#parametrize left (green) branch of parabola with cosh and sinh
x1 := -9/2 + sqrt(3)*sinh(t)/2 -sqrt(3)*cosh(t);

y1:=3+sqrt(3)*cosh(t)/2;


#parametrize right (branch) branch of parabola with cosh and sinh (change sign on cosh)
x2 := -9/2 + sqrt(3)*sinh(t)/2 +sqrt(3)*cosh(t);

y2:=3-sqrt(3)*cosh(t)/2;


#map left branch into projective plane
x3:=x1/sqrt(x1^2+y1^2+1);

y3:=y1/sqrt(x1^2+y1^2+1);



#map right branch into projective plane
x4:=x2/sqrt(x2^2+y2^2+1);

y4:=y2/sqrt(x2^2+y2^2+1);



#plot both branches in Euclidean plane (parametric plot)
plot({[x1,y1,t=-4..4],[x2,y2,t=-4..4]},-25..25,-25..25);


#plot both branches in projective plane (parametric plot)
plot({[x3,y3,t=-4..4],[x4,y4,t=-4..4]},-1..1,-1..1);