What are homogenuous coordinates and how do they relate to projective space? What even is projective space?

[Integrate]

I've been reading up on homogenous coordinates and it seems to orbit about projective space. However, I cannot seem to gather it into my brain. The description is too dense. How do parallel lines just magically converge? Is the point not that parallel lines cannot converge? To obtain the understanding of homogenous points it's apparent I need to understand projective space, but I have not been able to quite yet.

Here is a video I used to try to understand projective geometry. Where does that circle around the cartesian plane on the first slide come from?

Any simple lay person explanation that anyone might be able to offer?

Thank you in advance.

 

[tkhunny]

One word, Definition.

Don't drag some other definition or presupposition or personal bias into your system.