This may sound strange, and of course completely paradoxical, but what exactly are the properties of a triangle which has its points an infinite distance apart?
I have been toying with dimensions recently, kick started by an idea I had for a puzzle game using more than 3 dimensions, and being the sort of person that always needs a "how", I think that I'm in the process of developing a model for dimesions 3 and above.
I'll explain that when I think it works.
On a normal triangle, of the three vectors between the three points, only two of them are needed to pinpoint the location of a point inside its area. Is this true on such a paradoxical shape?
And please, keep it in laymans terms, I have only just finished my GCSEs...
In three dimensional space, lets say a cube, you need a maximum of three different vectors to be able to get from one point to another. That means lenth is not that important, just the direction. If trigonometry can prove that the same is true for a two dimensional shape, then my model might just work..
I mean an actual two dimensional shape. Not a three dimensional shape drawn in a two dimensional format.
I just want to check if my suspicions of the three vectors on said triangle can not equal each other are true, with a proof. If not, it would be helpful to have a proof of it being flase, but of course this is using infinity, so at most im hoping for some insight I can use.
What geometry are you using? Your "I mean an actual two dimensional shape" seems to imply Euclidean geometry but inn Euclidean Geometry there are NO "points at infinity" so the whole question is meaningless. In elliptic geometry or projective geometry, there are "ideal points" (points at infinity) but then the postulates you can use to prove things are, of course, different.
If two vectors meet at infinity, then they must be parallel, and so in this case, each of the three vectors in the triangle must be simultaneously parallel: a paradox.
This means, that each pair of vectors is simultaneously parallel, and the vector which connects them is the third vector, and so all three vectors act as a separate dimension, and so has the properties of a 3D shape.
At least, thats how it goes in my head. I think I may have posted this in the wrong forum, its not really a problem you can prove with triganometry XD
If you don't understand what I mean, I'll try and explain it a bit better.