Details

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.