# Curves

## Interactive Functy function viewing

One of the obvious but neat consequences of having the new STL export functionality from Functy is that the generated models can be imported in to other things. One of these things… well, p3d.in provides a clever HTML5 in-browser model renderer, which means the models can now be rendered interactively directly into this site (or indeed any others). Check out this version of a ball made from string, generated as a couple of curve functions in Functy. Just click and drag to rotate the model. And if you like it, you can even print a copy in 3D!

## Rendering tubes with shaders

One of the main aims with Functy has always been to allow functions to be rendered using shaders on the GPU and using the function derivative to generate normals. This should be faster than rendering on the CPU. Defining the normals mathematically should also give more accurate results, and since we have the functions to play around with, it just seems like the sensible thing to do.

This presented a bit of a challenge for the new *curve* functions though. As is so often the case when using shaders, the problem is one of parallelism. If you have a function, the position of each vector in the model should be independent of the others and therefore a prime target for parallelism. With a curve you have the path of the curve and the radius at a particular point determined mathematically as long as you have the position along the curve, *s* and the rotation around the curve *p* given to you. However, what isn’t necessarily pre-determined is the *orientation* of the curve.

To explain this a bit further, consider the curve in the diagram below. Notice how the vectors perpendicular to the curve change direction as you move along the curve. These vectors are used to define the thickness of the curve at a particular point. In two-dimensions this is fine, as there’s no ambiguity about which direction these vectors should be pointing in.

However, lets now consider this in 3D. Suddenly these vectors can rotate around the axis of the curve, and while the vector must always lie within the plane perpendicular to the curve, there’s still an infinite number of possible directions that the vector can point.

In Functy, we use all of these directions, because the *p* variable defines a full rotation around the curve, so that it becomes a tube (rather than a line). But we still need to decide which direction the zero angle should point.

Some how or other a choice has to be made for this. There are a number of possibilities. We could set it randomly. However, this means there will be no consistency from one piece of the curve to the next, and if the cross section isn’t a circle, will result in a random twisting of the curve. This would look rubbish, so it’s not an option.

In my 3D Celtic Knot program I came up against exactly the same problem. Since it was essential for the start and end of a curve to match up exactly, I used an iterative approach there. For each piece, the rotation between the previous and next point on the curve is calculated and the perpendicular vector transformed by this in order to establish its new position. At the end of the curve an adjustment is made to ensure pairs of curves will always fit perfectly together. This is possible because the maximum adjustment needed will never be more than 2π/*x* radians where *x* is the number of segments that make up the cross section of the curve (called the Radial Accuracy in Functy).

However, unfortunately this technique can’t be easily parallelised since the orientation of each piece depends on the last, meaning that it couldn’t be translated easily into shader code. For Functy I therefore needed a different solution.

Luckily for me this isn’t a new problem, and the solution came in the form of Frenet Frames. A Frenet Frame is an orthogonal set of axes that’s defined based on the curvature and torsion of the curve at a particular point. Since it’s (in general) canonically defined at each point on the curve, it can be calculated independently from the other points, by calculating the derivatives and second-derivatives of the curve. More specifically, it requires that the tangent, normal and binormal vectors of the curve be calculated. There’s a decent explanation in the Wikipedia section “Other expressions of the frame”, and there’s also a neat Wolfram Demonstration too.

Since these three vectors can be calculated using the derivative of the curve, there’s no need to iterate along the curve, which makes it perfect for calculation using shaders. This is now implemented in Functy, and it seems to work pretty well. On my laptop, which has a decent but not mindblowing graphics card, animating a Frenet curve on the GPU using shader code is considerably faster than using the CPU.

The only problem is that this method has a tendency to generate curves with twists in. That is, the axis can make sudden rotations around the direction of the curve. In general this isn’t a problem, but can cause the curve to ‘pinch’ if the resolution of the pieces is too low. Below is a particularly extreme example.

Usually it’s not as bad as this, but it’s a shame nonetheless. For the benefit to be had from parallelisation I’m willing to live with it.

## Working Curves

The latest version of Functy in the repository now has a (roughly) working version of the curve rendering code. This allows a cylindrical coordinate cross section to be extruded along the length of a parametrically defined curve. In other words, something like a tube.

The code is fairly incomplete. Trying to define functional colours will cause a crash, and vertices are all positioned on the CPU, so that animation isn’t particularly efficient. Both of these should be fixed soon, including GPU rendering of the entire curve.

Here’s a brief video demo of this early code to give a flavour of how it can be used.

## The Art of Good Design?

Designing a really good library is a difficulty task. Since around August last year I’ve been writing a program called Knot, which I also recently open sourced. This makes extensive use of 3D Bezier tubes to render Celtic Knots, and having written the code for this, it seemed sensible to try to integrate it into Functy, to allow arbitrary tubes to be rendered using curved cylindrical coordinates. So, that made it a perfect candidate - or so I thought - for implementing as a library. This way, improvements from the Knot code could be easily merged into Functy and *vice versa*.

Designing the library turned out to be tricky, especially the task of managing and animating all of the curves. I figured that if I put the effort in to getting the library right, it would make the process of integrating into Functy that much easier.

I was wrong. After painstakingly implementing everything needed in the library, I tried to integrate it into Functy, only to realise that Functy already has all of the management and animation functionality needed. Given that I wrote the Functy code, you’d have thought I might have know this.

The consequence is that I’ve ended up dropping all of the curve library methods and cut and paste the relevant code straight in to Functy. If I hadn’t done this Functy would have ended up with two totally separate lists of functions, two separate animation steps and two separate update steps. It would have been a mess.

It’s taken longer to modify the curve code for Functy as a result, but the final structure is much better. Unfortunately it also means the library is now redundant. I’ll have to find some other use for it.

There’s a design moral to this story I’m sure, but I’m not exactly sure what it is. I’ve ended up with a nice clean library implementation that doesn’t work for what it was designed for, a new implementation that doesn’t use this library, and what seems like a lot of wasted time. It feels like somewhere something went a bit wrong. At least Functy can now render curves, so even if it was a circuitous route, maybe the journey wasn’t entirely wasted after all.

## Curvy Screenshots

The Curves library is currently being integrated into Functy in order to provide parametric tubes to be added to function scenes. To give an idea about the sort of results this might allow, here are a few screenshots taken from the development version.

In all of these images the tubes follow a path constructed from cubic Bezier curves:

((((1 - *a*)^{3}) × *P1*) + (3 × ((1 - *a*)^{2}) × *a* × *D1*) + (3 × (1 - *a*) × (*a*^{2}) × *D2*) + ((*a*^{3}) × *P2*))

where *a* is the parametric variable, *P1*, *P2*, *D1* and *D2* are the control points. It’s a bit crazy using Bezier curves for this since the library could handle a spiral far more easily, but it was just a product of the way the demo developed. For the first image the function used for the radius was the following.

((2 - cos((((*s* × *a*) + *o*) × 6 × π)) + (sin ((12 × *p*)) / 5)) / 2).

Here *a* and *p* are the parametric variables for the position along the curve and angle around the radius respectively. The values *s* and *o* are offsets and scaling factors that are changed for each quarter-circle segment of the spring.

The intention is to allow these functions to be entered directly into Functy to generate curves such as these. There’s still quite a bit of work to be done. Although the Curves library is coming along, this still needs to be integrated with Functy, and the shader code for these parametric tubes also needs to be implemented (it’s certainly possible, but I’m anticipating problems!).

## Curves Library Joins the Functy Stable

The Curves library allows various types of tube-like object to be rendered in 3D. Things like Bezier tubes and parametrically defined curves. Although it’s still got a shed-load of functionality missing, it’s reached a more-or-less stable state, which means it’s time to integrate some of this functionality into Functy.

The idea is to allow parametrically defined curves to be added to Functy graph scenes. At the moment it’s not really possible to do this using the existing surface and spherical functions, but I’m hoping it’ll make a useful addition. For example, it would allow 3D charts (line charts or bar graphs) to be easily generated, which could be quite neat.

The library has grown out of Bezier curve rendering I needed for my Knot application. One of the key features of this is that it’ll output Stanford Triangle Format (PLY) files which can be imported into other modelling applications, or even used to create 3D prints.

The plan is to eventually add similar functionality to Functy, and hopefully the addition of this library will pave the way for a collection of new interesting features!