor $r\ne0$ is the transformation (mapping) sending a point $(x,y)$ to $(a+r(x-a),b+r(y-b))$. Note that it fixes the point $(a,b)$ (sends it to itself). (Why should $r\ne0$?)

To define the half-planes to the left (above) or right (below) a nonzero vector $(a,b)$, we'll need to rely on orientation from the previous section. But with orientation it's easy: the half-plane to the left (or above) $(a,b)$ is the set of points left of $(a,b)$. The half-plane to the right (or below) $(a,b)$ is the set of points right of $(a,b)$.