The Ida surface is an immersion of the Klein bottle in Euclidean space 
. It was produced by Ray Idaszak using a family of surfaces
developed by George Francis from Apéry's Romboy homotopy
(Francis 1987, pp. 178-179).
The family can be parametrized by
![p(theta,tau)=rho(theta,tau)[A(theta)costau+B(theta)sintau],](/images/equations/IdaSurface/NumberedEquation1.svg) |
(1)
|
where
 |  |  |
(2)
|
 |  |  |
(3)
|
and
 |
(4)
|
Here, 
and 
control the proportions of the surface,
and 
and 
are 
-periodic. With 
, 
, and 
, this gives the Ida surface illustrated above. The denominator is positive, and the parametrization satisfies
 |
(5)
|
which expresses the Klein bottle identification on the parameter torus.
The choices 
,
, 
, and 
give the Roman surface,
Boy surface, Etruscan
Venus surface, and Ida surface, respectively (Francis 1987, p. 179).
