Limiting case Sphere–cylinder intersection

viviani s curve intersection of sphere , cylinder

in case



r
=
r
+
a


{\displaystyle r=r+a}

, cylinder , sphere tangential each other @ point



(
r
,
0
,
0
)


{\displaystyle (r,0,0)}

. intersection resembles figure eight: closed curve intersects itself. above parametrization becomes

(
x
,
y
,
z
)
=

(
r
cos
⁡
ϕ
,
r
sin
⁡
ϕ
,
2


a
r


cos
⁡


ϕ
2


)

,


{\displaystyle (x,y,z)=\left(r\cos \phi ,r\sin \phi ,2{\sqrt {ar}}\cos {\frac {\phi }{2}}\right),}

where



ϕ


{\displaystyle \phi }

goes through 2 full revolutions.

in special case



a
=
r
,
r
=
2
r


{\displaystyle a=r,r=2r}

, intersection known viviani s curve. parameter representation is

(
x
,
y
,
z
)
=

(
r
cos
⁡
ϕ
,
r
sin
⁡
ϕ
,
r
cos
⁡


ϕ
2


)

.


{\displaystyle (x,y,z)=\left(r\cos \phi ,r\sin \phi ,r\cos {\frac {\phi }{2}}\right).}