Intersection of a Generalized Cylinder over a Rose Curve with a Circular Cylinder

This Demonstration shows the intersections of a circular cylinder with a generalized cylinder over a rose curve with a variable number of petals.

These intersections are composite space curves consisting of one or more closed sections.

Using the controls, you can vary the specific settings for the axial offset, circumradius and inclination of the circular cylinder and the axial rotation of the rose-curve-based cylinder.

Completely closed curves are possible when there is an exact fit of the rose curve cross section inside the circular cylinder; press A or B to see them.

The parametric equations of a circular cylinder with radius , inclined at an angle and offset by a distance from the horizontal are:

,

,

.

The parametric equations (with parameters and ) of a generalized upright cylinder over a rose curve in the - plane with petals and an angular offset of from the axis are:

,

,

.

To find the intersection, set the corresponding equations equal to get three equations with four unknown parameters: .

Eliminating all but gives the two sections of the parametric of the composite intersection curve:

and

with

.

Reference

[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.