The Complex Unit Circle

α/β parametrization range

4D rotation angles:

φ

x,y

φ

x,u

φ

x,v

φ

y,u

φ

y,v

φ

u,v

The points

{z,w}∈

2



of the complex unit circle

2

z

+

2

w

=1

can be parametrized:

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

2

z

+

2

w

=1

in

x,y,u,v

space. The angles

φ

a,b

denote the rotation angles inside the

a,b

hyperplane. In the limit, as

β0

, the complex unit circle becomes a circle in the

x,u

plane.

Details

For a detailed discussion of the complex unit circle, see

R. Hammack, "A Geometric View of Complex Trigonometric Functions," College Mathematics Journal, 38(3), 2007 pp. 210-217.

External Links

The Mathematica GuideBook for Graphics

Permanent Citation

Michael Trott, Michael Trott

"The Complex Unit Circle"
http://demonstrations.wolfram.com/TheComplexUnitCircle/
Wolfram Demonstrations Project
Published: June 13, 2007