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:
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Intersection of Two Cones (combinations)*
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Intersection of Cylinder and Plane (combinations)*
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Intersection of Cone and Plane (combinations)*
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:
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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.
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These 16 points are used as input for determining a best-fit plane (approximate unless axes are coincident).
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Ellipse is analytically constructed for the intersection of the
first (only) cone with the best-fit plane.
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Circle is constructed from ellipse by setting circle center to ellipse center and the radius = sqrt(major*minor) [constant area]
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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.