Does any sort of formal way to measure the amount of 'information' exist? We need 3 pieces of information for defining a circle. Basically we need to restrict 3 degrees of freedom, I think. By then with just a radius and centre, we can fully describe the circle. Where have I gonw wrong?
I've observed this property in a lot of math that to describe something you usually need a fixed amount of 'information' in my subjective opinion.
Someone more advanced can maybe help me out.
 A: My first thought is that the difference comes from the fact that you can specify the same circle with many different choices of 3 points, so that moving from those three points to the circle loses information as you can't recover the chosen points with just the circle.
I'd guess that the formalisation of this idea is in manifold theory (the study of "$n$-dimensional surfaces"). It would go something like (assuming you're considering circles in $\mathbb{R}^2$): there's the 6-dimensional space of all choices of 3 points in $\mathbb{R}^2$, and each possible circle is represented in that space by a $3$-dimensional submanifold (maybe parameterising each choice of points by the three angles between each point-to-centre line to the $x$-direction). Then quotienting out the 6-D space we started by this 3-dimensional relation leaves us with a 3-D manifold of all possible circles. This is exactly what you get with the other construction, where each circle is given by a 2-dimensional coordinate for the centre plus an extra real number for the radius.
(This is just a sketch, of course. There's still technicalities that need to be checked, like how to deal with choices of three points that coincide.)
