Sticky post Functy

A couple of overlayed circular sin/cosine waves with axes.

Screenshots

Functy is an open source 3D graph drawing package. The emphasis for the application is to allow Cartesian, spherical and parametric curve functions to be plotted and altered quickly and easily. This immediacy and the vivid results are intended to promote fun exploration of 3D functions.

Functy Updates – Shadows and UI Improvements

The last two weeks of summer break have given me the opportunity to give Functy some long-overdue attention. As I mentioned in my previous post, one of the improvements I’ve been working on has been dynamic shadows, and these are now fully implemented. The result is a big improvement in the visual fidelity and sense of depth in the visualisation. You can see the difference in the two screenshots below and I’ll include a proper comparison in a future post.

The next step will be to test out ambient occlusion, as kindly suggested by Andrea Bernabei (@faenil). I’m not sure how well this will work with the type of functional objects rendered by Functy, but it’s worth investigating. Ambient occlusion produces the sort of shadow effects more likely to occur indoors where the light tends to be more scattered, as compared to directed lights or outside direct sunlight, as the current version produces. However, research has shown that ambient occlusion gives a heightened sense of depth as compared to direct shadows, so it’ll be interesting to see the results.

The second major change has been to the user interface. This has been completely rebuilt from the ground up to fit with a single window workbench-style interface, as opposed to the previous multiple-window toolbox-style interface.

Personally I’m a big fan of multi-window interfaces; it should be the job of the window manager to allow you to arrange and configure your windows however you want. Sadly most window managers seem to be lacking in the flexibility department, and managing multiple windows becomes an exercise in hide-and-seek, trying to find windows that got lost behind others like a stack of papers on a desk. So, even though it’s not the perfect solution, I’ve converted Functy so that everything is visible and available in a single window.

I have to admit that the result does look more professional and I think it’s a positive change. I’m not quite sure what will happen if things get more complicated, but it works well for the timebeing.

There’s still a fair bit of work to be done before this can be given a full binary release, such as updating everything to work on Windows, and switching from libglade to GtkBuilder (this may be for a future time). Feel free to test out the version from source (it’s very easy to build, honest!) in the meantime.

Shadow mapping

One of the great benefits of teaching on computer game development modules at the university is that I have a justifiable reason for looking into interesting graphical effects for my work. It’s a real perk as far as I’m concerned. Last semester I covered shadow rendering as a new topic, which gave me the opportunity to find out about the subject in some depth.

Now it’s the summer and I have more time to experiment, so it only seemed right to apply some of these techniques to Functy; I’m hoping the function rendering will look a lot more realistic and solid once accurate shadows are being cast throughout the environment. Given the functions are all generated on the GPU, texture-based shadows seemed the most appropriate choice compared to stencil shadows. Texture-based shadows also strike me as more flexible and efficient in the long run. Below you can see the depth texture (greyscale) rendered alongside the existing function render (colour). Although they look similar, if you look closely you can see that they’re actually rendered from slightly different angles. That’s because the depth texture is rendered from the perspective of the light, rather than the camera.

Shadow depth map

Shadow depth map

The idea with texture-based shadow mapping is that by rendering the depth map from the perspective of the light, you get to see the parts of the world that the light is shining on. This is what the greyscale image in the screenshot above shows. The depth map is then used when rendering the functions to decide whether or not a given pixel is lit (visible from the light) or in shade (occluded from the light by another object). Put simply, if the distance of the pixel from the light is greater than the distance of the pixel transformed onto a point on the depth map, then the pixel is in shadow; otherwise it’s in light.

Once the shadow depth map has been rendered, this therefore needs to be fed into the fragment shaders used for rendering the functions. A quick texture lookup and a distance comparison are then baked into the shader to complete the process. Although it’s a really simple technique in theory, getting it to work in practice is turning out to be… intricate!

Sadly it’s been quite some time since I last had the chance to properly update the Functy codebase. Hopefully in time some of the latent improvements that haven’t yet made it into the release build can be rolled out, along with additions like this shadow rendering. It’s not quite there yet, but soon!

Projections

The sun was out in Liverpool today, creating crisp and long evening shadows. So it seemed like a great opportunity to take photos of recent 3D printed Functy objects. The full images are rather large, but show the grain of the printing, which I think is rather interesting in itself. Click on the images for the full views.

The original Lissajous is up on deviantArt and Shapeways; the alien egg is also on deviantArt and Shapeways.

Comparing renders with reality

Shapeways delivered a new batch of 3D prints recently. I was particularly pleased with the Alien Egg print of a spherical function with the formula:

radius = (3*(0.5+(sin(cos(a)+(p*((3+cos((pi*0.3)-1.5))/4))*10)**2)+((6/6.6)*((2+sin(8*a))/3)**4)))*sin(p)+(4.3*(1-(sin(p)**2))),

colour (R, G, B) = (r/8, (3+sin((a)*8))/5, (3+cos(a*8))/5).

The print is actually rather small (just 6 cm diameter) and the ridges of the shape are really quite delicate. In spite of this, the 3D print has come out really very similar to the original design. I guess you might expect it to be pretty similar, given the way it was produced directly from the model! However, if you look really closely at the original you can see the strata through the object created by the printing process. What I’m really impressed with, though, is the colour produced. I’d expected this to be a bit washed out, but in practice it’s a pretty impressive match.

Below is a comparison of (from top to bottom) the Functy render, the 3D print and a render done using Blender Cycles. In case you’re interested and your browser supports APNGs, there’s also a peculiar animated version!

Alien Egg Comparison (Functy, 3D print, Blender; click to enlarge)

Alien Egg Comparison: Functy, 3D print, Blender (click to enlarge)

Lissajous Looping

Following on from my previous post, I thought it’d be interesting to make an animated render of the Lissajous figure. If you have an APNG-capable browser (e.g. Firefox) you can see the result on DeviantArt.

While it’s neat to be able to print static versions of these Lissajous figures, in the future I’m sure it’ll be possible to make the fully moving version as well. Now that would be really something!

Static version of the render - click for the animated version

Static Lissajous render - click for the animated version

Lissajous Loops

Sines and Cosines have been responsible for some of the most elegant mathematical constructs. Lissajous curves are a particularly simple, yet elegant example. Put simply, a Lissajous is a parametric curve where each axis follows a sinusoidal path. By tweaking the amplitude and cycle length for each axis, a myriad of different patterns can be generated, from circles to intricately woven lattices.

The parametric curves in Functy are particularly suitable for generating nice Lissajous curves, and as usual, they can be output for 3D printing. The results of pumping them through a 3D printer, courtesy of Shapeways, can be seen in the photos below, along with a Blender Cycles render of one of the curves.

If you fancy getting really up-close-and-personal with them, you can order your own copies as unusual desk ornaments, from the Shapeways site.

3D printed Lissajous curves

3D printed Lissajous curves

3D rendered Lissajous curve

Rendered Lissajous curve

Network Video

I finally managed to get the video to record properly, so here’s a video version of the shader-based depth-of-field focus-blur.

P.S. This post was actually made using Ubuntu phone. The browser’s still a bit “experimental” so it’s been a bit tricky, but it does seem to work!

Less focus for more clarity

One of the difficulties with 3D visualisations - and network visualisations seem to suffer especially badly from this - is that while they’re a great way to represent complex data, they rely on animation for much of their clarity. This stems from the fact that the third dimension is illusory. The fact is we’re still largely stuck with 2D screens that our 3D images get projected onto. Until we move to proper volumetric displays, even the current attempts at 3D (stereoscopic) displays won’t fix this.

For the moment then, the best way for us to understand the lost depth component is by shifting our perspective, or in other words, by moving things around.

However, there are other ways to approach this and in the past I’ve been incredibly impressed by the way blur can be used to improve the perception of depth. This can give a really nice effect of camera focus, where different parts of the image have different levels of blur applied depending on whether they’re in or out of focus.

Testing this on some of the network graphs that I’m currently working with has produced some really quite nice effects. Ironically, by making parts of the image less clear, the overall coherence and clarity of the image is improved substantially. Here are a couple of screenshots of a random network with focus blur applied, both from up close and from a further distance.

The effect is achieved by rendering the whole image to a texture framebuffer. A very simple fullscreen shader is then applied, which increases or decreases the level of blur for a given pixel depending on the z-buffer depth value at that point. It’s a really very simple technique, but in my opinion produces some quite effective result.

This is one of the reasons why shaders are so phenomenally powerful. The depth blur is a very short program, but it needs to be executed for every single pixel of the image. That’s a lot of pixels, and a lot of computing power is needed to do this. Applying a shader program in parallel across the whole image is hugely more efficient than getting the CPU to do it. This allows it to be applied in realtime for each rendered frame, without stretching resources, even on my relatively underpowered laptop.

Check out Lighthouse3d.com and open.gl for some nice tutorials about using framebuffers for applying full screen shaders.

Network structures

As part of an experimental game project I’ve been trying to use the Functy rendering routines to visualise network structures. At the moment it’s at a very early stage, but has - I think - already generated some interesting results.

The screenshot below shows a network of 60 nodes, each one rendered as a spherical co-ordinate function, joined together using links rendered as curves. I just plucked some simple functions out of the air to see what the results would be like but am hoping to extend it with more interesting shapes as things progress.

Functy rendered network structure

Functy rendered network structure

The various parts of the network are a little hard to discern with a static image, but when I tried to capture a video the result was a mess of fuzzy artefacts (I think there must be something going wrong with my screen capture software), so I gave up on that.

The next step, after neatening up the code, is to arrange better animation of the nodes and links, with dynamic movement based on things like the forces between the nodes. I’m hoping this will produce some really nice effects, and if anything comes of it I’ll put a bit more effort into getting a successful video capture.

Comment Spam

  • December 29, 2012 at 4:00 am in

Unfortunately the site is receiving so much comment spam (at the last count it was up to 26000 spam comments in just a few months) that I’ve had to turn commenting off. Please do email me if you have thoughts about Functy, or would like me to turn commenting back on.
I’ll try to figure out a solution, but it’ll probably involve having to find a different way to host Wordpress on Sourceforge. Very frustrating! Thank you to ARJ though, for helpfully explaining how to clear all of the comments from the Wordpress database!

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