The Complex Unit Circle
The Complex Unit Circle
The points of the complex unit circle +=1 can be parametrized:
{z,w}∈
2
2
z
2
w
z=x+iy=cos(α)cosh(β)-isin(α)sinh(β)
w=u+iv=sin(α)cosh(β)+icos(α)sinh(β)
This Demonstration shows 3D projections of the surface +=1 in space. The angles denote the rotation angles inside the hyperplane. In the limit, as , the complex unit circle becomes a circle in the plane.
2
z
2
w
x,y,u,v
φ
a,b
a,b
β0
x,u