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.

Enveloped abstracts over things which have an envelope.

Get the envelope of a diagram.

Replace the envelope of a diagram.

Use the envelope from some object as the envelope for a diagram, in place of the diagram's default envelope.

sqNewEnv =
    circle 1 # fc green
    |||
    (    c # dashing [0.1,0.1] 0 # lc white
      <> square 2 # withEnvelope (c :: D R2) # fc blue
    )
c = circle 0.8
withEnvelopeEx = sqNewEnv # centerXY # pad 1.5

phantom x produces a "phantom" diagram, which has the same envelope and trace as x but produces no output.

pad s "pads" a diagram, expanding its envelope by a factor of s (factors between 0 and 1 can be used to shrink the envelope). Note that the envelope will expand with respect to the local origin, so if the origin is not centered the padding may appear "uneven". If this is not desired, the origin can be centered (using, e.g., centerXY for 2D diagrams) before applying pad.

extrudeEnvelope v d asymmetrically "extrudes" the envelope of a diagram in the given direction. All parts of the envelope within 90 degrees of this direction are modified, offset outwards by the magnitude of the vector.

This works by offsetting the envelope distance proportionally to the cosine of the difference in angle, and leaving it unchanged when this factor is negative.

intrudeEnvelope v d asymmetrically "intrudes" the envelope of a diagram away from the given direction. All parts of the envelope within 90 degrees of this direction are modified, offset inwards by the magnitude of the vector.

Note that this could create strange inverted envelopes, where diameter v d < 0 .

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.

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.