Turning an irregular shape into regular

[R C-R]

1 hour ago, Alfred said:

But what is a circle, if not a regular polygon with an infinite number of sides? 

A circle is an idealized geometric abstraction, the collection of all points equidistant from a common point, but a point is not the same thing as a side -- a point exists in one dimension while a side must exist in two. That's why a collection of 2D Bezier curves, even if they are infinitely many of them, can only approximate a perfect Euclidean circle.

 

As the Cut The Knot page says, any rigorous definition of a circle involves some theory of limits, which in more practical terms means our vector shapes can be perfect only to the extent resolution independence can be maintained. So while this is not a problem in the abstract vector (metric?) spaces of AD & similar apps, rendering those shapes with real world devices that are inherently limited to finite resolutions will always be to some extent problematic.

 

In the final analysis, it all reduces to the same thing each of us has been saying in one way or another: perfection is unattainable so it is all about how much imperfection we can tolerate.

[Alfred]

1 hour ago, R C-R said:

a point is not the same thing as a side -- a point exists in one dimension while a side must exist in two

 

You mentioned limits in your next paragraph. The limit, as the number of sides tends to infinity, of the length of each side is zero, surely?

 

[dutchshader]

Going round in circles now, think i reached my limit.

[R C-R]

1 minute ago, Alfred said:

The limit, as the number of sides tends to infinity, of the length of each side is zero, surely?

Yes, but that limit can never actually be reached, which is why contemporary mathematical definitions of limits now rely on topological concepts like "nearness" & "neighborhoods." This is necessary because "infinity" is not a number or a metric, it is the idea that something is without end.