# Introduction

This document explains the low-level inner workings of diagrams-core. Casual users of diagrams should not need to read this (although a quick skim may well turn up something interesting). It is intended more for developers and power users who want to learn how diagrams actually works under the hood; there is quite a lot that goes on behind the scenes to enable the powerful tools provided in diagrams-lib.

Chris Mears has written an article giving a quick walkthrough of some of the internals which is useful for getting started.

The remainder of this document is organized around the modules in diagrams-core. At some level, there is no substitute for just diving in and reading the source code (see the diagrams-core repository), which is generally well-commented, but the hope is that this document can serve to orient you and supply useful commentary.

# Diagrams.Core

This module simply re-exports many things from the other modules for convenience.

# Diagrams.Core.V

Diagrams.Core.V contains the definition of the V type family, which maps from types to their associated vector space. (See the relevant section from the user manual) along with some basic instances.

# Diagrams.Core.Points

The linear package defines a Point type in Linear.Affine along with some functions for working with points.

The Diagrams.Core.Points module simply re-exports a few things from linear, defines an instance of V and N for Point, and adds a few utility functions for points.

# Diagrams.Core.Names

Diagrams.Core.Names defines the infrastructure for names which can be used to identify subdiagrams.

AName, representing atomic names, is an existential wrapper, allowing (almost) any type to be used for names, as the user finds convenient. Strings may be used of course, but also numbers, characters, even user-defined types. The only restriction is that the wrapped type must be an instance of the following three classes:

• Typeable (so values can be pulled back out of the wrappers in a type-safe way),

• Show (so names can be displayed, for debugging purposes), and

• Ord (in order to be able to create a Map from names to subdiagrams).

Equality on atomic names works as expected: two names are equal if their types match and their values are equal.

The Ord instance for atomic names works by first ordering names according to (a String representation of) their type, and then by value for equal types (using the required Ord instance).

A qualified name (Name) is a list of atomic names. The IsName class covers things which can be used as a name, including many standard base types as well as ANames and Names. Most user-facing functions which take a name as an argument actually take any type with an IsName constraint, so the user can just pass in a String or an Int or whatever they want.

The motivation for having names consist of lists of atomic names is that it is not always convenient or even feasible to have globally unique names (especially when multiple modules by different authors are involved). In such a situation it is possible to qualify all the names in a particular diagram by some prefix. This operation governed by the Qualifiable class, containing the function (|>) :: IsName a => a -> q -> q for performing qualification.

# Diagrams.Core.HasOrigin

This module defines the HasOrigin class (containing the moveOriginTo method) as well as related functions like moveOriginBy, moveTo, and place. It also defines instances of HasOrigin for a number of types, including Points, tuples, lists, sets, and maps.

See the section of the type class reference on HasOrigin for more information.

# Diagrams.Core.Transform

This module defines a type of generic affine transformations parameterized over any vector space, along with a large number of methods for working with transformations.

First, the (:-:) type consists of a pair of functions, which are assumed to be linear and inverse to each other.

A Transformation type is then defined to contain three components:

• a linear map and its inverse (stored using (:-:))

• the transpose of the linear map, with its inverse (again stored using (:-:))

• a vector, representing a translation

The point is that we need transposes and inverses when transforming things like Envelopes and Traces. While it would be possible in theory to simply store a transformation as a matrix and compute its transpose or inverse whenever required, this would be computationally wasteful (especially computing inverses). Instead, we simply package up a transformation along with its inverse, transpose, and inverse transpose (which we can think of as a little 2x2 matrix of functions). Such a representation is closed under composition, and we can compute its inverse or transpose by just flipping the matrix along the appropriate axis.

Along with the definition of the Transformation type itself, this module exports many functions for generically creating, transforming, querying, and applying Transformation values. For example, in addition to straightforward things like composing and applying transformations, this is where you can find code to convert a Transformation to a matrix representation or to compute its determinant. (On the other hand, converting a matrix to a Transformation is only implemented specifically for 2 or 3 dimensions, and can be found in the diagrams-lib package, in Diagrams.Transform.Matrix.)

This module also defines the important Transformable class of things to which Transformations can be applied, along with many generic instances.

Finally, the module defines a few specific transformations which are polymorphic over the vector space, namely, translation and scaling. Other specific transformations (e.g. scaleX and so on) are defined in diagrams-lib.

# Diagrams.Core.Envelope

This module defines the Envelope type; see the user manual section on envelopes for a general overview of what envelopes are and how to use them.

For an explanation of the specific way that Envelope is defined, see Brent Yorgey's paper on diagrams and monoids.

The real meat of this module consists of the definitions of HasOrigin and Transformable instances for the Envelope type. The fact that packaging transformations together with their transpose and inverse makes it possible to correctly compute the affine transformation of an envelope is one of the key insights making the diagrams framework possible. The source code has extensive comments explaining the instances; consult those if you want to understand how they actually work.

Finally, this module defines the Enveloped class for things with Envelopes, a number of functions like envelopeV, envelopePMay, and so on for querying envelopes, and size-related functions like diameter, extent, and size that are defined in terms of envelopes.

# Diagrams.Core.Juxtapose

This module defines the Juxtaposable class, the default implementation juxtaposeDefault for instances of Enveloped and HasOrigin, and generic instances for Envelope, pairs, lists, maps, sets, and functions.

# Diagrams.Core.Measure

This module defines the Measured type along with a number of utility functions and instances for working with it. See the user manual section on measurement units.

Measured values are implemented as functions from a triple of scaling factors to a final value: the local scaling factor, global scaling factor, and normalized scaling factor. XXX write about how these are computed

# Diagrams.Core.Trace

This module implements the trace which is associated with every diagram. A trace is essentially an "embedded raytracer" which can compute an intersection with a diagram in any direction from any given base point. Note that a trace needs to be able to answer a trace query from any given base point, not just from some chosen particular base point (e.g. the origin), since we need to be able to apply affine transformations, including translations.

Often when one thinks about raytracing the basic idea is that you follow a ray and return the first intersection that occurs. However, to allow for also computing the last intersection and other generalizations, the base framework in this module actually computes a sorted list of all the intersection points. Hence this module defines a small abstraction for sorted lists, as well as the Trace abstraction itself. A number of functions for querying Trace values are defined here, as well as the Traced class for things which have a Trace.

# Diagrams.Core.Query

A Query is a function that associates a value of some (monoidal) type to each point in a diagram; see the user manual section on queries. There is not much in this module besides a great many type class instances for the Query type.

# Diagrams.Core.Style

This module implements styles, which are collections of attributes (such as line color, fill color, opacity, ...) that can be applied to diagrams. Diagrams takes a dynamically typed approach to attributes and styles. This is in contrast to the approach with backends and primitives, where the type of a diagram tells you what backend it is to be rendered with—or, if it is polymorphic in the backend, there are type class constraints that say what primitives the backend must be able to render. But the type of a diagram never says anything about what attributes a backend must support; indeed, by looking only at the type of a diagram it is impossible to tell what types of attributes it contains. In general, backends pick out the attributes they can handle and simply ignore any others.

## Attributes

Attributes are the primitive values out of which styles are built. Almost any type can be used as an attribute, with only a few restrictions: attributes must be Typeable, to support the use of dynamic typing, and a Semigroup, so there is some sensible notion of combining multiple attributes of the same type (which is used to combine attributes applied within the same scope; as we will see, for many standard attributes the semigroup is simply the one which keeps one attribute and discards the other). AttributeClass is defined as a synonym for the combination of Typeable and Semigroup.

The Attribute type is then defined as an existential wrapper around AttributeClass types. In a simpler world Attribute would be defined like this:

> data Attribute where
>   Attribute :: AttributeClass a => a -> Attribute

Historically, it did indeed start life defined this way. However, as you can see if you look at the source, by now the actual definition is more complicated:

> data Attribute (v :: * -> *) n :: * where
>   Attribute  :: AttributeClass a => a -> Attribute v n
>   MAttribute :: AttributeClass a => Measured n a -> Attribute v n
>   TAttribute :: (AttributeClass a, Transformable a, V a ~ v, N a ~ n) => a -> Attribute v n

This looks like the simpler definition if you ignore the type parameters and consider only the Attribute constructor. So let's consider each of the other constructors.

• MAttribute is for attributes that are Measured, i.e. whose values depend on the size of the final diagram and/or the requested output size; the primary examples are line width and font size. Recall that a Measured n a is actually a function that can produce a value of type a once it is provided some measurement factors of type n. The unmeasureAttribute function is provided to turn MAttribute constructors into Attribute constructors; this is typically used when preparing a diagram for rendering.

• TAttribute is for attributes that are Transformable, i.e. which are affected by transformations applied to the objects to which they are attached. The primary examples are line and fill texture (e.g. gradients), and clipping paths. (Note that MAttributes can actually be affected by transformations too, in the case of Local units.)

The Attribute type has instances of Semigroup (combine attributes of the same type, otherwise take the rightmost) and Transformable (ignore Attribute constructors and do the appropriate thing for the other constructors). There are also various lenses/prisms for accessing them.

Note that one does not typically construct an Attribute value directly using the constructors; instead, the functions applyAttr, applyMAttr, and applyTAttr are provided for applying an attribute directly to any instance of HasStyle.

## Styles

A Style is just a dynamically-typed, heterogeneous collection of attributes. The attributes are actually stored as values in a hash table, keyed by their type (specifically, a TypeRep), so at most one attribute of any given type can be stored in a Style.

Style is an instance of Semigroup, with union as a combining operation. If the two styles both have an attribute of a given type, those attributes are combined according to the Semigroup instance for that attribute type.

There are a number of other functions for creating Style values and extracting attributes from them, which are mostly self-explanatory. One function worth mentioning is unmeasureAttrs, which maps over a Style and changes all MAttributes into Attributes (based on the provided scaling factors). This is typically done as a final step before rendering.

Finally, the HasStyle class governs types which "have a Style", specifically, types to which a Style can be applied. Style itself has an instance of HasStyle, corresponding to the semigroup operation on Style. This module also defines a number of other instances for applying styles to entire data structures such as lists, tuples, functions, maps, and sets.

# Diagrams.Core.Types

This is an unfortunately large module which contains definitions and utility functions for many of the core data structures of diagrams. In principle, it would be nice to break it up into smaller pieces, but in fact a lot of things in this module end up cyclically depending on one another, so easier said than done.

## QDiagram

The central type in diagrams is the definition of QDiagram:

> newtype QDiagram b v n m
>   = QD (D.DUALTree (DownAnnots v n) (UpAnnots b v n m) Annotation (QDiaLeaf b v n m))

DUALTree is defined in the dual-tree package. A value of type DUALTree d u a l consists of an n-ary (rose) tree, with:

• Values of type l at the leaves.

• Monoidal annotations of type u which "travel up" the tree. Each leaf of type l has a corresponding value of type u, and the u values are combined as one travels up the tree, so that the root would contain the mconcat of the u values in all the leaves. We think of each tree (and subtree) as intrinsically posessing a u value, which is often some kind of "summary" or "measurement".

• There are also values of another monoid, of type d, which can be applied to a tree at the root and accumulate as one travels down any path from the root to a leaf. These d values may act on the u values, that is, there is a function d -> u -> u satisfying certain coherence properties with respect to the monoid structures of d and u (though for a particular d and u the action may be trivial, i.e. const id).

• Finally, there are values of type a which can be stored at internal nodes. They are simply "along for the ride": they do not have associated u values nor are they affected by d values. a values will never be moved (on the other hand, it is permissible to push d values up or down the tree in a way that preserves all monoid compositions).

The QDiagram type specifically instantiates these types as follows:

• l values at leaves are QDiaLeaf values. One might think these consist simply of primitives, but actually they are a bit more complicated, because they also handle the case of "delayed" diagrams that need to know something about their context before they can be generated; this is explained in more detail below.

• The inert internal a values are of type Annotation. Currently these consist solely of information about hrefs (for backends that support hyperlinks) and opacity grouping.

• The upwards-traveling u values are of type UpAnnots, explained in more detail below, which record various summary information about a diagram (envelope, trace, named subdiagrams, etc.).

• The downwards-traveling d values are of type DownAnnots, explained in more detail below, which record information imposed on a diagram (affine transformations, attributes, etc.).

### QDiaLeaf

The QDiaLeaf type has two cases:

> data QDiaLeaf b v n m
>   = PrimLeaf (Prim b v n)
>   | DelayedLeaf (DownAnnots v n -> n -> n -> QDiagram b v n m)

The first case, PrimLeaf, is simple enough: it contains a Prim (also defined in this module), which is simply an existential wrapper around a Transformable, Typeable, Renderable thing.

The second case is trickier. It represents a diagram which needs to know its global context, represented by the accumulated DownAnnots values along the path from the root to this delayed leaf, together with the normal-to-output scaling factor and the global-to-output scaling factor. Note that we cannot possibly know this information until just before the diagram is to be rendered, at which point it becomes fixed and we know we are not going to insert this diagram into a yet bigger diagram, and we know the requested output size and hence can compute the scaling factors. So just before rendering, delayed leaves are recursively expanded into QDiagrams. For an example of DelayedLeaf in action, see Diagrams.TwoD.Arrow in diagrams-lib, where it is needed since arrowhead sizes can depend on this global context.

### Annotation

The Annotation type has two constructors, one for hyperlinks (Href) and one for opacity groups. In both of these cases, it's important that these annotations stay exactly where the user places them in the tree, and they are unaffected by transformations and the like.

### UpAnnots

UpAnnots is a heterogeneous list of monoidal values (with an elementwise monoidal operation; see Data.Monoid.MList in the monoid-extras package) which serves as the "upwards-traveling" monoid in a QDiagram. That is, every primitive diagram at a leaf has an intrinsic associated UpAnnots value, and these get combined as one moves up the tree. An UpAnnots value consists of the following components:

• An Envelope (in a Deletable wrapper so we can implement setEnvelope monoidally; see Data.Monoid.Deletable in monoid-extras),

• a Trace (similarly Deletable),

• a SubMap (also Deletable) which maps names to subdiagrams (more on these later), and

• a Query, which associates a monoidal value of some type m to every point in the diagram.

### DownAnnots

DownAnnots is