# A project: Constraints

I’ve recently started making some tiny contributions to Brent Yorgey’s Haskell diagrams library, mostly bug fixes. The bug fixes are primarily just a way of familiarising myself with the diagrams codebase though. I have my eye on a rather chunkier task in the open issues list. That’s writing a constraints solver for diagram layout. I’ve been interested in this sort of problem for some time, and have never had a good reason to get down to it, so I’m going to see what I can come up with.

And I’m going to try something I’ve not done before, which is to document experiments, design and development as blog articles. It may all turn out to be hideously embarrassing, it may end up that none of what I do actually makes it into the mainstream of the diagrams library, but having to write about what I’m doing will keep me honest, force me to be clear about what I’m doing, and will provide an incentive to work on this stuff.

## Constraints 101

OK, so what do we mean here when we talk about “constraints”? The word “constraint” is used in lots of ways: we have type class constraints in Haskell, people talk about constraint logic programming, in mechanics constraints are extra side-conditions on Newton’s equations of motions imposed by the structure of the problem we’re solvingFor examples, see Section 1.3 of Goldstein’s Classical Mechanics. (think bead-on-wire problems), and so on. Here, we’re talking about something closer to the last case. Given a bunch of geometrical objects (points, lines, polygons, circles, arcs, Bézier curves, whatever), we want to be able to specify geometrical relationships between our objects, for instance that the end-point of one line segment is coincident with the corner of a rectangle, or that a certain circle is tangent to a certain line segment.

Why do this? My first exposure to constraint-based drawing was through AutoDesk’s Mechanical Desktop, which was then part of the AutoCAD product suiteNow obsolete. Do I feel old? No way!. This had a sketching tool that was just magical. Draw a very rough sketch of approximately what a component looked like, then say, “OK, this line should be horizontal, this one should be perpendicular to this one. This curve is a fillet and should be tangential to this line and this line. These two circles are concentric, these two lines are parallel. This point lies on this line, not quite sure where yet.” What you end up with is a drawing that has a smaller number of free parameters (usually the major dimensions of whatever part you’re designing). Specify those dimensions and the rest of your drawing jumps around to make everything right with the constraints you definedIn Mechanical Desktop, the dimensions were often pulled from a spreadsheet or a database so that you could, for example, draw one metric standard bolt, parameterised by the relevant dimensions, and have an M8 bolt produced from the same base drawing as an M10 bolt. More recent Autodesk and comparable products go a lot further on this front. The other day, I was reading some publicity for Autodesk Inventor, which claims to have a library of some 700,000 standard components to choose from, many of which are presumably parameterised in just this way. (I sure hope so. If not, some poor schmuck had to draw them all by hand...). I was using Mechanical Desktop for work, but I got a real kick out of just playing with the constraint-based drawing tool. It really seemed like magic. Set up your drawing with some constraints, drag some points around and watch as AutoCAD updates the rest of the drawing to maintain the constraints you specified. Cute and powerful.

The same power can be applied to layout in the diagrams library. You might not want to go the whole hog with the complex geometric constraints available in CAD packages (although why not?), but you can certainly say things like: “These three items lie on a horizontal line; the gaps between them should be equal; I don’t know how wide the items are yet.” This is a very common situation in GUI layout and lots of GUI toolkits provide the capability to specify layout in this fashion. Since the diagrams library is intended to produce images more complex than typical GUI layout, it seems to me that something closer to the AutoCAD experience is what we want. (It will also be more fun!)

## Goals

Given all that spiel, how do I get from nothing to a usable constraints module for the diagrams library?

1. The first thing to do is probably to implement some existing constraint solver algorithms in Haskell and do some experiments. Cassowary seems like a good starting point. Other interesting possibilities are the system used in Juno-2 and Mike Gleicher’s snap-together mathematics. This shouldn’t be too hard, and would result in a constraint solver working at the level of individual equations and inequalities for sets of real variables.

2. The diagrams library already has primitives for drawing various geometric objects, but to start with, I’d like to abstract away from that. Two reasons for this. The first is to reduce the dependence on the details of the representations used in the base diagrams library in order to get an idea of exactly what is required in the constraints system. The second reason is that there are multiple ways of looking at geometrical objects. A point can be identified by its Cartesian coordinates or by its polar coordinates. A line segment can be identified by two points or by a single point, a direction and a length. A rectangle can be identified by giving two corner points and a rotation angle, or by a single corner point, a width and height and a rotation angle. Each of these “views” of these objects is equivalent, and it should be possible to express constraints in any convenient view. For instance, I might want to constrain the centre of a circle $C$ to lie on the circle $r=1$. I should be able to give a constraint $r(\mathrm{centre}(C)) = 1$ to achieve this. I have an idea of some sort of “constrainable lens” floating around in my head but it’s going to require some experimentation to pin down just how this should work, and keeping this independent from diagrams to start with should help to reduce my confusion to manageable levels. (There are some relevant thoughts about this idea of different views of geometrical objects in Mike Gleicher’s PhD thesis.)

3. The most important part of a constraint system from the user’s point of view is the interface for defining and managing constraints. The constraint that two lines are perpendicular, or that a circle is tangent to a particular line segment, can be expressed as sets of equations and/or inequalities that can be handled by a constraint solver, and we’d like a way to work at this higher level: in the most trivial case, I want to be able to say “Point $P_1$ is coincident with point $P_2$”, not “$x(P_1) = x(P_2) \wedge y(P_1) = y(P_2)$”. More complex geometrical constraints can be broken down and expressed as equations in similar ways. Providing an easy-to-use interface between this geometrical view of the world and the equational view required by the constraints solver will be key to making the constraint system useable. Part of that will also involve finding a good way to deal with underconstrained and overconstrained systems.

4. The last job to tackle will be taking whatever I come up with and trying to massage it into a state where it’s usable in the diagrams library. I’m going to keep this final goal in mind, but I’m not going to let it constrain what I do too muchHa ha. I made a joke.. Chances are, someone else will develop something usable for diagrams before I’m too far out of the starting gate on this, and it’s mostly for my own amusement anyway...

## First things first

OK, so I’ll start with reading the Cassowary papers and will implement the relevant algorithms in Haskell to act as a baseline for experimentation. Off we go!