Diagrams.Core.Envelope

diagrams-core defines the core library of primitives forming the basis of an embedded domain-specific language for describing and rendering diagrams.

The Diagrams.Core.Envelope module defines a data type and type class for "envelopes", aka functional bounding regions.

Envelopes

newtype Envelope v n

Every diagram comes equipped with an envelope. What is an envelope?

Consider first the idea of a bounding box. A bounding box expresses the distance to a bounding plane in every direction parallel to an axis. That is, a bounding box can be thought of as the intersection of a collection of half-planes, two perpendicular to each axis.

More generally, the intersection of half-planes in every direction would give a tight "bounding region", or convex hull. However, representing such a thing intensionally would be impossible; hence bounding boxes are often used as an approximation.

An envelope is an extensional representation of such a "bounding region". Instead of storing some sort of direct representation, we store a function which takes a direction as input and gives a distance to a bounding half-plane as output. The important point is that envelopes can be composed, and transformed by any affine transformation.

Formally, given a vector v, the envelope computes a scalar s such that

• for every point u inside the diagram, if the projection of (u - origin) onto v is s' *^ v, then s' <= s.
• s is the smallest such scalar.

There is also a special "empty envelope".

The idea for envelopes came from Sebastian Setzer; see http://byorgey.wordpress.com/2009/10/28/collecting-attributes/#comment-2030. See also Brent Yorgey, Monoids: Theme and Variations, published in the 2012 Haskell Symposium: http://ozark.hendrix.edu/~yorgey/pub/monoid-pearl.pdf; video: http://www.youtube.com/watch?v=X-8NCkD2vOw.

Constructors

Envelope (Option (v n -> Max n))

Instances

Show (Envelope v n)

Methods

showsPrec :: Int -> Envelope v n -> ShowS

show :: Envelope v n -> String

showList :: [Envelope v n] -> ShowS

Ord n => Semigroup (Envelope v n)

Envelopes form a semigroup with pointwise maximum as composition. Hence, if e1 is the envelope for diagram d1, and e2 is the envelope for d2, then e1 `mappend` e2 is the envelope for d1 `atop` d2.

Methods

(<>) :: Envelope v n -> Envelope v n -> Envelope v n

sconcat :: NonEmpty (Envelope v n) -> Envelope v n

stimes :: Integral b => b -> Envelope v n -> Envelope v n

Ord n => Monoid (Envelope v n)

The special empty envelope is the identity for the Monoid instance.

Methods

mempty :: Envelope v n

mappend :: Envelope v n -> Envelope v n -> Envelope v n

mconcat :: [Envelope v n] -> Envelope v n

Wrapped (Envelope v n)

Associated Types

type Unwrapped (Envelope v n) :: *

Methods

_Wrapped' :: Iso' (Envelope v n) (Unwrapped (Envelope v n))

(Metric v, Fractional n) => HasOrigin (Envelope v n)

The local origin of an envelope is the point with respect to which bounding queries are made, i.e. the point from which the input vectors are taken to originate.

Methods

moveOriginTo :: Point (V (Envelope v n)) (N (Envelope v n)) -> Envelope v n -> Envelope v n

(Metric v, Floating n) => Transformable (Envelope v n)

Methods

transform :: Transformation (V (Envelope v n)) (N (Envelope v n)) -> Envelope v n -> Envelope v n

(Metric v, OrderedField n) => Enveloped (Envelope v n)

Methods

getEnvelope :: Envelope v n -> Envelope (V (Envelope v n)) (N (Envelope v n))

(Metric v, OrderedField n) => Juxtaposable (Envelope v n)

Methods

juxtapose :: Vn (Envelope v n) -> Envelope v n -> Envelope v n -> Envelope v n

Rewrapped (Envelope v n) (Envelope v' n')

type Unwrapped (Envelope v n)

type Unwrapped (Envelope v n) = Option (v n -> Max n)

type N (Envelope v n)

type N (Envelope v n) = n

type V (Envelope v n)

type V (Envelope v n) = v

appEnvelope :: Envelope v n -> Maybe (v n -> n)

"Apply" an envelope by turning it into a function. Nothing is returned iff the envelope is empty.

onEnvelope :: ((v n -> n) -> v n -> n) -> Envelope v n -> Envelope v n

A convenient way to transform an envelope, by specifying a transformation on the underlying v n -> n function. The empty envelope is unaffected.

mkEnvelope :: (v n -> n) -> Envelope v n

Create an envelope from a v n -> n function.

pointEnvelope :: (Fractional n, Metric v) => Point v n -> Envelope v n

Create a point envelope for the given point. A point envelope has distance zero to a bounding hyperplane in every direction. Note this is not the same as the empty envelope.

class (Metric (V a), OrderedField (N a)) => Enveloped a where

Enveloped abstracts over things which have an envelope.

Minimal complete definition

getEnvelope

Methods

getEnvelope :: a -> Envelope (V a) (N a)

Compute the envelope of an object. For types with an intrinsic notion of "local origin", the envelope will be based there. Other types (e.g. Trail) may have some other default reference point at which the envelope will be based; their instances should document what it is.

Instances

Enveloped b => Enveloped [b]

Methods

getEnvelope :: [b] -> Envelope (V [b]) (N [b])

Enveloped b => Enveloped (Set b)

Methods

getEnvelope :: Set b -> Envelope (V (Set b)) (N (Set b))

Enveloped t => Enveloped (TransInv t)

Methods

getEnvelope :: TransInv t -> Envelope (V (TransInv t)) (N (TransInv t))

(Enveloped a, Enveloped b, (~) (* -> *) (V a) (V b), (~) * (N a) (N b)) => Enveloped (a, b)

Methods

getEnvelope :: (a, b) -> Envelope (V (a, b)) (N (a, b))

Enveloped b => Enveloped (Map k b)

Methods

getEnvelope :: Map k b -> Envelope (V (Map k b)) (N (Map k b))

(OrderedField n, Metric v) => Enveloped (Point v n)

Methods

getEnvelope :: Point v n -> Envelope (V (Point v n)) (N (Point v n))

(Metric v, OrderedField n) => Enveloped (Envelope v n)

Methods

getEnvelope :: Envelope v n -> Envelope (V (Envelope v n)) (N (Envelope v n))

(OrderedField n, Metric v, Monoid' m) => Enveloped (Subdiagram b v n m)

Methods

getEnvelope :: Subdiagram b v n m -> Envelope (V (Subdiagram b v n m)) (N (Subdiagram b v n m))

(Metric v, OrderedField n, Monoid' m) => Enveloped (QDiagram b v n m)

Methods

getEnvelope :: QDiagram b v n m -> Envelope (V (QDiagram b v n m)) (N (QDiagram b v n m))

Utility functions

diameter :: (V a ~ v, N a ~ n, Enveloped a) => v n -> a -> n

Compute the diameter of a enveloped object along a particular vector. Returns zero for the empty envelope.

radius :: (V a ~ v, N a ~ n, Enveloped a) => v n -> a -> n

Compute the "radius" (1/2 the diameter) of an enveloped object along a particular vector.

extent :: (V a ~ v, N a ~ n, Enveloped a) => v n -> a -> Maybe (n, n)

Compute the range of an enveloped object along a certain direction. Returns a pair of scalars (lo,hi) such that the object extends from (lo *^ v) to (hi *^ v). Returns Nothing for objects with an empty envelope.

size :: (V a ~ v, N a ~ n, Enveloped a, HasBasis v) => a -> v n

The smallest positive axis-parallel vector that bounds the envelope of an object.

envelopeVMay :: Enveloped a => Vn a -> a -> Maybe (Vn a)

Compute the vector from the local origin to a separating hyperplane in the given direction, or Nothing for the empty envelope.

envelopeV :: Enveloped a => Vn a -> a -> Vn a

Compute the vector from the local origin to a separating hyperplane in the given direction. Returns the zero vector for the empty envelope.

envelopePMay :: (V a ~ v, N a ~ n, Enveloped a) => v n -> a -> Maybe (Point v n)

Compute the point on a separating hyperplane in the given direction, or Nothing for the empty envelope.

envelopeP :: (V a ~ v, N a ~ n, Enveloped a) => v n -> a -> Point v n

Compute the point on a separating hyperplane in the given direction. Returns the origin for the empty envelope.

envelopeSMay :: (V a ~ v, N a ~ n, Enveloped a) => v n -> a -> Maybe n

Equivalent to the norm of envelopeVMay:

 envelopeSMay v x == fmap norm (envelopeVMay v x)

(other than differences in rounding error)

Note that the envelopeVMay / envelopePMay functions above should be preferred, as this requires a call to norm. However, it is more efficient than calling norm on the results of those functions.

envelopeS :: (V a ~ v, N a ~ n, Enveloped a) => v n -> a -> n

Equivalent to the norm of envelopeV:

 envelopeS v x == norm (envelopeV v x)

(other than differences in rounding error)

Note that the envelopeV / envelopeP functions above should be preferred, as this requires a call to norm. However, it is more efficient than calling norm on the results of those functions.

Miscellaneous

type OrderedField s = (Floating s, Ord s)

When dealing with envelopes we often want scalars to be an ordered field (i.e. support all four arithmetic operations and be totally ordered) so we introduce this constraint as a convenient shorthand.