Maintainer | diagrams-discuss@googlegroups.com |
---|---|

Safe Haskell | None |

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

The `Envelope`

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

- newtype Envelope v = Envelope (Option (v -> Max (Scalar v)))
- appEnvelope :: Envelope v -> Maybe (v -> Scalar v)
- onEnvelope :: ((v -> Scalar v) -> v -> Scalar v) -> Envelope v -> Envelope v
- mkEnvelope :: (v -> Scalar v) -> Envelope v
- pointEnvelope :: (Fractional (Scalar v), InnerSpace v) => Point v -> Envelope v
- class (InnerSpace (V a), OrderedField (Scalar (V a))) => Enveloped a where
- getEnvelope :: a -> Envelope (V a)

- diameter :: Enveloped a => V a -> a -> Scalar (V a)
- radius :: Enveloped a => V a -> a -> Scalar (V a)
- envelopeVMay :: Enveloped a => V a -> a -> Maybe (V a)
- envelopeV :: Enveloped a => V a -> a -> V a
- envelopePMay :: Enveloped a => V a -> a -> Maybe (Point (V a))
- envelopeP :: Enveloped a => V a -> a -> Point (V a)
- envelopeSMay :: Enveloped a => V a -> a -> Maybe (Scalar (V a))
- envelopeS :: (Enveloped a, Num (Scalar (V a))) => V a -> a -> Scalar (V a)
- class (Fractional s, Floating s, Ord s, AdditiveGroup s) => OrderedField s

# Envelopes

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://www.cis.upenn.edu/~byorgey/pub/monoid-pearl.pdf; video: http://www.youtube.com/watch?v=X-8NCkD2vOw.

Action Name (Envelope v) | |

Show (Envelope v) | |

Ord (Scalar v) => Monoid (Envelope v) | |

Ord (Scalar v) => Semigroup (Envelope v) | |

(InnerSpace v, Fractional (Scalar v)) => HasOrigin (Envelope v) | The local origin of an envelope is the point with respect to
which bounding queries are made, |

(HasLinearMap v, InnerSpace v, Floating (Scalar v)) => Transformable (Envelope v) | |

(InnerSpace v, OrderedField (Scalar v)) => Enveloped (Envelope v) | |

(InnerSpace v, OrderedField (Scalar v)) => Juxtaposable (Envelope v) | |

(~ * (Scalar v) s, ~ * (Scalar v') s', ~ * s s') => Wrapped (Option (v -> Max s)) (Option (v' -> Max s')) (Envelope v) (Envelope v') | |

Wrapped (DUALTree (DownAnnots v) (UpAnnots b v m) () (QDiaLeaf b v m)) (DUALTree (DownAnnots v') (UpAnnots b' v' m') () (QDiaLeaf b' v' m')) (QDiagram b v m) (QDiagram b' v' m') |

appEnvelope :: Envelope v -> Maybe (v -> Scalar v)Source

mkEnvelope :: (v -> Scalar v) -> Envelope vSource

pointEnvelope :: (Fractional (Scalar v), InnerSpace v) => Point v -> Envelope vSource

Create an envelope for the given point.

class (InnerSpace (V a), OrderedField (Scalar (V a))) => Enveloped a whereSource

`Enveloped`

abstracts over things which have an envelope.

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

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.

Enveloped b => Enveloped [b] | |

Enveloped b => Enveloped (Set b) | |

(OrderedField (Scalar v), InnerSpace v) => Enveloped (Point v) | |

Enveloped t => Enveloped (TransInv t) | |

(InnerSpace v, OrderedField (Scalar v)) => Enveloped (Envelope v) | |

(Enveloped a, Enveloped b, ~ * (V a) (V b)) => Enveloped (a, b) | |

Enveloped b => Enveloped (Map k b) | |

(OrderedField (Scalar v), InnerSpace v, HasLinearMap v, Monoid' m) => Enveloped (Subdiagram b v m) | |

(HasLinearMap v, InnerSpace v, OrderedField (Scalar v), Monoid' m) => Enveloped (QDiagram b v m) |

# Utility functions

diameter :: Enveloped a => V a -> a -> Scalar (V a)Source

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

radius :: Enveloped a => V a -> a -> Scalar (V a)Source

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

envelopeVMay :: Enveloped a => V a -> a -> Maybe (V a)Source

Compute the vector from the local origin to a separating
hyperplane in the given direction, or `Nothing`

for the empty
envelope.

envelopeV :: Enveloped a => V a -> a -> V aSource

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

envelopePMay :: Enveloped a => V a -> a -> Maybe (Point (V a))Source

Compute the point on a separating hyperplane in the given
direction, or `Nothing`

for the empty envelope.

envelopeP :: Enveloped a => V a -> a -> Point (V a)Source

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

envelopeSMay :: Enveloped a => V a -> a -> Maybe (Scalar (V a))Source

Equivalent to the magnitude of `envelopeVMay`

:

envelopeSMay v x == fmap magnitude (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 magnitude. However, it is more
efficient than calling magnitude on the results of those functions.

envelopeS :: (Enveloped a, Num (Scalar (V a))) => V a -> a -> Scalar (V a)Source

Equivalent to the magnitude of `envelopeV`

:

envelopeS v x == magnitude (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 magnitude. However, it is more
efficient than calling magnitude on the results of those functions.

# Miscellaneous

class (Fractional s, Floating s, Ord s, AdditiveGroup s) => OrderedField s Source

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 class as a convenient shorthand.

(Fractional s, Floating s, Ord s, AdditiveGroup s) => OrderedField s |