The name lemniscatic torus is used for several objects involving a lemniscate. In geometric modeling it denotes a surface of revolution generated from a lemniscate in a plane offset from the axis (Iglesias et al. 2013). In elliptic function theory it denotes the square complex torus (Cecotti and Vafa 2013, Eremenko 2023). A limiting solid of revolution whose meridian is a lemniscate also occurs in gravitational modeling (Zavizion 2006).
For the geometric construction, let 
and 
, and define
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(1)
|
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(2)
|
The plane curve 
satisfies the Bernoulli lemniscate
equation
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(3)
|
where 
is the half-width and the distance from the center to either focus
is 
.
Rotating the curve 
about the z-axis
gives
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(4)
|
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(5)
|
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(6)
|
One lobe, with 
and its endpoints identified, generates the entire surface
as 
.
Using 
covers the surface twice. The formulas above correct
the inconsistent half-width normalization and the first coordinate of the printed
rotation formula in Flores-Córdova et al. (2026, p. 118).
Eliminating the parameters gives the quartic surface
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(7)
|
In cylindrical coordinates, with 
, a meridian
is parametrized by
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(8)
|
and the bounds are 
and 
.
This meridian is generally not a Bernoulli lemniscate.
The generating lemniscate lies in the offset plane 
, which does not contain the axis of
revolution.
For 
,
the surface becomes an ordinary ring
torus with major radius 
and tube radius 
, so 
(Iglesias et al. 2013). Its equation can be written
 |
(9)
|
Curves on the geometric surface have been used to choose vertices for polyhedra
(Velezmoro-León et al. 2021, 2022) and, after orthogonal
projection onto a plane, control points for a wire
antenna (Flores-Córdova et al. 2026). Writing the parametric
equations above as 
, one useful family is
 |
(10)
|
where 
and 
are positive integers and 
. Its orthogonal
projection onto the 
-plane has the complex
number representation
![w_(p,q)(t)=e^(ipt)[B+if(qt)],](/images/equations/LemniscaticTorus/NumberedEquation6.svg) |
(11)
|
and satisfies
 |
(12)
|
Thus relatively prime 
and 
give 
-fold rotational symmetry.
If 
is a common integer factor, write 
and 
. Then 
, so the common factor repeats the
parametrization. Flores-Córdova et al. (2026) use 
, 
, 
, and 
. Their actual antenna is a polygonal wire constructed
from sampled points of the projected curve, rather than
the smooth projected curve itself.
In the complex torus usage, the defining lattice is square,
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(13)
|
where 
is a complex number, and the quotient
space 
has half-period ratio 
. Its Weierstrass
elliptic function has invariants 
and 
, giving the lemniscate
case. The normalized Klein's absolute
invariant is 
,
whereas the usual j-function has 
. An equivalent elliptic
curve, completed by its point at infinity,
is
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(14)
|
This usage specifies a complex structure and does not select an embedding as a particular surface of revolution in three-dimensional space (Cecotti and Vafa 2013, Appendix B.3; Eremenko 2023).
Zavizion (2006) uses the name for a homogeneous solid of revolution bounded by rotating a Bernoulli lemniscate
about its symmetry axis in a study of gravitational fields. Its meridian
is a lemniscate, and its boundary is the 
limit of the geometric family above,
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(15)
|
It has a singular point at the origin. The 
family has an open central hole.
Supersymmetric Theories in 4 Dimensions." Surveys in Differential Geometry 18,
19-101, 2013.