Intersections

Most intersections have a clear mathematical solution such as the intersection of a line and a plane. However others require more explanation including the following:

Cone/cone and cone/cylinder intersections can be computed as a plane, ellipse or circle depending on the customers needs. These are computed as follows:

  1. An exact analytical solution for 16 points spaced at uniform angular rotation about the axis of the first cone are developed for the intersection between the cone/cone or cone/cylinder.

  2. These 16 points are used as input for determining a best-fit plane (approximate unless axes are coincident).

  3. Ellipse is analytically constructed for the intersection of the

first (only) cone with the best-fit plane.

  1. Circle is constructed from ellipse by setting circle center to ellipse center and the radius = sqrt(major*minor) [constant area]

  2. Circle normal is set to best-fit plane normal, first cone axis, or second cone (cylinder) axis.

Its important to realize that, as cone/cone or cone/cylinder axes deviate further from coincident, the approximate solution degrades and as cone/plane or cylinder/plane deviate from perpendicularity, the solutions for circles created by intersection degrades. Therefore these intersections are intended primarily for situations where the axes are constrained such as measurements of shafts.