When I’m focused on other things and doodling, one of the textures that regularly emerges is a fill pattern of concentric irregular circles.

(click to enlarge)

Now, doodling is a way of letting my mind idle. Why I’d want to automate that, I couldn’t say. But for some reason, I did.

So the question was: how do I draw a “sloppy circle,” or one that’s not exactly regular, especially if I want to then draw another similar circle inside of it, and so on. I settled on the idea of picking a set of points around a circle, with roughly the same radius. If I just drew a line between these points, it would be a regular polygon, and not as gentle as I’d like. So, instead of lines, I needed gentle Bezier curves from point to point. To get the Bezier control points, I need to create a tangent to the radial line, and put the control points for the curve on that tangent. But how far away from the original point? Better make it a variable! So, with some playing, I was able to get reasonable values. You can play with it yourself! You can easily get results like these: (click to enlarge any of them)

So I got a few questions about that Javascript experiment, so I thought I’d add some explanation. In the process, I added some features to it.

So what’re all those variables, I was asked. And what do they mean?

To start with, this whole experiment was based on a traditional Islamic tiling pattern. I don’t know the actual origin, but I’m sure it goes way back.

So, step 1, you tile a plane with equilateral triangles. The number of triangles that fit across our page is what I called “Spacing.”

Next, you draw a circle at each intersection of lines. The size of the circle is “Radius” — in this model, a radius of 100% means that the neighboring circles touch.

Last, you divide each circle into twelve equal pieces. Then, according to some predetermined pattern, you connect the points on neighboring circles. Here’s a traditional pattern:

The only other variables of interest are “Column Radius Growth” and “Row Radius Growth” which is the percentage by which the radius of each circle changes depending on the number of the column or row (starting in the upper left-hand corner).

This time around, I’ve added a few different connection patterns.

I was thinking about textures that are traditionally called Islamic or Moorish tilings.

One simple pattern is built by placing circles on a staggered grid, placing points around their circumferences, and then connecting points to neighboring circles in a pre-defined pattern. Here’s one example:

I was thinking – hey, what if I vary the radius of those circles from row to row?

Three hours of clumsy Javascripting later, I found the answer:

(click on image to see detail)

Unfortunately, it often takes hours of coding to learn that an idea’s not much good.

You can play with some of the variables yourself, or (horrors) look at the source code to see how it works.

Updated:
I couldn’t leave well enough alone, and have added a few features. And I’m getting slightly more interesting stuff now.
(click on image to see detail)

Consider a photograph or painting. We see a range of colors (or shades of gray) reflected that form an image. But since our eyes and brain are highly optimized for teasing coherent images out of what we see, a great deal of manipulation can be done to those colors and we will still see the original. Artists have exploited this capability for millennia, using areas of light and dark paints to hint at the details which our brains happily provide.

With the introduction of printing, techniques for “half-toning” emerged to convert a continuous image into a two-color (normally black ink and white paper) image that maintained as much as possible the appearance of the original continuous tone image. There are many photographic processes for making such conversions. I’ll discuss one simple example here, using digital instead of photographic techniques.

We start with an piece of a photograph. This picture is sepia toned and not extremely contrasty.

The first step we take is to convert the picture to purely gray tones. When we do this, we also adjust the contrast by shifting the the tones so that the darkest shade in the image is assigned to pure black and the lightest shade in the image is assigned to pure white, and all other shades are adjusted according to their position between the lightest and darkest shade.

The second step involves superimposing a pattern over the picture. The pattern should have be a continuous repeating variation between light and dark — when we superimpose, we multiply the two images, so that the darker of the pattern or the photo dominates. In this case, I used simple straight lines, which are made with a sine wave luminance cycle between pure white and pure black.

The remaining step is to look at each pixel, and decide whether it should be black or white. We do this by simply comparing it to a threshold. Is it lighter than, say, 50%? If so, then it’s white. Darker? Then it’s black. But 50% may not be the best place to position our threshold. We can try various thresholds to see how it comes out:

Here are a few observations that may be relevant at this point. At each of these steps, we made decisions which I glossed over. For example, when we adjust the contrast of this image, we chose a linear conversion. We could, instead, have used different curves to emphasize bright areas, dark area, or middle tones. We could have used histogram equalization which adjusts the image such that there are roughly the same number of pixels for each shade used (often used to bring out details).

Similarly, our overlay pattern needn’t go from pure black to pure white; by changing the ranges of this overlay pattern we are doing the equivalent of adjusting the tonal curves of the original image. We can also have a strong influence on how the final output looks. With a pattern that includes shades darker than our threshold, we will end up with the pattern throughout the final image (as in this case, our final image has lines across all parts of it). By having a pattern of only half maximum density, the lighter areas will not show the pattern:

The overlay pattern can be many shapes other than lines (like concentric circles), and there can even be multiple overlays. Traditional newspaper half-toning uses two linear patterns like the one we used, but set at an angle with respect to one another, thereby creating diamond patterns. Newspapers chose this diamond pattern because the size of the pattern relative to the detail in the image determines how much detail winds up in the final image.

I tried to use the above techniques for generating 3D halftones or etchings. While it’s probably a project best suited for use with a laser cutter, I don’t have a laser cutter. I do, however, have a Nomad CNC router!

I wrote a short script that analyzes an image file, and converts it into a set of 3D ridges. My first approach looked at the image row by row, and created a groove with a thickness inversely proportional to the luminosity of the pixels in the row.

Result (click to view)	Detail (click to view)

This works well in theory, but neglected to take into consideration some limits of my machine: the work area is 20cm x 20cm, and the smallest end-mill (cutting bit) I have is 1mm in diameter. This functionally limits my smallest detail to somewhere around 1.05mm. Add the fact that the wood stock I had on hand was around 8cm on its narrow dimension, and this results in an image that I can’t carve.

My next algorithm analyzes three rows of the image at a time. As it steps along the rows, it uses the average of the three pixels at each column (call them a, b, and c, where a is the top row). If the combined density is greater than 50%, a 1mm ridge is created. The ridge is thickened on the top by the average density of a + b, and thickened on the bottom by the average density of of b + c.

Result (click to view)	Detail (click to view)

This algorithm provides something that’s within the resolution I can carve, but loses an enormous amount of detail. Furthermore, it requires harder wood than the birch plywood I tested on. I did some minor tweaking of the threshold, and here’s what I got:

So at this point, I have a set of 0.5mm cutters on order, and need to track down some good hardwood stock to try carving. As always, details will be posted here if notable.

When I last posted, I was playing with plastic and metal prints from Shapeways.com. So here’s an experience printing in a different material — gold-plated brass. As with last time, I used MoI modeler. I wanted to make a butterfly pendant for my wife for our anniversary. My initial try used an actual photograph of a monarch butterfly that I had taken in our garden. I traced the outlines, and created extruded forms for the wings. I then created the shape of the body, and a ring for the necklace chain to pass through.

When I uploaded it to Shapeways, however, I immediately ran into printability problems. I had misunderstood the limitations for different materials. When they say that the minimum dimension for an unsupported wire is 1mm in a given material, I had for some reason interpreted that as having a minimum cross-section of 1mm². So, for example, I thought I could get away with a 0.5mm width if the thickness was 2mm. This interpretation was not based on their actual requirements. So it was back to the drawing board.

It turns out that using the actual dimensions was not going to be a successful approach. In the end, I bit the bullet, and did a complete redesign inspired by — but not matching — an actual butterfly. Since I was now dispensing with reality, I chose to make it more dramatic by having the wings all spread apart rather than overlapping as monarch actually hold their wings. The design that would be printable looks like this:

I ordered the print three weeks before our anniversary, since the estimated turn-around was 10 days. In the end, it shipped in five days before our anniversary, but it didn’t actually arrive until the day after. Bummer. Then, while it was en route, Shapeways sent out an email extolling their new approach to precious-metal-plated items. Same price, better results. Well, thanks, guys. I mean, I know there has to be a transition between approaches at some point, but I wish it wasn’t between the time I’d placed an order and received my print.

Still, as even this shaky iPhone picture shows, the final result was OK.

(click on images for larger versions)

Coming up soon: some posts on a completely different path I’ve taken to 3D production. Here’s a hint.