Intersection of Two Polygonal Cylinders
Intersection of Two Polygonal Cylinders
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
Details
The and functions define the composite curve of the -gonal cross section of the polygonal cylinder[1]:
radius
angle
n
radius(θ,,,n)=tan
θ
0
r
p
r
p
π
n
tancos(p(θ,,n))+sin(p(θ,,n))
π
n
θ
0
θ
0
angle(θ,,n)=-
θ
0
π
n
2cotn(θ-)
-1
tan
1
2
θ
0
n
The parametric equation of a polygonal cylinder with sides and radius rotated by an angle around its axis is:
n
r
p
θ
0
pcyl(θ,,,n)={cos(θ)radius(angle(θ,,n),,,n),sin(θ)radius(angle(θ,,n),,,n),v}
θ
0
r
p
θ
0
θ
0
r
p
θ
0
θ
0
r
p
θ
v
References
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
External Links
Permanent Citation
Permanent Citation
Erik Mahieu
"Intersection of Two Polygonal Cylinders"
http://demonstrations.wolfram.com/IntersectionOfTwoPolygonalCylinders/
Wolfram Demonstrations Project
Published: December 4, 2017