Qi-Hardware logo: cutting corners

Werner Almesberger werner at almesberger.net
Tue Aug 23 20:05:36 EDT 2011

Last but not least, here's a technical issue: the logo has several
right angles. Right angles are something that doesn't really exist
in nature and some machining/production processes only allow a
rough approximation.

For example, a CNC mill can make very pointy convex corners but
concave corners are always round. This is because the rotating
mill bit can only cut a circular area, and we can't reach the
inside of the corner without cutting into the adjacent walls. (*)

(*) Which is sometimes done, but it's probably something we don't
    want to use in the case of the logo.

The drawing on the right side of
illustrates this.


The circular bit (black circles and half-circles) moves along the
dark blue edge. By doing so, it removes material in the area
shown in pink. The resulting edge, shown in light blue, is curved
at the second corner.

Furthermore, which corners are convex or concave changes with the
side from which we approach them. If we want to engrave the logo,
we'd remove material from the inside of it. This is shown on the
left side of the second row. The thin read line is the path along
which a tool with radius 1 unit would move.

This leaves various rounded corners. I've marked some of them with
small red arrows.

If we want to make a relief ("embossed") of the logo, we remove
material surrounding it. This is shown in the right side, with the
tool path shown in green. Most of the corners are convex and thus
nice and pointy, but there are still two rounded ones left.

I called these "simple paths", because they're just a set of
straight lines. They minimize the difference from the original
shape but have the perhaps undesirable property that engraving and
embossing produce different results.

We can change the shape such that is can be produced both ways.

On the left side, all sharp corners have been rounded to arcs with
the same radius. It's basically a combination of the weaknesses of
both of the simple paths.

On the right side, a constant stroke width is maintained by
varying the radii of the arcs along which the tool moves.

These drawings show the worst case in which the overall spacing
can still be maintained. With a tool radius smaller than one unit,
the rounded corners could be made sharper than shown.

Thi leaves us with two questions:

- first of all, do we want to make rounding an integral part of
  the design ? This could be in the form of having two variants of
  the logo. (We we mandated rounding in general, this would then
  raise issues with techniques that prefer right angles over
  curves, such as low-resolution raster images.)

- if we consider rounding an implementation issue and not part of
  the underlying concept, what would be the advice to people who
  mill the logo ? Use the simple paths ? Or pick one of the
  "universal" shapes ?

- Werner

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