Analytic geometry

What analytic geometry is

Analytic geometry, also called coordinate geometry or Cartesian geometry, is the branch of mathematics that studies geometric figures using a coordinate system and algebra. It represents points by ordered pairs of numbers, and describes lines, curves and shapes with equations and inequalities.

That is exactly how a game sees the world. A point is two numbers. A circle is a centre and a radius. A rectangle is a range of values on each axis. The rest of this book is about turning those numbers into collision tests.

Coordinates of a point on a line

A number axis is a line $x$ with a chosen unit and a chosen origin $O$. The origin divides the line into two opposite half-lines, the negative and the positive half-axis. Every point on the line then has exactly one real number, its coordinate.

The Cartesian coordinate system

The Cartesian coordinate system in a plane is two perpendicular number axes, the $x$-axis and the $y$-axis, that intersect at the origin $O$. Every point in the plane is represented by exactly one ordered pair of real numbers $(x, y)$: how far it is along the $x$-axis, and how far along the $y$-axis. In the figure below, the point $A$ has the coordinates $(4, 3)$.

Shapes are numbers too. The circle is its centre $S(-5, 2)$ and its radius, $2$. The rectangle is every point whose $x$ lies between $3$ and $8$ and whose $y$ lies between $-4$ and $-1$.

Screen coordinates

Screens flip the $y$-axis. On a canvas, $(0, 0)$ is the top-left corner and $y$ grows downwards. None of the formulas in this book change because of that, since they only ever use differences and squares; it only matters when you think about which way is up. The interactive demos in the following chapters use screen coordinates.