If you calculate toolpaths around a very narrow and 'pointy' triangle you will get toolpaths in the shape of the inverse cutter - the "Inverse Tool Offset". Here I've plotted the basic operations, in red/blue **drop-cutter** which drops the cutter down along the z-axis until it contacts the triangle, and in cyan waterlines which are the results of a **push-cutter** operation that pushes the cutter at constant z-height along the x/y axis into contact with the triangle.

There are four basic cutter shapes: (1) Cylinder, (2) Sphere('Ball'), (3) Toroid('Bull'), and (4) Cone. The triangle contact can be divided into tests for contact with (a) the three vertices of the triangle, (b) the facet of the triangle, and (c) the three edges of the triangle. That's 4x3 = 12 contact/collision-test functions that have to be written (a few, particularly the facet-tests, can be combined into one base-class method).

Once the basic cutter shapes work it is possible to combine them through CompositeCutter. The bottom row shows cutters with a central part corresponding to the top row of cutters, and a conical outer part.

The point of this exercise is of course not only to plot inverse-tool-shapes, but to be able to calculate toolpaths for these and other CompositeCutter tool shapes. This will become more interesting if/when the cutting-simulation starts working and it will be possible to compare for example surface-finish vs. cutting-time of a BallCutter operation vs. a BullCutter operation.

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