How come there isn’t an imaginary system for dividing a number by 0, while there is one for taking the square root of a negative number?
In: Mathematics
Projective Geometry uses Homogeneous Cartesian Coordinates (HCC), which can be mapped with conventional coordinates by an additional “scaling” number at the end. So (3,5) would be (3,5,1) or (6,10,2) or (4.5, 7.5, 1.5), for example. But if you have (3,5,0) it refers to the point at infinity along the line from (0,0,1) and (3,5,1). The General Projective Transformation (GPT) is used to handle these coordinates.
If you “draw” a line between any two points at infinity, they define the line at infinity, which goes through all the points at infinity. BTW, (3,5,0) refers to the single point at infinity in both “directions.” Using an appropriate GPT, you can transform the point at infinity to become local, but another point, previously local, becomes inaccessible, i.e., at infinity.
Because of the convenience of using HCC via the GPT, this has taken over computer graphics internally. I learned this at MIT in 1960 when our math was mostly done by hand.
If you have 3D coordinates, (3,4,5) becomes (3,4,5,1) and so on.
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