@Carl Love 

I like that evala@evalc function, I'll be sure to use it every single chance I get. I know where I went wrong with the pythagorean quadruple. It's funny how this bug sent me down the wrong path and ultimately confused me. 

@vv So simplify is probably bugged. Can evalf(u) also be bugged? I got this example off a pythagorean quadruple of some sort. I don't remember what I did as I lost the function. They are the zeros of some kind of rotation involving the pythagorean quadruple I used.

@Doug Meade zero_vector.mw

The function I used to make this thing was part of a pythagorean n-tuple. I think I modfied it so that z is 1/y and I don't know what I did with x. The function was supposed to be zero on all three, so it does kind of make sense that this is a zero vector. I seem to have misplaced the function. Anyway, I just made the assumption that there is some serious canceling going on and it ends up being a zero vector after it's properly reduced. I am trying to figure out how to properly reduce it. 

Someone above said the problem might be a bug in the simplification itself and this is not a zero vector. That does make sense and I'm willing to accept that.

@J4James Oh, sorry. You were right on it. I just misunderstood what you were doing. I would give you a thumbs up but I can't.

I found the answer, it wasn't quite what I was looking for. It's a sphere. They are all spheres.

mypoint := proc(p1, p2,p3)

          if (p1^2+p2^2+p3^2)<>0 then        return [2*p1*p3/(p1^2+p2^2+p3^2) ,2*p2*p3/(p1^2+p2^2+p3^2),(p3^2-(p1^2+p2^2))/(p1^2+p2^2+p3^2)];      else        return [0, 0, 0];      end if;    end proc;

all_points := [seq([seq([seq(mypoint(x, y, z), y = -47 .. 47)], x = -47 .. 47)], z = -47 .. 47)]; plots[surfdata](all_points)

[2*p1*p3/(p1^2+p2^2+p3^2), 2*p2*p3/(p1^2+p2^2+p3^2), (p3^2-p1^2-p2^2)/(p1^2+p2^2+p3^2)]

[-(4*(p1^2+p2^2-p3^2))*p1*p3/(p1^2+p2^2+p3^2)^2, -(4*(p1^2+p2^2-p3^2))*p2*p3/(p1^2+p2^2+p3^2)^2, (p1^4+2*p1^2*p2^2-6*p1^2*p3^2+p2^4-6*p2^2*p3^2+p3^4)/(p1^2+p2^2+p3^2)^2]

[(8*(p1^2+p2^2-p3^2))*p3*p2*(p1^4+2*p1^2*p2^2-6*p1^2*p3^2+p2^4-6*p2^2*p3^2+p3^4)/(p1^2+p2^2+p3^2)^4, (8*(p1^2+p2^2-p3^2))*p3*p1*(p1^4+2*p1^2*p2^2-6*p1^2*p3^2+p2^4-6*p2^2*p3^2+p3^4)/(p1^2+p2^2+p3^2)^4, (p3^2-p1^2-p2^2)/(p1^2+p2^2+p3^2)]

@J4James Getting there. It looks like we need a procedure to jump through p. Range can be integers btw. I'm going to play around and try and figure it out. I don't know why it's looking like a surface in the case where p=2 but I think it will look completely different when it goes through its complete range.

@tomleslie I have no idea what you just did, but it definitely got me interested in implicit plots. It's not quite what I'm expecting. I'm looking for something along the lines of a lissajous curve but it's entirely 3d. Every point should be less than 1 and greater than -1. That one goes from -2 to 2. I'm hoping if I put a bunch of them together it will be a spherical surface.