Non-trivial cases Sphere–cylinder intersection

1 non-trivial cases

1.1 intersection consists of 2 closed curves
1.2 intersection single closed curve
1.3 limiting case

non-trivial cases

subtracting 2 equations given above gives

z

2


+
(

r

2


−

r

2


+

a

2


)
=
2
a
x
.


{\displaystyle z^{2}+(r^{2}-r^{2}+a^{2})=2ax.}

since



x


{\displaystyle x}

quadratic function of



z


{\displaystyle z}

, projection of intersection onto xz-plane section of orthogonal parabola; section due fact



−
r
<
x
<
r


{\displaystyle -r<x<r}

. vertex of parabola lies @ point



(
−
b
,
0
,
0
)


{\displaystyle (-b,0,0)}

, where

b
=




r

2


−

r

2


−

a

2




2
a



.


{\displaystyle b={\frac {r^{2}-r^{2}-a^{2}}{2a}}.}



intersection consists of 2 closed curves

if



r
>
r
+
a


{\displaystyle r>r+a}

, condition



x
<
r


{\displaystyle x<r}

cuts parabola 2 segments. in case, intersection of sphere , cylinder consists of 2 closed curves, mirror images of each other. projection in xy-plane circles of radius



r


{\displaystyle r}

.

each part of intersection can parametrized angle



ϕ


{\displaystyle \phi }

:

(
x
,
y
,
z
)
=

(
r
cos
⁡
ϕ
,
r
sin
⁡
ϕ
,
±


2
a
(
b
+
r
cos
⁡
ϕ
)


)

.


{\displaystyle (x,y,z)=\left(r\cos \phi ,r\sin \phi ,\pm {\sqrt {2a(b+r\cos \phi )}}\right).}

the curves contain following extreme points:

(
−
r
,
0
,
±



r

2


−
(
r
+
a

)

2




)

;


(
0
,
±
r
,
±



r

2


−
(
r
−
a
)
(
r
+
a
)


)

;


(
+
r
,
0
,
±



r

2


−
(
r
−
a

)

2




)

.


{\displaystyle \left(-r,0,\pm {\sqrt {r^{2}-(r+a)^{2}}}\right);\quad \left(0,\pm r,\pm {\sqrt {r^{2}-(r-a)(r+a)}}\right);\quad \left(+r,0,\pm {\sqrt {r^{2}-(r-a)^{2}}}\right).}



intersection single closed curve

if



r
<
r
+
a


{\displaystyle r<r+a}

, intersection of sphere , cylinder consists of single closed curve. can described same parameter equation in previous section, angle



ϕ


{\displaystyle \phi }

must restricted



−

ϕ

0


<
ϕ
<
+

ϕ

0




{\displaystyle -\phi _{0}<\phi <+\phi _{0}}

,



cos
⁡

ϕ

0


=
−
b

/

r


{\displaystyle \cos \phi _{0}=-b/r}

.

the curve contains following extreme points:

(
−
b
,
±



r

2


−

b

2




,
0
)

;


(
0
,
±
r
,
±



r

2


−
(
r
−
a
)
(
r
+
a
)


)

;


(
+
r
,
0
,
±



r

2


−
(
r
−
a

)

2




)

.


{\displaystyle \left(-b,\pm {\sqrt {r^{2}-b^{2}}},0\right);\quad \left(0,\pm r,\pm {\sqrt {r^{2}-(r-a)(r+a)}}\right);\quad \left(+r,0,\pm {\sqrt {r^{2}-(r-a)^{2}}}\right).}



limiting case

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).}