Implicit equations

Implicit equations are relations between two variables. If you insert a value for one variable, you can calculate the value of the other. So the difference to functions is that you don't get the value of the second variable immediately. You first have to calculate it. However implicit equations are still linked very strong to the functions. Actually there is a theorem that says that for every implicit equations, there's an function which describes exactly the same thing. Unfortunately, the theorem just says that this function exists. It doesn't tell us what it looks like or how we could find it. Most of the time it is even so difficult that there's no way to find the corresponding function.

If you look at the second function and at the first implicit equation, you will notice that they have identical graphical representations. The reason for this is that the one is the corresponding function of the implicit equation. This was one example where it was more or less easy to find it. Here are a few examples of implicit equations and their graphical representations.

The following equation has beside its variables x and y two other parameters p and q. You can actually give any value to these parameters you want. They are completely independent from x and y. The top graph was drawn with p=1 and q=0.5, the lower one with p=0.5 and q=0.5.

 So even here we found examples of very interesting equations with strong messages hidden deep at the inner of the equations. Who would have believed that a simple equation could have given such an interesting result? As I already said, mathematics are full of wonderful mysteries, waiting for somebody to discover them. 

You don't need much to go on an exploration through the world of mathematics. Patience and curiosity should be enough. Patience for being able to wait until the a structure reveals its secrets and curiosity for never giving up searching or asking. The question "What would happen if...?" is fundamental in mathematics. It should never been left unanswered.