A new milling strategy for an elliptical cylinder
11/07/2026
How it works:
At every iteration:
- look at all current vertices of the outer polygon,
- select one with the largest distance to the ellipse,
- from that vertex, compute its orthogonal projection P onto the ellipse,
- at point P, compute the tangent line to the ellipse,
- make a cut along the direction defined by that tangent.
Repeat until all vertices are closer to the ellipse than a given threshold.
By the way, computing the distance from a point to the ellipse (which also yields its orthogonal projection onto the ellipse) is not straightforward. It is performed by the same dichotomy-based algorithm described here.
However, a theoretical approach must cope with real-life constraints, as shown in the figure below:
The first cut in the example above would fail as the tool's length is too short to make it; the shaft holder (i.e. the white rectangle) would hit the stock. Moreover, the cutting part of the tool, the tool's flute, is shorter than the tool's length. Both constraints must be handled.
The chosen solution is to shift the tangent to the minimal distance from the ellipse that solves these two constraints when they arise and, if needed, cut in two steps, one from above and the second from below.
The cut from above is done by rotating the stock such that the tangent is vertical. The cut from below is done by rotating the stock by an additional 180° and cutting on the other side.
The sequence of cuts is illustrated by the animation below:
The red segment is where the stock is cut. The dot is where the cut is split when it's within the red segment.
It looks obvious, doesn't it?
Work in progress…