A paraboloid geodesic is a geodesic on a paraboloid. The surface of revolution 
has parametrization
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
.
The coefficients of its first fundamental form
are
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
Writing a geodesic locally as 
, with 
, gives the Euler-Lagrange
differential equation
 |
(10)
|
Since 
,
,
and 
are independent of 
,
this integrates once to
 |
(11)
|
where 
is a constant of integration. For 
, 
and the geodesics are meridians 
. For 
and 
, this equation gives
 |
(12)
|
which integrates to
 |
(13)
|
where 
is a constant of integration and the inverse tangent takes values in 
. The branches with constants 
and 
meet at 
, 
.
These branches satisfy the implicit relation given by Weinstock (1974),
![u-c^2=u(1+4c^2)sin^2{v-2cln[k(2sqrt(u-c^2)+sqrt(4u+1))]},](/images/equations/ParaboloidGeodesic/NumberedEquation5.svg) |
(14)
|
where 
.
The differential equation for 
selects the geodesic branches
of this relation. Squaring the trigonometric expression introduces additional branches
that do not satisfy the Euler-Lagrange
differential equation.
