WOLFRAM NOTEBOOK

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

Permanent Citation

Wolfram Cloud

You are using a browser not supported by the Wolfram Cloud

Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.


I understand and wish to continue anyway »

You are using a browser not supported by the Wolfram Cloud. Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.