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THE DOCUMENT

• code-able definitions
• minimal (for agreement)
• task-based

goals

• understand structure (why) of OCM, OCD, and other metrics
• propose new metrics

cool new things we can do now:

• construct only the possible contours
• construct OCDs by basis normal form

big questions

• what happens as L increases? as n increases? w/r/t direction and magnitude?

less big questions

• given two morphs, what is OCD
• given a morph all OCDs a given distance
• given morph distribution of OCDs
• ...etc

TERMS

morph

• the values

contour

• differences between values
  • (assume combinatorial for now)

spaces

• morph-space
• contour-space
• basis-space

normal form

morph normal form

morph normal form is a convention for representing morphs (in morph form) that have the same contour (are equivalent under some contour relation). it allows us to identify morphs that have the same contour w/o converting to contour form. this is useful when enumerating morphs (in morph form) that are unique under some contour relation. the criteria change according to the contour relation:

a morph is in normal form (under ternary contour) iff:

1. zeroed: minimum element is 0
2. compact: number of unique elements = max - min (+1)

to find the normal form of a morph (under ternary contour):

1. zero: move minimum elem to 0
2. compact: compress so number of unique elements = max - min (+1)

• ex use: filter for normal form as generate all morphs of a given L
• how to generate just the normal forms (w/o filtering) (https://github.com/cardinalaleph/morphometrics/issues/1)
• a way of enumerating all morphs of a given normal form [more implementation...]
• extend to n-ary and beyond

basis normal form

basis normal form is a convention for representing basis contours (in basis form) that are equivalent under some contour relation. it allows us to identify basis contours that are equivalent under some contour relation w/o converting contour form. this is useful when enumerating basis contours (in basis form) that are equivalent under some contour relation. the basis normal form produces the "stellated form." the criteria change according to the contour relation:

important idea: basis space cannot explicitly represent every contour under a squashed contour relation (-> language). for example, basis space cannot represent all ternary contours, but each point has a ternary contour, and there exists a point corresponding to each ternary contour. (-> example to illustrate?)

a basis contour is in normal form (under ternary contour) iff:

• for n != [-1,0,1]: (all combinations of +/- 1s and 0s are OK)
  • if we have n (positive), then we have n-1
  • if we have n (negative), then we have n+1

to find the normal form of a basis contour (under ternary contour):

1. do some stuff
2. do some more stuff

sta: bounds of L is L-1 (-> proof)

• prove this (we have a numerical verification for ternary)
• ex use: enumerate unique ternary contour
• how to generate just the normal forms (w/o filtering)
• todo: a way of enumerating all basis of a given normal form [more implementation]
• extend to n-ary and beyond

basis

• construction
• change of basis
• intuition
• terms basis have rank (number of rows) and length (number of cols)

direction

magnitude

n-ary

stellated