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.

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

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

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

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.

Enveloped abstracts over things which have an envelope.

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

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

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.

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

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

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

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

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

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.

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.

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.