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#### rashtastic

##### New member

- Jan 3, 2013

- 11

We can identify a circle with three points on the circle, or six parameters $(x_1,y_1,x_2,y_2,x_3,y_3)$

where, keeping order, every circle can be uniquely identified with six reals. So the above tuple is a point in the set of circles on the plane, encoded in a certain way.

We can do this more efficiently by using the center and the radius, $(x_1,y_1,r)$, bringing us down to three parameters to uniquely identify any circle. Can we go any further?

Defining a circle in terms of real parameters involves finding a parameter set $\mathcal{B}=\{(\theta_1,...,\theta_n)\}$ where the set of all circles $\mathcal{C}$ can be mapped injectively into $\mathcal{B}$. We have done this for $\mathcal{B}$ with order 6 and 3. Can we do it for order 2?

Yes... ignoring decimal points and negatives for the moment, we can take the center coordinates and make a new number from interpolating the center coordinates' digits. So (11111,22222) can be (1212121212). This is quite silly but allows us to define an injective function from $\mathcal{C}$ into $\mathcal{B}$ where $\mathcal{B}$ has order 2- one parameter for the combined center, one for the radius. Again, given that you produce rules for decimals, you can then combine the same thing for the radius and create a single parameter that can be used to uniquely identify a circle. Or a single parameter can identify any conic section, or a list of conic sections, and so on. If five points is sufficient to identify a conic section, then ten parameters (two reals per point) can be uniquely mapped into a single real that identifies the conic. Consider that $\mathcal{R}^{10}\sim\mathcal{R}$.

You might notice that somewhere along this line we have basically lost touch with reality.

Can you think of a good argument for why $(x_1,y_1,r)$ is the most efficient way to uniquely express any circle

*in a way that is geometrically sensible*?

Or can you find a two-parameter solution (reals only) that can uniquely identify any circle in a reasonable way?