Coordinate Systems

Coordinate Systems

N Dimensional systems

Definition:

 

A set of "n" numbers such that all elements of the set are members of all Real numbers (everything except complex).


For a 1D coordinate system, the definition will be as such:



 

It will simply be a line that contains all real numbers in the x(1) axis.

For a traditional 2D coordinate system, the definition will be as such:

This is how we usually define our (x,y) system. It contains all real numbers in the x(1) axis and all real numbers in the x(2) axis.


If you have already spotted the pattern, for a 3D coordinate system, it will be as such:

It now have 3 axes[x(1), x(2), x(3)] and all of them contain all real numbers in all axes.

In short, we can have an infinite amount of axes but it will be hard to imagine anything above 3-dimensional as we are living in a 3-dimensional world.


Coordinate Systems

Question: Can you tell me the coordinates of this point?

Obviously, you can't. There are no axes and no reference as to where the point is.

How about now?

If you answered (4,1), you are wrong. If (4,1) is valid, don't you think that (1,4) is valid too? This is so as the axes are not defined in this picture! Traditionally, you will think that the horizontal axis will be the x-axis and the vertical axis is the y-axis. However, it is undefined now so its impossible to tell which one comes first. No axis information can reverse the point location.

Finally:

It is very clear now that the point is (4,1) in a [x(1) , x(2)]-system. You might be surprised that why not x and y axes? This is because they are completely arbitrary, it will get more complicated as you go into more dimensions, so using incremental alphabet is preferred.

It is clear now that a proper coordinate system gives a name to any point.

Therefore, the proper way to define a point is:
A point is (    ) in a [  ]-system.

All the examples above utilized what we call a Cartesian coordinate system. All its axes will be perpendicular against one another and all the ticks(1,2,3,4,5..etc) are of equal distances.

Why do we use this system most of the time? The answer lies in the Pythagoras theorem. Give any 2 points in a Cartesian coordinate system, it is very easy to create a right angle triangle and in return we can compute distances conveniently.

The question is, must we follow this system? No! It is totally possible to distort the axes!!

Example:

How do you plot the point (4,1) in this [x(1) , x(2)]-system?

To do this effectively, since the axes are distorted, we will just have to realign the point coordinates parallel to their respective axes as shown in the picture.


Summary:

1. A proper coordinate system gives a name to any point.
2. Proper way to define a point is: A point is (    ) in a [  ]-system.
3. No axis information can reverse the point location.4
4. Axes can be distorted.