The smoother the better!
It’s been a while I haven’t written anything on my blog. A bit of refreshment doesn’t hurt much, what do you think?
As a demoscener, I attend demoparties, and there will be a very important and fun one in about a month. I’m rushing on my 3D application so that I can finish something to show up, but I’m not sure I’ll have enough spare time. That being said, I need to be able to represent smooth moves and transitions without any tearing. I had a look into a few Haskell spline libraries, but I haven’t found anything interesting – or not discontinued.
Because I do need splines, I decided to write my very own package. Meet smoothie, my BSD3 Haskell spline library.
A spline is a curve defined by several polynomials. It has several uses, like vectorial graphics, signal interpolation, animation tweening or simply plotting a spline to see how neat and smooth it looks!
Splines are defined using polynomials. Each polynomials is part of the curve and connected one-by-one. Depending on which polynomial(s) you chose, you end up with a different shape.
For instance, 1-degree polynomials are used to implement straight lines.
As you can see, we can define a few points, and interpolate in between. This is great, because we can turn a discrete set of points into lines.
Even better, we could use 3-degree polynomials or cosine functions to make each part of the spline smoother:
We still have discrete points, but in the end, we end up with a smooth set of points. Typically, imagine sampling from the spline with time for a camera movement. It helps us to build smooth moves. This is pretty important when doing animation. If you’re curious about that, I highly recommend having a look into key frames.
Using splines in Haskell
So I’ve been around implementing splines in Haskell the most general way as possible. However, I don’t cover – yet? – all kinds of splines. In order to explain my design choices, I need to explain a few very simple concepts first.
A spline is often defined by a set of points and polynomials. The first point has the starting sampling value. For our purpose, we’ll set that to 0:
let startSampler = 0
The sampling value is often
Float, but it depends on your spline and the use you want to make of it. It could be
Int. The general rule is that it should be orderable. If we take two sampling values
t, we should be able to compare
t (that’s done through the typeclass constraint
Ord in Haskell).
So, if you have a spline and a sampling value, the idea is that sampling the spline with
startSampler gives you the first point, and sampling with
t > startSampler gives you another point, interpolated using points of the spline. It could use two points, three, four or even more. It actually depends on the polynomials you use, and the interpolating method.
In smoothie, sampling values have types designed by
A spline is made of points. Those points are called control points and smoothie uses
CP s a to refer to them, where
s is the sampling type and
a the carried value.
Although they’re often used to express the fact that the curve should pass through them, they don’t have to lie on the curve itself. A very common and ultra useful kind of spline is the B-spline.
With that kind of spline, the property that the curve passes through the control points doesn’t hold. It passes through the first and last ones, but the ones in between are used to shape it, a bit like magnets attract things around them.
Keep in mind that control points are very important and used to define the main aspect of the curve.
Polynomials are keys to spline interpolation. They’re used to deduce sampled points. Interpolation is a very general term and used in plenty of domains. If you’re not used to that, you should inquiry about linear interpolation and cubic interpolation, which are a very good start.
Polynomials are denoted by
Polynomial s a in smoothie, where
a have the same meaning than in
CP s a.
Getting started with smoothie
Types and constraints
smoothie has then three important types:
CP s a, the control points
Polynomial, the polynomials used to interpolate between control points
Spline s a, of course
The whole package is parameterized by
a. As said earlier,
s is very likely to require an
Ord constraint. And
a… Well, since we want to represent points, let’s wonder: which points? What kind of points? Why even “points”? That’s a good question. And this is why you may find smoothie great: it doesn’t actually know anything about points. It accepts any kind of values. Any? Almost. Any values that are in an additive group.
I won’t go into details, I’ll just vulgarize them so that you get quickly your feet wet. That constraint, when applied to Haskell, makes
a to be an endofunctor – i.e.
Functor – and additive – i.e.
Additive. It also requires it to be a first-class value – i.e. its kind should be
* -> *.
Additive, we can do two important things:
- First, with
Functor. It enables us to lift computation on the inner type. We can for instance apply a single function inside
- Then, with
Additive. It enables us to add our types, like
a + b.
We can then make linear combinations, like ak + bq. This property is well known for vector spaces.
The fun consequence is that providing correct instances to
Additive will make your type useable with smoothie as carried value in the spline! You might also have to implement
Ord as well, though.
Creating a spline
Creating a spline is done with the
spline function, which signature is:
spline :: (Ord a, Ord s) => [(CP s a, Polynomial s a)] -> Spline s a
It takes a list of control points associated with polynomials and outputs a spline. That requires some explainations… When you’ll be sampling the spline, smoothie will look for which kind of interpolation method it has to use. This is done by the lower nearest control point to the sampled value. Basically, a pair
(cp,polynomial) defines a new point and the interpolation method to use for the curve ahead of the point.
Of course, the latest point’s polynomial won’t be used. You can set whatever you want then – protip: you can even set
undefined because of laziness.
Although the list will be sorted by
spline, I highly recommend to pass a sorted list, because dealing with unordered points might have no sense.
A control point is created by providing a sample value and the carried value. For instance, using linear’s
let cp0 = CP 0 $ V2 1 pi
That’s a control point that represents
V2 1 pi when sampling is at
0. Let’s create another:
let cp1 = CP 3.341 $ V2 0.8 10.5
Now, let’t attach a polynomial to them!
Hold that for me please
The simplest polynomial – wich is actually not a polynomial, but heh, don’t look at me that way – is the 0-degree polynomial. Yeah, a constant function. It takes the lower control point, and holds it everwhere on the curve. You could picture that as a staircase function:
You might say that’s useless; it’s actually not; it’s even pretty nice. Imagine you want to attach your camera position onto such a curve. It will make the camera jump in space, which could be desirable for flash looks!
Polynomial to use such a behavior.
Oh yeah, it’s straightforward!
1-degree functions often describe lines. That is,
linear is the
Polynomial to use to connect control points with… straight lines.
Have fun on the unit circle!
One very interesting
cosine, that defines a cosine interpolation, used to smooth the spline and make it nicer for moves and transitions.
Highway to the danger zone
If you’re crazy, you can experiment around with
linearBy, which, basically, is a 1-degree polynomial if you pass
id, but will end up in most complex shapes if you pass another function –
(s -> s). Dig in documentation on hackage for further information.
Sampling our spline
Ok, let’s use a linear interpolation to sample our spline:
let spl = spline [(cp0,linear),(cp1,hold)]
Note: I used
holdas a final polynomial because I don’t like using
Ok, let’s see how to sample that. smoothie exports a convenient function for sampling:
smooth :: Ord s => Spline s a -> s -> Maybe a
smooth spl s takes the sampling value
s and maybe interpolate it in the
“Maybe? Why aren’t you sure?”
Well, that’s pretty simple. In some cases, the curve is not defined at the sampling value you pass. Before the first point and after, basically. In those cases, you get
That’s not the end
I wrote smoothie in a few hours, in a single day. You might have ideas. I want it to be spread and widely used by awesome people. Should you do graphics programming, sound programming, animation or whatever implying splines or smoothness, please provide feedback!
If no one comes up with, I’ll try to add some cubic interpolation methods, like hermitian splines, and one of my favorite, the famous Catmull Rom spline interpolation method.
As always, have fun hacking around, and keep doing cool stuff and sharing it!