This chapter goes into cartesian coordinates, which is a handy idea to use in many games. The next chapter features the source code to Sonar, a game that uses cartesian coordinates.
A problem in many games is how to talk about exact points on the board. A common way of solving this is by marking each individual row and column on a board with a letter and a number. Here is a chess board that has each row and each column marked.
In chess, the knight piece looks like a horse. The white knight is located at the point e, 6 and the black knight is located at point a, 4. We can also say that every space on row 7 or every space in column c is empty.
A grid with labeled rows and columns like the chess board is called a Cartesian coordinate system. By using a row label and column label, we can give a coordinate that is to exactly one and only one space. This can really help us describe to a computer the exact location we want. If you have learned about Cartesian coordinate systems in math class, you may know that usually we have numbers for both the rows and columns. This is handy, because otherwise after the 26th column we would run out of letters. That board would look like this:
The numbers going left and right that describe the columns are part of the X-axis. The numbers going up and down that describe the rows are part of the Y-axis. When we describe coordinates, we always say the X coordinate first, followed by the Y coordinate. That means the white knight in the above picture is located at the coordinate 5, 6. The black knight is located at the coordinate 1, 4.
Notice that for the black knight to move to the white knight's position, the black knight must move up two spaces, and then to the right by four spaces. (Or move right four spaces and then move up two spaces.) But we don't need to look at the board to figure this out. If we know the white knight is located at 5, 6 and the black knight is located at 1, 4, then we can just use subtraction to figure out this information.
Subtract the black knight's X coordinate and white knight's X coordinate: 5 - 1 = 4. That means the black knight has to move along the X-axis by four spaces.
Subtract the black knight's Y coordinate and white knight's Y coordinate: 6 - 4 = 2. That means the black knight has to move along the Y-axis by two spaces.
Another concept that Cartesian coordinates use is negative numbers. Negative numbers are numbers that are smaller than zero. We put a minus sign in front of a number to show that it is a negative number. -1 is smaller than 0. -2 is smaller than -1. -3 is smaller than -2. If you think of regular numbers (called positive numbers) as starting from 1 and increasing, you can think of negative numbers as starting from -1 and decreasing. 0 itself is not positive or negative. In this picture, you can see the positive numbers increasing to the right and the negative numbers decreasing to the left:
The number line is really useful for doing subtraction and addition with negative numbers. The expression 4 + 3 can be thought of as the white knight starting at position 4 and moving 3 spaces over to the right (addition means increasing, which is in the right direction).
As you can see, the white knight ends up at position 7. This makes sense, because 4 + 3 is 7.
Subtraction can be done by moving the white knight to the left. Subtraction means decreasing, which is in the left direction. 4 - 6 would be the white knight starting at position 4 and moving 6 spaces to the left:
The white knight ends up at position -2. That means 4 - 6 equals -2.
If we add or subtract a negative number, the white knight would move in the opposite direction. If you add a negative number, the knight moves to the left. If you subtract a negative number, the knight moves to the right. The expression -6 - -4 would be equal to -2. The knight starts at -6 and moves to the right by 4 spaces. Notice that -6 - -4 has the same answer as -6 + 4.
The number line is the same as the X-axis. If we made the number line go up and down instead of left and right, it would model the Y-axis. Adding a positive number (or subtracting a negative number) would move the knight up the number line, and subtracting a positive number (or adding a negative number) would move the knight down. When we put these two number lines together, we have a Cartesian coordinate system.
The 0, 0 coordinate has a special name, the origin.
Subtracting negative numbers or adding negative numbers seems easy when you have a number line in front of you, but it can be easy when you only have the numbers too. Here are three tricks you can do to make evaluating these expressions easier to do.
The first is if you are adding a negative number, for example; 4 + -2. The first trick is "a minus eats the plus sign on its left". When you see a minus sign with a plus sign on the left, you can replace the plus sign with a minus sign. The answer is still the same, because adding a negative value is the same as subtracting a positive value. 4 + -2 and 4 - 2 both evaluate to 2.
The second trick is if you are subtracting a negative number, for example, 4 - -2. The second trick is "two minuses combine into a plus". When you see the two minus signs next to each other without a number in between them, they can combine into a plus sign. The answer is still the same, because subtracting a negative value is the same as adding a positive value.
A third trick is to remember that when you add two numbers like 6 and 4, it doesn't matter what order they are in. (This is called the commutative property of addition.) That means that 6 + 4 and 4 + 6 both equal the same value, 10.
Say you are adding a negative number and a positive number, like -6 + 8. Because you are adding numbers, you can swap the order of the numbers without changing the answer. -6 + 8 is the same as 8 + -6. But when you look at 8 + -6, you see that the minus sign can eat the plus sign to its left, and the problem becomes 8 - 6 = 2. But this means that -6 + 8 is also 2! We've rearranged the problem to have the same answer, but made it easier to solve.
Of course, you can always use the interactive shell as a calculator to evaluate these expressions. It is still very useful to know the above three tricks when adding or subtracting negative numbers. After all, you won't always be in front of a computer with Python all the time!
The absolute value of a number is the number without the negative sign in front of it. This means that positive numbers do not change, but negative numbers become positive. For example, the absolute value of -4 is 4. The absolute value of -7 is 7. The absolute value of 5 (which is positive) is 5.
We can find how far away two things on a number line are from each other by taking the absolute value of their difference. Imagine that the white knight is at position 4 and the black knight is at position -2. To find out the distance between them, you would find the difference by subtracting their positions and taking the absolute value of that number.
It works no matter what the order of the numbers is. -2 - 4 (that is, negative two minus four) is -6, and the absolute value of -6 is 6. However, 4 - -2 (that is, four minus negative two) is 6, and the absolute value of 6 is 6. Using the absolute value of the difference is a good way of finding the distance between two points on a number line (or axis).
It is common that computer monitors use a coordinate system that has the origin (0, 0) at the top left corner of the screen, which increases going down and to the right. There are no negative coordinates. This is because text is printed starting at the top left, and is printed going to the right and downwards. Most computer graphics use this coordinate system, and we will use it in our games. Also it is common to assume that monitors can display 80 text characters per row and 25 text characters per column. This used to be the maximum screen size that monitors could support. While today's monitors can usually display much more text, we will not assume that the user's screen is bigger than 80 by 25.
Things Covered In This Chapter:
- Cartesian coordinate systems.
- The X-axis and Y-axis.
- Absolute values and the abs() function.