 # Intersection of Circular and Polygonal Cylinders

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Circular and polygonal cylinders intersect in interesting 3D curves. Mathematica's built-in function RegionFunction shows that the cylinders make realistic pipe connections.

Contributed by: Erik Mahieu (February 2014)
Open content licensed under CC BY-NC-SA

## Snapshots   ## Details

The parametric equation of a circular cylinder with radius inclined at an angle from the vertical is: , with parameters and .

Define the functions and . The and functions define the composite curve of the -gonal cross section of the polygonal cylinder .

The parametric equation of a polygonal cylinder with sides and radius rotated by an angle around its axis is: with parameters and .

To find the equation of the intersection curve, put . This gives the three equations: , , .

These are equations with four variables, , , , and . Eliminating , , and by solving the equations gives the parametric curve of the intersection with θ as the only parameter (choosing gives the upper or lower half of the intersection curve): .

Reference

 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.

## Permanent Citation

Erik Mahieu

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