Euclidea 3 solver

I like puzzles, and I like geometry. So no wonder I really enjoyed the Euclidea game.

While solving the geometric problems there, you can see hints for the number of moves for minimal solutions, including the number of minimal moves when solving the problem with straightedge and compass only. So after you successfully solve the puzzle, you can try solving it again in the most efficient way known for given problem. And that's where it became interesting for me from a different perspective as well. For sometimes, I was able to solve the problem, but I missed the straightedge/compass minimal solution.

I also like programming. After I spent some time thinking about the minimal solution for a problem, and was still clueless, I thought I could hack a script that would help me find it. As it can't be that hard to calculate intersections of lines and circles, right? :-)

But as the problems became harder, the solving logic needed to reflect that as well. And what used to be a couple of lines of code, has over time grown into a solving engine for whole class of geometry problems, not only limited to the Euclidea game.

Given initial set of objects (points / lines / circles), you want to come up with constructions using straightedge and compass only that yield some result (described as a set of points). And with this type of problem, the solving engine here can help.

Engine

The engine is written in vanilla JavaScript, so you don't need any build process or tool-chain to get it up and running in node.js.

Exported objects

There is one cornerstone type: Point, and two advanced types: Line, and Circle. A Line is defined by two Points that belong to it, and a Circle is defined by two Points as well: the center and another Point on the Circle.

The most important function for Line and Circle is calculating intersections with each other (links to corresponding algorithms on wikipedia are in the source code). With Point you can only calculate distance from another Point.

Besides the three types described above, the engine library also exports function solve that is the entry point to kick off the searching for solution of given problem.

Arguments of solve function

Let's have a look at the arguments the solve function takes one by one, let's start with the mandatory ones...

Mandatory

The first argument is an array of Points we want to construct, i.e. our goals. Often times, however, the goal is to create a Line, or a Circle with some properties. Then you have to be creative :-). For example, when you know what Line you are looking for, you can add another Line to the initial setup (see below), and set the goal to be the intersection of the Line you are trying to construct, and the Line you just artificially added to the initial setup. Which means that whenever you manage to construct the Line you were looking for, you'll get that intersection Point with the artificial Line. The same you can do when the goal is to create Circle with some properties.

The second argument is again an array of Points. This time the array represents starting set for object constructions, i.e. we'll be taking pairs of Points from this set to create new Lines and/or new Circles.

The third argument is an array of Lines and/or Circles that describe the initial problem setup. Let's say the goal is to inscribe a Circle into given triangle. This initial setup for this problem is the triangle, defined with three Lines.

The rest of the arguments of the solve function are optional.

Optional

The first of optional arguments, and the fourth one in total, is mask of the problem solution. Remember that our solution to given problem consists of a sequence of Line and Circle constructions. So in each step you create either one of the two object types. Providing the mask parameter, you hint the solving algorithm about what the resulting sequence should look like. The mask is in form of array of strings, with the following meanings:

• "line": single Line object,
• "circle": single Circle object,
• "*": single object (Line or Circle),
• "line**": any number of Line objects (including zero instances),
• "circle**": any number of Circle objects (including zero instances),
• "**": any number of objects (Lines or Circles, including zero instances).

There is one limitation on the "generator" mask parts (the ones with two asterisks, i.e. "line**", "circle**", and "**"), and that is that you can specify up to one of them when providing the mask argument. It limits the power of the masks you could construct, but simplifies the solving logic a lot.

The default value for the mask is [ "**" ], i.e. you search for the minimal solution and you don't care what the solution looks like in advance.

Note: I added the support for the mask argument only after I run into the solve-this-problem-using-compass-only puzzles in Euclidea, so I needed better control over the resulting solution.

The second optional argument is ignore options object. It has two (again) optional properties: min and max. When min value (number) is provided, we consider new intersection found useful for future constructions only if its distance from existing Points we already consider is at least min. When max value (number) is provided, we ignore all intersections found that are further than max from the origin, i.e. Point(x = 0, y = 0).

Note: I added the ignore options argument to speed up the searching algorithm. In the Euclidea game, you can zoom the scene in to some extend, but if two intersections will be close to each other, you probably won't be able to select them anyway. The same goes with intersections too distant from the origin: you can zoom the scene out only to some extend, so the solution won't make use of intersections far, far away.

The last optional argument is verification set of Points. This is an array similar to mandatory starting set (and must be of the same length!), and is used to verify the solution using different set of Points than was used for actual finding the solution candidate. When the verification set is provided, the solution is claimed correct only if the solution candidate can be replayed using different input set of Points while still yielding the desired goals Points.

Note: this was also added as an after-thought, when I used the engine to find minimal solutions for advanced Euclidea problems. As it turned out that often times the engine found a solution, but that solution worked only for the initial set of Points, and failed on another set, i.e. that solution was not general (it was rather a coincidental solution). When you have a combinatorial problem and going though millions of combinations, you'd be surprised how many solutions like that you may find (but still miss the general one you are looking for...)

Algorithm

And here comes the most important part of this README: the description of how the solving algorithm works.

First of all, we want to find the minimal solution for given problem. For given solution mask (defaulting to [ "**" ]), I decide what is the length of the shortest sequence possible that matches the mask. With the fixed length of the solution sequence, I generate all possible line / circle combinations and then let another function find a solution according to that prescribed combination. When all combinations of given length are explored without success, I increase the tested solution sequence length by 1 (if that still matches the solution mask) and start generating the combinations for that length (and testing those combinations in turn again). And this I keep doing until the solution is (hopefully) found.

Example

So let's say I use the default solution mask, which is [ "**" ]. The shortest sequence that matches the mask is of length 0, which means the goals would already be in the starting set of Points provided. So we can skip that solution sequence length.

Next one is of length 1. There we have two combinations:

• line
• circle

So first we try line and explore if we can find a solution using just one Line. If so, we are done. If not, we try to find a solution using one Circle only. Again, if we find the solution, we are done, if not, we are out of combinations for sequences of length 1, so it is time to explore possible sequences of length 2. There we have these four options:

• line, line
• line, circle
• circle, line
• circle, circle

And again, we try finding solutions taking the combinations one by one. If there is no 2-step solution, we try sequence of length 3. Then 4, 5, and so on... until we find the solution.

With different solution mask provided, the generated combinations are different (so that each of the generated combination matches the mask), but the idea is essentially the same.

One level down

Now we know what our expected solution looks like (the sequence of Lines and Circles is given). At the beginning, we have only the starting set of Points to work with, so we generate the first object according to the sequence by taking two points from our set. We make sure the object is new (that we don't construct a Line or a Circle that we have among our objects already), and if so, we calculate the intersections with other objects we already have. We evaluate the calculated intersection Points (again, they need to be new to our working set and we may discard some based on ignore options), add them to our working set of Points, and we proceed to generating next new object according to our given solution sequence.

This again is a combinatorial problem, so we need to make sure that when generating an object, we take into account all possible couples of Points available in current working set, i.e. that we consider all couples of Points that could be used for constructing the new object.

When we reach the end in generating the sequence, we simply check if the current working set of Points contains all the Points from the goals set. If so, then we have a solution. If not, we need to generate another combination (possible object sequence) and verify that one. And we keep iterating until we have a solution, or we exhaust all our options for given sequence, and it's time to try another one (going one level up again).

Note: when verification set is provided, it is not good enough to find all the goals Points in the current working set, but the same constructions steps must yield the goals Points again when applied on the verification set and the same intersection Points as in examined solution candidate are used. Only then we are happy and consider the construction sequence to be our solution. If not, we keep searching.

Pretty printing

OK, we have a solution! The last step is to make the solution human-readable and thus useful for actual construction. The sequence of Lines and/or Circles is analyzed, all the intersections that are used for constructing objects later in the sequence are named and everything is then pretty printed.

Example 1: Midpoint

Goal

Construct the midpoint between the given points using only a compass.

Idea

As the starting set, just provide two Points for which you can calculate the midpoint easily. E.g. [-1,0] and [1,0], the resulting midpoint is obviously [0,0]. There are no initial objects in the scene besides the two points, so pass empty array as the initial problem setup parameter. Instruct the algorithm to use circles only (passing the mask argument set to "circle**") and let the algorithm find the solution for you.

Code

var engine = require('./engine');
var Point  = engine.Point;
var solve  = engine.solve;

solve(
  /* goals         */ [new Point(0,0)],
  /* starting set  */ [new Point(1,0), new Point(-1,0)],
  /* problem setup */ [],
  /* mask          */ ['circle**']
);

Output

Solution found!
-----
Initial set of points:
-----
# point1 (x: -1, y: 0)
# point2 (x: 1, y: 0)
-----
Construction:
-----
1: circle1 [point1 ---> point2]
2: circle2 [point2 ---> point1]
   ... point3 = circle1 x circle2 (x: 2.220446049250313e-16 , y: 1.7320508075688774)
   ... point4 = circle1 x circle2 (x: -2.220446049250313e-16 , y: -1.732050807568877)
3: circle3 [point3 ---> point4]
   ... point5 = circle2 x circle3 (x: 3 , y: 2.220446049250313e-16)
4: circle4 [point5 ---> point1]
   ... point6 = circle1 x circle4 (x: -0.5000000000000004 , y: 1.9364916731037083)
5: circle5 [point6 ---> point1]
   ... point7 = circle2 x circle5 (x: 1.4999999999999996 , y: 1.9364916731037085)
6: circle6 [point7 ---> point5]
-----
Goals:
-----
# (x: 0, y: 0) === circle5 x circle6

So with 6 circles (which is the minimal known solution) you can construct the midpoint using only a compass.

Example 2: Napoleon's problem

Goal

Construct the center of the given circle using only a compass.

Idea

Let's take unit circle (the center of it will then obviously be [0,0]). Pick 2 (reasonably "random") points on the circle, instruct the algorithm to use only circles, and make yourself a cup of coffee. Before you finish it, you'll get the solution.

Code

var engine = require('./engine');
var Point  = engine.Point;
var Circle = engine.Circle;
var solve  = engine.solve;

var center = new Point(0,0);
var radius = new Point(1,0);

// for x in [-1,1] return positive y on unit circle
function getY(x) { return Math.sqrt(1.0 - x*x); }

var pointA = new Point(-1/2, getY(-1/2));
var pointB = new Point( 1/5, getY( 1/5));

solve(
  /* goals         */ [center],
  /* starting set  */ [pointA, pointB],
  /* problem setup */ [new Circle(center, radius)],
  /* mask          */ ['circle**']
);

Output

Solution found!
-----
Initial set of points:
-----
# point1 (x: 0.2, y: 0.9797958971132712)
# point2 (x: -0.5, y: 0.8660254037844386)
-----
Initial construction:
-----
1: circle1 [(x: 0, y: 0) ---> (x: 1, y: 0)]
-----
Construction:
-----
2: circle2 [point1 ---> point2]
   ... point3 = circle1 x circle2 (x: 0.7994112549695427 , y: 0.6007841920590293)
3: circle3 [point3 ---> point1]
   ... point4 = circle1 x circle3 (x: 0.99676363543604 , y: -0.08038815256198717)
4: circle4 [point4 ---> point3]
   ... point5 = circle2 x circle4 (x: 0.397352380466497 , y: 0.2986235524922546)
5: circle5 [point5 ---> point3]
   ... point6 = circle3 x circle5 (x: 0.7018610913724186 , y: -0.10165989164761002)
6: circle6 [point6 ---> point3]
   ... point7 = circle4 x circle6 (x: 0.8992134718389153 , y: -0.7828322362686265)
7: circle7 [point7 ---> point5]
-----
Goals:
-----
# (x: 0, y: 0) === circle6 x circle7

Note: this is not the nicest solution for Napoleon's problem, but still it is a solution with 6 circles only, which is the minimal solution known.

Example 3: Third Proportional

Goal

Given ray with endpoint A and with points B and C on the ray: construct a point D on the ray such that the segment AD is the third proportional to the given line segments. I.e. that |AC| / |AB| = |AB| / |AD|.

Idea

Pick some points A, B, and C, calculate the expected position of the point D, and let the algorithm to solve the problem.

Code

var engine = require('./engine');                                               
var Point  = engine.Point;                                                      
var Line   = engine.Line                                                        
var solve  = engine.solve;                                                      
                                                                                
var A = new Point(0,0);                                                         
                                                                                
var b = Math.sqrt(3);                                                           
var B = new Point(b,0);                                                         
                                                                                
var c = Math.sqrt(5);                                                           
var C = new Point(c,0);                                                         
                                                                                
var d = b * b / c;                                                              
var D = new Point(d,0);                                                         
                                                                                
solve(                                                                          
  /* goals         */ [D],                                                             
  /* starting set  */ [A, B, C],                                                       
  /* problem setup */ [new Line(A,B)]                                                  
);

Output

Solution found!
-----
Initial set of points:
-----
# point1 (x: 0, y: 0)
# point2 (x: 1.7320508075688772, y: 0)
# point3 (x: 2.23606797749979, y: 0)
-----
Initial construction:
-----
1: line1 [point1 ---> point2]
-----
Construction:
-----
2: circle1 [point3 ---> point1]
3: circle2 [point1 ---> point2]
   ... point4 = circle1 x circle2 (x: 0.6708203932499368 , y: -1.5968719422671314)
4: circle3 [point4 ---> point1]
-----
Goals:
-----
# (x: 1.3416407864998736, y: 0) === line1 x circle3

Your turn now!

Feel free to use the engine to solve your construction problems using straightedge and compass. I hope you'll find the engine useful. Remember though that searching for the solution of your problem is a combinatorial problem at several levels and that the algorithm may need to go through millions of options before finding the solution for you...