Intersection of a Cone and a Sphere

​
cone
base semimajor axis
1.
base semiminor axis
0.76
height
1.5
apex offset from sphere center,
z
0
0
opacity
1
This Demonstration explores the shape of the difference between a right elliptic cone and a sphere.

Details

The parametric equation of a right elliptic cone of height
h
and an elliptical base with semi-axes
a
and
b
(
z
0
is the distance of the cone's apex to the center of the sphere) is
x
c
=
a(h-v)cos(ϕ)
h
,
y
c
=
b(h-v)sin(ϕ)
h
,
z
c
=v+
z
0
,
where
ϕ
and
υ
are parameters.
The parametric equation of a sphere with radius
1
is
x
s
=cos(θ)sin(u)
,
y
s
=sin(θ)sin(u)
,
z
s
=cos(u)
,
where
θ
and
u
are parameters.
The intersection curve of the two surfaces can be obtained by solving the system of three equations
x
c
=
x
s
,
y
c
=
y
s
,
z
c
=
z
s
for three of the four parameters
u,υ,ϕ,θ
.
In this Demonstration, solving for
u
,
υ
, and
ϕ
gives the parametric equations for the intersection curve with parameter
θ
. The curve consists of four parts of similar form, depending on the sign of some parts of the equations:
{x,y,z}=±
4f(g±k)cos(θ)
m
,
4f(g±k)sin(θ)
m
,
2
3
a
3
b
h+
3
a
3
b
z0±gh
n

,
where
f=
2
a
2
sin
(θ)+
2
b
2
cos
(θ)
,
k=
3
a
b
2
h
2
sin
(θ)+
3
a
bh
z
0
2
sin
(θ)+a
3
b
h(h+
z
0
)
2
cos
(θ)
,
m=((
2
b
-
2
a
)cos(2θ)+
2
a
+
2
b
)(-
2
h
(
2
a
-
2
b
)cos(2θ)+
2
a
(2
2
b
+
2
h
)+
2
b
2
h
)
,
n=ab(-
2
h
(
2
a
-
2
b
)cos(2θ)+
2
a
(2
2
b
+
2
h
)+
2
b
2
h
)
,
g=
6
a
2
b
2
sin
(θ)
2
h
2
sin
(θ)-
2
b

2
h
+2h
z
0
+
2
z
0
-1-
4
a
4
b
2
cos
(θ)
2
b

2
h
+2h
z
0
+
2
z0
-1+
2
h
cos(2θ)-
2
h
+
2
a
6
b
2
h
4
cos
(θ)
.

External Links

Elliptic Cone (Wolfram MathWorld)
Sphere (Wolfram MathWorld)

Permanent Citation

Erik Mahieu
​
​"Intersection of a Cone and a Sphere"​
​http://demonstrations.wolfram.com/IntersectionOfAConeAndASphere/​
​Wolfram Demonstrations Project​
​Published: April 2, 2014