Variation in the Nil Circle
This note books shows a circle draw in nil space with different theta

Varioustheta = {Pi/2, Pi/8, Pi/16};
listList = {};
For[j =1, j < 4, j ++, 
{x1,y1,z1} = {0,0,0};
point = {x1, y1,z1};
pointList = {point};
θ = Varioustheta[[j]];
u = {1,0, 2*y1};
v = {0,1, -2*x1};
v1 = {1,0,2*y1};

interations= 1000;
For[i=0, i < interations, i ++,
u = {1.0,0, 2*point[[2]]};
v = {0,1, -2*point[[1]]};
n = Cross[u,v];
n = n/Norm[n];
vtemp = ((v1 -(v1.n)*n)/Norm[(v1 -(v1.n)*n)])*Norm[v1];
v2 = vtemp*Cos[θ] + (Cross[n,vtemp])*Sin[θ];
newpoint = point + v2;
pointList = Append[pointList, newpoint];
point = newpoint;
v1 =v2];
listList = Append[listList, pointList]]

(*ListPlot3D[listList[[1]]]*)	
ListPointPlot3D[listList[[1]]]
Graphics3D[Line[listList[[1]]]]
ListPointPlot3D[listList[[2]]]
Graphics3D[Line[listList[[2]]]]
ListPointPlot3D[listList[[3]]]
Graphics3D[Line[listList[[3]]]]