## Shape and Representation (Part 1)

Let R^N be N-dimensional euclidian space.

How to think about 2-dimensional shape representation?

Clearly (R^2)^R (the set of all curves in R^2) is the maximal representation. But that’s so fucking huge. Let’s look at smaller representative spaces:

R^3 can be used to describe all circles (center + radius).

{0, 1} x R^4 can be used to describe all circles and squares as follows: if first coordinate == 0, then let the first 3 coordinates of the 4-D space on the right define the circle position, else let the first two coordinates describe one corner of the square and let the latter two describe the other corner.

E.g.

{0, 1, 1, 1, 0} -> circle at (1,1) with radius = 1

{1, 1, 1, 2, 2} -> square with one corner at (1,1) and other at (2, 2)

If we bulk up to {0, 1, 2} x R^6 we can get rectangles (since they require us to define 3 points).

Let’s refer to the set on the left as the “index set” and the set on the right as the “parameter space.”

We can keep going, adding all the “types” of shapes that we’re familiar with – rhombuses, ellipses/conics in general, various curves. Perhaps the index set will grow somewhat large and the parameter space, too, before we reach the end of “types” of shapes.

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Having a finite parameter space and a finite-dimensional parameter space is fairly restrictive, however.

For instance, suppose we want to describe regular n-gons. These are nice because they maximally require 6 parameters (2 points on a line segment and an “interior” point) to describe. However, they require a countable set of indices.

For an example where finite-dimensional parameter space is restrictive, consider the set of arbitrary convex polygons. We don’t want to limit the number of vertices unnecessarily.

In light of this, let our standard space for shape representations be Z x R^Z.

In some sense, the index set is a set of pointers to “programs” which use the parameter space as “inputs.”

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Given this, the question naturally arises as to what is the “best” scheme for associating an element of Z x R^Z with a geometric shape.

## Maximal Cat

There exists a cat who is bright-eyed and simple.

He waits in a forest,

Knowing where are the bright streams,

Smelling if the snow approaches,

His paws picking mice out from deeper than the brush.

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