What is the smallest possible amount of data to describe a polygon?

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ihaveaquston

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Hello,
what is the smallest possible amount of data to describe a polygon? I wonder if there is any theorem that could help answering my question. As describing I mean giving enough information to accurately recreate the polygon. As amount of data I mean a number of given values and each value can represent one of the following:
- length of a specific side
- measure of a specific angle
For triangles the number would be 3 (for example lengths of each side), but for a quadrangle you would need 6, I guess (for example 4 side lengths and 2 angles). How a formula for minimum number of values depending on numer of sides looks? How does it change if we know that a polygon is convex?
 
ihaveaquston said:
Hello,
what is the smallest possible amount of data to describe a polygon? I wonder if there is any theorem that could help answering my question. As describing I mean giving enough information to accurately recreate the polygon. As amount of data I mean a number of given values and each value can represent one of the following:
- length of a specific side
- measure of a specific angle
For triangles the number would be 3 (for example lengths of each side), but for a quadrangle you would need 6, I guess (for example 4 side lengths and 2 angles). How a formula for minimum number of values depending on numer of sides looks? How does it change if we know that a polygon is convex?
Click to expand...
For a quadrilateral polygon to be uniquely defined, we need five (5) data points.

I think we need (2n - 3) data points where n is the #of sides.
 
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I agree with 2n-3, if we know it's convex.

Sticking with quadrilaterals, here is ABCD specified by four lengths (blue); since A'BCD' is noncongruent, clearly 4 facts are not enough:

1622426631736.png

Adding an angle as the fifth fact, it looks good:

1622426691351.png

But if we allow it to be nonconvex, there is another noncongruent figure with the same data:

1622426746426.png

So with n=4, we actually need 5 values plus an extra bit. So the suggestion of 6 values is correct, but a little more than is really needed.

I suspect that with more sides, there may be more such extra bits needed if convexity is not required.
 
Perhaps a sequence of length followed by (internal) angle, 2n-3 data. (Final side, 2 angles of final side omitted).
 
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