Qi-Hardware logo: cutting corners

Allin Kahrl qbject at gmail.com
Fri Aug 26 18:50:40 EDT 2011

The solution I initially imagined while reading Werner's post was the
"constant width" option as well, and I agree with Ron that it
certainly does look nice. However, how do we then reflect it back into
the rasterized domain? By aliasing it and knocking a pixel out of the
tail like my attached .png?

I don't really love that option because when the process used to
generate it is, in turn, reflected back into the machining/vector
domain you don't get that pretty curved tail back.

Doing so would mean that our description of how to make the logo has
to include a conditional at the end that the design must be aliased to
fit on a rasterized display. Strikes me as insufficient simplicity.

So I suggest it would be better to adopt the "simple paths" approach.
Then we can write a single description, essentially pseudocode, that
works whether the logo is being drawn with pixels milled with an
endmill, or executed through other means.


On Tue, Aug 23, 2011 at 6:13 PM, Ron K. Jeffries <rjeffries at gmail.com> wrote:
> To my eye, "constant width" looks simple and elegant.
> ---
> Ron K. Jeffries
> On Tue, Aug 23, 2011 at 17:05, Werner Almesberger <werner at almesberger.net>
> wrote:
>> 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
>> http://downloads.qi-hardware.com/people/werner/logo/corners.pdf
>> illustrates this.
>> Source:
>> http://projects.qi-hardware.com/index.php/p/wernermisc/source/tree/master/logo/corners.fig
>> 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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