Type classes are the approach to constrain type variables, to bound polymorphism and limit the set of types the variables can be instantiated with. If we have two type variables to constrain, multi-parameter type classes are the natural answer then. Let's take this solution and see where it leads to. Here is the original type class
class XyConv a where toXy :: a b -> [Xy b]
and the problematic instance
data CircAppr a b = CircAppr a b b -- number of points, rotation angle, radius deriving (Show)
instance Integral a => XyConv (CircAppr a) where toXy (CircAppr divns ang rad) = let dAng = 2 * pi / (fromIntegral divns) in let angles = map ((+ ang) . (* dAng) . fromIntegral) [0..divns] in map (\a -> am2xy a rad) angles
To be more explicit, the type class declaration has the form
class XyConv a where toXy :: forall b. a b -> [Xy b]
with the type variable 'b' universally quantified without any constraints. That means the user of (toXy x) is free to choose any type for 'b' whatsoever. Obviously that can't be true for (toXy (CircAppr x y)) since we can't instantiate pi to any type. It has to be a Floating type. Hence we have to constrain b. As I said, the obvious solution is to make it a parameter of the type class. We get the first solution:
class XYConv1 a b where toXy1 :: a b -> [Xy b]
instance (Floating b, Integral a) => XYConv1 (CircAppr a) b where toXy1 (CircAppr divns ang rad) = let dAng = 2 * pi / (fromIntegral divns) in let angles = map ((+ ang) . (* dAng) . fromIntegral) [0..divns] in map (\a -> am2xy a rad) angles
The type class declaration proclaims that only certain combinations of 'a' and 'b' are admitted to the class XYConv1. In particular, 'a' is (CircAppr a) and 'b' is Floating. This reminds us of collections (with Int keys, for simplicity)
class Coll c where empty :: c b insert :: Int -> b -> c b -> c b
instance Coll M.IntMap where empty = M.empty insert = M.insert
The Coll declaration assumes that a collection is suitable for elements of any type. Later on one notices that if elements are Bools, they can be stuffed quite efficiently into an Integer. If we wish to add ad hoc, efficient collections to the framework, we have to restrict the element type as well:
class Coll1 c b where empty1 :: c insert1 :: Int -> b -> c -> c
Coll1 is deficient since there is no way to specify the type of elements for the empty collection. When the type checker sees 'empty1', how can it figure out which instance for Coll1 (with the same c but different element types) to choose? We can help the type-checker by declaring (by adding the functional dependency c -> b) that for each collection type c, there can be only one instance of Coll1. In other words, the collection type determines the element type. Exactly the same principle works for XYConv.
class XYConv2 a b | a -> b where toXy2 :: a -> [Xy b]
instance (Floating b, Integral a) => XYConv2 (CircAppr a b) b where toXy2 (CircAppr divns ang rad) = let dAng = 2 * pi / (fromIntegral divns) in let angles = map ((+ ang) . (* dAng) . fromIntegral) [0..divns] in map (\a -> am2xy a rad) angles
The third step is to move to associated types. At this stage you can consider them just as a different syntax of writing functional dependencies:
class XYConv3 a where type XYT a :: * toXy3 :: a -> [Xy (XYT a)]
instance (Floating b, Integral a) => XYConv3 (CircAppr a b) where type XYT (CircAppr a b) = b toXy3 (CircAppr divns ang rad) = let dAng = 2 * pi / (fromIntegral divns) in let angles = map ((+ ang) . (* dAng) . fromIntegral) [0..divns] in map (\a -> am2xy a rad) angles
The step from XYConv2 to XYConv3 is mechanical. The class XYConv3 assumes that for each convertible 'a' there is one and only Xy type 'b' to which it can be converted. This was the case for (CircAppr a b). It may not be the case in general. But we can say that for each convertible 'a' there is a _class_ of Xy types 'b' to which they may be converted. This final step brings Tillmann Rendel's solution.