Elliptical Spheres

function parameters:

α

m

embedding parameters:

x-direction Re/Im ratio

y-direction Re/Im ratio

z-direction Re/Im ratio

Elliptical spheres are neither spheres nor elliptic. Elliptical spheres are complex vectors

r={x,y,z}

such that

r.r=1

and

x

,

y

,

z

are elliptic functions. A possible parametrization of these vectors is given through Neville theta functions as

r

α,m

(z)=

1

1-m

ϑ

n

(αm)

ϑ

n

(zm)



1-m

m

ϑ

s

(αm)

ϑ

s

(zm),

m

ϑ

c

(αm)

ϑ

c

(zm),

ϑ

d

(αm)

ϑ

d

(zm)

,

z∈

,

α∈

,

m∈

.

In this Demonstration,

z

runs over part of the complex plane and

α

and

m

are complex parameters. The graphic displays variable relative fractions of the real and imaginary parts of

x

,

y

, and

z

.