The evolution of PotterDraw: a math holiday

Chris Korda — Mon, 12 Feb 2018 21:07:26 -0000

While on vacation in Playa del Carmen I encountered something unexpected nestled between the palm trees: a paint-your-own pottery shack. It was a small thatched hut containing several plain wooden tables and benches, with molded pottery painted in traditional styles hanging on the walls. Unpainted pieces were available on shelves in a wide variety of shapes, already fired and coated with a white gesso-like base. The tables were crowded with a profusion of small jars of acrylic paint in many colors, along with tin cans containing brushes. You picked out a piece, received a dish of water and a paper towel, and painted, taking as long as you liked. When you finished, the staff stored your piece, so it could be clear-coated overnight.

Perhaps needing occupational therapy, I was drawn to this process, and decided to paint S-curved stripes on a small vase. I tried very hard to keep the stripes symmetrical, but after a few hours it was obvious that I wasn't skilled enough to avoid glaring discontinuities when the pattern finally reached all the way around the vase. [see photo below] I resolved to try again, but with improved methods. I spent the rest of the day drawing and cutting out a long succession of paper templates and attempting to match them to the curves on my ineptly painted vase. Admittedly this was an odd way to spend one's vacation but workaholics relax in strange ways.

It dawned on me that what I was making was a spline. It needed to be flexible enough to conform to the vase's curvature, but strong enough so that I could trace along its edge with a pencil without tearing it. If it was too wide, it wouldn't lie flat on the vase, but if it was too narrow, it wouldn't hold its shape. A suitable plastic might have worked better, but paper was the best material I had access to. The biggest problem was finding the right shape to produce the desired S-curve on the vase. There was no simple relationship between how a given curve appeared in 2D (drawn on paper) and how it appeared lying on the curved surface of the vase. A better mathematician might have calculated the right shape, but being unschooled in topology, I resorted to trial and error. After many iterations I finally arrived at a paper spline that looked good enough. In retrospect a laptop, a video projector, and a printer would have been handy.

The next day I showed up at the pottery shack early in the morning, paper spline in hand. The staff were amused, but encouraging. Using homemade paper rulers, I drew twenty-five equally-spaced tick marks at the top and bottom of the vase. These let me place my spline accurately enough to avoid excessive variation in stripe thickness. I spent hours carefully tracing the spline's edge onto the vase. It took a lot of fussing with the paper to get the shape right and trace it without slippage, but the spline held up pretty well. Once the curved outlines were sufficiently precise and evenly spaced, I spent the rest of the day painting in the twenty-five curved stripes, using a repeating palette of five colors. The result was decent, given that I'm not a professional pottery painter, but by this point it was obvious to me that this was a problem computer programming could solve. Thus was launched PotterDraw, a project that consumed the next nine months of my life.

My lifelong programming career predisposes me to apply computers to almost any problem, but I was particularly well-situated for virtualizing pottery, having spent the last eighteen years developing software for the 3D printing industry. I was already familiar with Direct3D (Microsoft's 3D graphics API), and had considerable experience with the application of geometry, trigonometry, polar coordinates, and periodic functions to digital art, in previous open-source projects such as my Whorld visualizer.

Using the classical engineering technique of divide and conquer, I divided the problem into three sub-problems: 1) Make a 3D view capable of showing a cylindrical mesh at any scale and from any angle, with realistic shading and lighting, rotation, and zoom. 2) Make 3D mesh effects by dividing the cylinder into a vertical stack of rings, and modifying the radii of the rings via parameterized trigonometric functions. 3) Make a spline editor, and generate the radii of the rings from a 2D spline, consisting of one or more Bézier curves.

Each of these problems spawned myriad further sub-problems. For example, initially I used a Direct3D function to calculate the vertex normals required for correct lighting, but this turned out to be prohibitively slow. To calculate the normals myself, I needed to generate adjacency data for each vertex, a complex task. A triangular facet's surface normal can be calculated directly from its vertices via the cross product of two of its sides, but a vertex normal is the average of the surface normals of all facets that share that vertex, and thus requires adjacency data.

Other challenges included how to make the properties user-editable (property sheets), how to store the properties in a document (INI file format), and how to coax the standard Microsoft color picker dialog to thrive on a control bar, and how to allow a palette's color swatches to be reordered via drag and drop. And so many more! Many user interface features--undo and redo, clipboard support, low-frequency oscillators--could be reused to some extent from my previous projects. The elephant in the room was the spline editor, arguably the single most complex component, but since the rest of the program could be developed without it, I left the spline editor for last.

Rolling my own spline editor was pure wheel reinvention. The interface is well-defined by now, due to the prevalence of vector graphics. This reduces the problem to painstaking emulation of existing interfaces, a tedious but familiar and soothingly predictable process. I used CorelDraw and Adobe Illustrator as my main examples. They're broadly similar but have subtle differences in behavior and terminology, perhaps due to the need to avoid patent infringement. I built a whole separate app to develop and test my spline editor, mostly just to keep the code size and compilation time reasonable. Luckily Windows includes considerable graphics support for Bézier curves, greatly facilitating the development of features such as altering the curvature by dragging the spline or its control points.

Once I had a basic spline editor up and running, the next step was to convert the spline into a series of radii for the 3D mesh. This was a deep problem, not only because it necessitated the computation of precise intercepts with a Bézier curve (via iterative subdivision), but also because many Bézier curves won't generate a valid 3D surface, e.g. curves that loop around and intersect themselves. Later additions included full clipboard support (copy, cut, paste) and transformations (translate, scale, rotate, and horizontal and vertical mirroring). The spline editor took a month to completely solve.

A major decision was how to handle the pot's inner wall. One option was to have the spline only specify the outer wall, and have the program automatically generate the inner wall. The other option was to have the spline specify both walls. The latter would give the user more control over the shape, but would also require the user to keep the two walls in sync, making the program harder to use. For this reason I went with the former. The algorithm to generate the inner wall took considerable time.

For a plain cylinder, the inner wall can be computed trivially, by subtracting the desired wall thickness from the outer wall's radius. This crude method is unsatisfactory for more complex shapes however, because the wall will get thinner the more it angles away from vertical. One solution is to compensate the wall thickness for the wall's slope. My approach is to divide the wall thickness by the cosine of the arctangent of the slope. This works well in many cases, but has a vexing weakness: if the wall approaches horizontal, the result of the division approaches infinity. This overcompensation can produce unwanted ridges or spikes on the inner wall, or even close off the pot's interior altogether. I was able to mitigate the overcompensation using averaging techniques, but only to a limited extent. An alternative technique that might produce less artifacts would be to generate the inner wall by reversing the outer wall's vertex normals, but more research is needed.

PotterDraw 1.0.6 adds Ruffle, a new mesh effect with great creative potential

Chris Korda — Sun, 11 Feb 2018 07:25:15 -0000

Version 1.0.6 features a new mesh effect called Ruffle. Ruffles are essentially wavy ripples. Imagine a corrugated drain pipe, standing on end. The corrugations are ripples. Now picture a periodic wave applied to the ripples, as they progress around the circumference of the pipe. The result is ruffles, so named because they resemble the ruffled collar that was the height of men's fashion during the Elizabethan period. The Ruffle effect opens up many new creative possibilities, some of which are shown below.

PotterDraw is free software that lets you easily design pottery for 3D printing in full color. You draw the pot's outer wall using a spline, which consists of curved or linear segments, while the inner wall is generated automatically. You can also apply effects to the pot's 3D mesh and texture. The effects can be modulated with oscillators to create complex shapes, and the modulations can be animated to create shape-morphing videos.

PotterDraw 1.0.5 features new mesh effects, new texture mappings, transparency

Chris Korda — Fri, 05 Jan 2018 21:40:29 -0000

PotterDraw 1.0.5 introduces many new features including 1) exponential properties for the Ripple and Bend mesh effects, 2) new texture mapping algorithms that take the surface normals as input, and 3) transparency. The images below demonstrate some of these features.

For Ripple and Bend, Operation, Power, and Power Type properties were added. Setting Operation to Exponentiate causes ripples or bends to be proportional, such that their depth scales with radius, which wasn't possible before. Power and Power Type allow the underlying ripple or bend waveform (a sine wave) to be distorted into various types of exponential curves.

The new texture mappings include Azimuth and Incline. They map texture according to how the surface normals deviate from those of an ideal shape (a cylinder), hence they're more properly called Azimuthal Deviation and Inclinational Deviation. For each vertex, Azimuth shows how "non-round" the pot is at that point, while Incline shows how "non-vertical" it is at that point. It's also possible to combine them, by using Azimuth as the U coordinate and Incline as the V coordinate, and doing so effectively reproduces the 3D curvature in texture coordinates.

Another new texture mapping is Edges, which does edge detection. Here's how it works. For each vertex, we know the face normals of the adjacent faces that intersect that vertex. So we compute the standard deviation of the face normals. Actually we compute three standard deviations, one for each axis, which gives us a 3D standard deviation vector. Then we take the magnitude of the standard deviation vector. This is effectively the geometric mean of the standard deviations for X, Y, and Z. Finally we log scale the mean, in order to amplify small differences, and that's the U texture coordinate.

Finally, transparency is now supported as a render style. It's not perfect by any means, but it works pretty well.

PotterDraw 1.0.4 adds polygon properties, radius color mapping, and more

Chris Korda — Tue, 28 Nov 2017 21:02:16 -0000

Version 1.0.4 of PotterDraw adds several major new features. Most of important of these is the Polygon effect, a new group of mesh properties that facilitate the creation of polygonal pots. The polygon can have any number of sides greater than two, and since fractional values are supported, the number of sides can be smoothly modulated. The Roundness property lets you round the polygon’s corners, while the Bulge property lets you bulge or pinch the polygon’s sides. In previous versions, roughly polygonal pots could be made by tweaking the Scallop properties, but the new polygon properties are much more flexible and precise, and can be combined with the Scallop properties to make scalloped polygonal pots, which was previously impossible.

Another major new feature is the addition of the Radius color pattern, which uses the pot’s radius as the U coordinate of the texture mapping. Because the pot’s shape directly controls the texture mapping, the resulting texture automatically illustrates the pot’s shape. This is fundamentally different from the other color patterns, in which the texture coordinate function is unrelated to the pot’s shape. The radius color pattern responds to any change of radius, but it’s particularly effective if the radius varies within each ring, as is the case for the Polygon, Scallop and Bend mesh effects. The new color pattern is ideal for coloring polygonal pots. It can also create kaleidoscope-like effects if the U offset is animated.

Still another new feature is the Power Type property, which selects how exponential power is applied to waveforms. The options are asymmetric (the only option in previous versions), or symmetric. Asymmetric produces lopsided waves, whereas symmetric preserves any vertical symmetry the wave may have, by applying the power function while the wave is bipolar.

Other new features include configurable lighting, palette import/export, and the organization of mesh properties into subgroups that be hidden. Additionally, several bugs were fixed. One of these fixes made texture updates 16x faster, greatly improving the performance of texture animation.

PotterDraw 1.0.3 adds more scallop options and new waveforms

Chris Korda — Wed, 06 Sep 2017 20:36:50 -0000

PotterDraw 1.0.3 adds some new features that allow more radical pottery designs. The biggest change is the addition of several new scallop-related properties, including Scallop Waveform, Operation, Range, and Power. In previous versions, the scallops were always sine waves, but now there are many waveforms to choose from. The curvature of the scallops can also be changed, using the Scallop Power property, and whereas previously scallops could only be added to the pot's radius, now subtraction, multiplication, division, and exponentiation are also possible. All of the new scallop properties can be also modulated, opening up many new avenues for creativity.

The second big change is the addition of new modulation waveforms, including Pulse, Rounded Pulse, Triangular Pulse, Sine Cubed, and Flame. The new pulse waveforms have variable Pulse Width and Slew, and are especially useful for creating envelopes that smoothly fade effects in and out. The Sine Cubed and Flame waveforms make organic, leaf-like shapes. In addition, modulation now supports the Exponentiate operation, which allows a target property to be scaled logarithmically (like the fret spacing on a guitar), for a more natural appearance.

Enclosed are images that demonstrate some of the new features. There's still a lot to learn about how to put them to best use, so stay tuned!

New version of PotterDraw adds Scallop Phase and other useful features

Chris Korda — Wed, 06 Sep 2017 20:35:04 -0000

A new version of PotterDraw (1.0.2) is now available. It adds a new property called Scallop Phase, which is useful for making wavy scallops, as shown in the images below. This property was added at the suggestion of user Juan Bustos at Filar3D. Other new features in this version include a Load Texture command added to the File menu, the ability to load a texture file via drag and drop from Explorer, automatic checking for updates, better-optimized texture mapping for improved animation frame rates, and an option to continuously update the 3D view during spline dragging. Download the new version here.

PotterDraw is a free software application that lets users design pottery for full-color 3D printing. The pot's outer wall is drawn using a spline editor, while the inner wall is generated automatically. The pot’s texture can be synthesized or loaded from an image file. Various trigonometric effects can be applied to the mesh and the texture, and the effects can be modulated with low-frequency oscillators to create animated videos.

The developer, Chris Korda, has designed 3D printing firmware and software for nearly twenty years, first at Z Corporation (now 3D Systems), and later at Viridis3D (now EnvisionTEC).

Filar3D finds unexpected use for PotterDraw

Chris Korda — Fri, 25 Aug 2017 19:08:28 -0000

Buenos Aires-based Filar3D is using PotterDraw to design... wait for it... lampshades! According to Filar3D's Juan Jose Bustos, "It makes my work super fast ... I have to take several steps to accomplish the same results with other software." Thanks to Juan for using PotterDraw in an unanticipated way, and for his suggestions, which will be addressed in subsequent releases of PotterDraw.

Filar3D's Facebook page is here and their 3D printed lampshades will soon be available at their online shop

Interview with Chris Korda: 3D printing pottery, open-source software and activism

Chris Korda — Thu, 24 Aug 2017 21:42:10 -0000

PotterDraw received another good review today, this one from global media company 3D Printing Industry (3DPI). The review highlighted PotterDraw's ease-of-use, pointing out that "Typically, would-be-designers would have to digitally sculpt such designs or build them from the ground-up using polygons/stock shapes. By comparison, Korda’s software automatically generates vessel interiors."

The review was also more wide-ranging than usual, covering topics such as open-source soffware, digital artwork, and environmental activism, including this entertaining excerpt: "When asked about the probability of a gulping Martini glass [Chris Korda said]: 'I have no idea. The holy grail of 3D printing is to print a mobile phone all at once, microchips and moving parts and all, so that no assembly is required. If that happens before the waves wash over southern Florida, look out for gulping martini glasses.'"

PotterDraw reviewed in All3DP: "Great Specialist for 3D Printing and 3D Designing Cylindrical Shapes"

Chris Korda — Wed, 16 Aug 2017 19:58:16 -0000

PotterDraw received an excellent review today from the Munich-based 3D printing news organization All3DP. Under the heading "PotterDraw: Great Specialist for 3D Printing and 3D Designing Cylindrical Shapes", the review states that "PotterDraw really is a great case of 3D specialist software – it gets the job done with ease, and even if you have never designed before, you can use the software. It also includes a concise and helpful documentation." The revew includes images and video demonstrating PotterDraw's capabilities.

PotterDraw featured on 3ders.org

Chris Korda — Tue, 15 Aug 2017 22:16:09 -0000

The 3D printer and 3D printing news organization 3ders.org published an excellent review of PotterDraw today, complete with pictures and video.