The bicuspid curve is the quartic curve given by the implicit equation
 |
(1)
|
so-named because of its resemblance to a tooth.
The bicuspid curve has cusps at 
and 
.
For 
,
the horizontal tangents are located at 
, and the vertical
tangents at 
,
, and 
, where 
is the positive real root of
.
For 
,
a parameterization of the upper half of the
curve is
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
,
 |
(4)
|
and 
is the unique real root of 
. This gives an exact elliptic
integral representation for the enclosed area
 |
(5)
|
For an exact expression in terms of the Lauricella functions 
and 
,
define
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
where 
is the root of 
with positive imaginary
part, 
is the imaginary unit, an overbar denotes the complex conjugate, and 
and 
are taken on the branch defined by their one-dimensional
Euler integrals over 
. The area is then
 |  | ![(pia^2sqrt(-alpha-1))/2[B_0D_2+2Re[BD_3]]](/images/equations/BicuspidCurve/Inline48.svg) |
(12)
|
 |  |  |
(13)
|
(OEIS A193625), where 
denotes the real part. For the
arc length, differentiating the parameterization
gives
 |
(14)
|
where
 |
(15)
|
and hence the perimeter is
 |  | ![4a[int_(-infty)^alpha(|t|)/((1+t^2)^2)sqrt((R(t))/(Q(t)))dt+int_(-1)^infty(|t|)/((1+t^2)^2)sqrt((R(t))/(Q(t)))dt]](/images/equations/BicuspidCurve/Inline55.svg) |
(16)
|
 |  |  |
(17)
|
 |  |  |
(18)
|
(OEIS A193626), where
 |
(19)
|
and
 |
(20)
|
with 
on the intervals of integration. Because the polynomial 
has degree
10 and no multiple roots, the equation defines a
hyperelliptic curve of genus
4 and the expression for 
in terms of 
is an exact genus-4 Abelian
integral.
