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data-structuresfunctional-programmingschemephysicscalculus

What is an "up" or "down" structure in Scheme?

Very early on in Functional Differential Geometry, Sussman & Wisdom start using an "up structure"... but I haven't the slightest idea what this could be.

(print-expression
  ((compose P2-chi R2-chi-inverse)
  (up ’x0 ’y0)))

I cannot find the description of this structure anywhere in the text, and I cannot find it in a standard version of Scheme or the language documentation... so I'm wondering what exactly these "up structure" and "down structure" things are. I get that they correspond to the derivative and the integral in basic calculus. Just haven't the slightest idea how they're put together in Scheme.

Solution

  • From the scmutils reference manual:1

    We identify the Scheme vector data type with mathematical n-dimensional vectors. These are interpreted as up tuples when a distinction between up tuples and down tuples is made. We inherit from Scheme the constructors VECTOR and MAKE-VECTOR, the selectors VECTOR-LENGTH and VECTOR-REF, and zero-based indexing.

    And, I think, the mathematical interpretation is: Covariance and contravariance of vectors.

    1 (go to Up Tuples and Down Tuples section and scroll down to multiplication explanation, to see what it is all about).