Intersection of Two Polygonal Cylinders

​
vertical cylinder
number of vertices
3
4
5
6
36
axial rotation
1.
inclined cylinder
number of vertices
3
4
5
6
36
radius
1
inclination
1.571
axial offset
0
show
cylinders
Steinmetz solid
hide
vertical cylinder
inclined cylinder
none
cylinder opacity
0.95
view direction
default
 vertical cylinder
 inclined cylinder
This Demonstration shows the intersection of two polygonal cylinders. The built-in Mathematica function RegionFunction is used to make cutouts and show that the cylinders make possible pipe connections.
If the inequalities used in the RegionFunction are inverted, we get a instance of what is known as a Steinmetz solid, formed by the intersection of two solid cylinders.

Details

The
radius
and
angle
functions define the composite curve of the
n
-gonal cross section of the polygonal cylinder[1]:
radius(θ,
θ
0
,
r
p
,n)=
r
p
tan
π
n

tan
π
n
cos(p(θ,
θ
0
,n))+sin(p(θ,
θ
0
,n))
,
angle(θ,
θ
0
,n)=
π
n
-
2
-1
tan
cot
1
2
n(θ-
θ
0
)
n
.
The parametric equation of a polygonal cylinder with
n
sides and radius
r
p
rotated by an angle
θ
0
around its axis is:
pcyl(θ,
θ
0
,
r
p
,n)={cos(θ)radius(angle(θ,
θ
0
,n),
θ
0
,
r
p
,n),sin(θ)radius(angle(θ,
θ
0
,n),
θ
0
,
r
p
,n),v}
with parameters
θ
and
v
.

References

[1] E. Chicurel-Uziel, "Single Equation without Inequalities to Represent a Composite Curve," Computer Aided Geometric Design, 21(1), 2004 pp. 23–42. doi:10.1016/j.cagd.2003.07.011.

External Links

Steinmetz Solid (Wolfram MathWorld)

Permanent Citation

Erik Mahieu
​
​"Intersection of Two Polygonal Cylinders"​
​http://demonstrations.wolfram.com/IntersectionOfTwoPolygonalCylinders/​
​Wolfram Demonstrations Project​
​Published: December 4, 2017