CS in Algebra: Coordinate Plane

You're going to learn how math, like algebra
and geometry, can be used to develop video

games.

Some modern video games are incredibly complex,
with realistic graphics, physics, artificial

intelligence and so on.

Major video games can take large teams of
developers years to produce but you can apply

many of the same concepts that the big developers
use to create a simple game of your own.

Let's start by looking at a fairly simple
2-dimensional game.

Each of the actors, or sprites, in this game,
can be described by their location or movement

on the screen.

Your job is to figure out how we can describe
the actions of each of these sprites mathematically

in relation to the coordinate plane.

Let's take a quick look at the game Plants
vs. Zombies as an example.

If we have a zombie and a flower on the screen,
there are a few things we need to know about

them in order to make the game work.

Where's the zombie?

In which direction is it moving?Where's the
flower?

How far apart are they?

As a player of the game, you might say in
general terms that they're on opposite sides

of the screen.

Or that the zombie is moving to the left.

Or that the zombie and the flower are pretty
close to each other.

These might be ok approximations but they
really aren't specific enough and they definitely

aren't stated in a way that a computer can
understand.

Suppose you were talking to your friend on
the phone, trying to tell them exactly where

the dragon is.

You could use words like on the left but that
isn't specific enough.

If you had a ruler, you could measure from
the left side of the screen and tell your

friend exactly how many inches away the dragon
is.

That is exactly what computers do.

Using a number line that starts with a 0 on
the far left and moving across the right measuring

the number of pixels on the screen.

For our video game, we'll place the number
line so that the screen runs from 0 on the

left to 400 on the right.

We can imagine the image of the dragon, stick
it anywhere on the line and measure the distance

back to the left hand edge of the screen from
our dragon.

Anyone else who knows about our number line
will be able to duplicate the exact position

of the dragon know only the number.

On the right side of the screen, the dragon
is at 400.

In the center, he's at 200.

What if we wanted the dragon to be off the
edge of the screen?

We could use numbers bigger than 400 to place
him past the right hand side and negative

numbers to place him past the left hand side.

But even with a number line, we aren't being
quite specific enough.

Even at 400, the dragon could be at the top
of the screen or at the bottom or anywhere

in between.

By adding another number line, we can locate
a character anywhere on the screen in either

dimension.

The first line is called the x axis which
runs from left to right.

The second line which runs up and down is
called the y axis.

A 2-dimensional coordinate consists of both
the x and y locations on the axis.

Suppose we want to locate ninja's position
on the screen.

We can find the x coordinate by dropping a
line down from the ninja and read the position

on the number line.

The y coordinate is found by running a line
to the y axis.

With two numbers, x and y coordinates, we
can describe the location of any sprite on

our screen.

And by changing those numbers, we can get
our sprites to move around on the screen.

What we've created is actually quadrant one
of the coordinate plane!

If we zoom out, we can see that there are
four different quadrants to the plane: quadrant

I, which we're using as our screen, contains
the points of positive values for x and y.

Moving counterclockwise we get to quadrant
II which contains the points of a negative

X and positive Y.

Quadrant III contains all points with negative
X and Y and quadrant IV contains all points

with a positive X and a negative Y.