Succinct Trig-Free Parametric Circles

Posted on Sep 14, 2021

Introductory Remarks

Recently, I've been studying asymmetric elliptical cryptography so that I can implement a bitcoin client that can read and engage with the bitcoin blockchain.

I'm also studying ECC to better enable me to reason about an election system that uses public and private keys, and signs hashed votes with private keys.

The Description

I came across the following description of a circle when working through this material.

\[ (\frac {1 - t^2}{1 + t^2}, \frac {2t}{1 + t^2}) \]

Let's be more precise and sprinkle in some set notation:

\[ \{ x = \frac{1 - t^2}{1 + t^2}, y = \frac{2t}{1 + t^2}, x^2 + y^2 = 1 \vert x,y,t \in \R \} \]

The original text I linked to had \( x,y,t \in \mathbf{Q} \) , but I decided to lift the restriction of rationals so that we can have a circle that continuous in \( \R^2\), which is what most people are used to when they think of circles.

I don't have much more time to type out all my thoughts, but the attentive reader will quickly discover that the circle doesn't fully close. In fact, the circle only closes as \( t \) approaches both infinity and minus infinity.

This initially surprised me, but perhaps I can provide some intuition for this observed behavior.

Let's rewrite the x-component of EQ2.

\[ x = \frac{1}{1 + t^2} - \frac{t^2}{1 + t^2} = f_1(t) + f_2(t)\]

\[f_1(t) = \frac{1}{1 + t^2} \]

\[f_2(t) = - \frac{t^2}{1 + t^2}\]

And apply some limits to \( f_1(t) \):

\[ \lim_{t \to \infty} f_1(t) = 0\]

And use l'hopitals rule to reveal the limits on \( f_2(t) \):

\[ \lim_{t \to \infty} f_2(t) = 1\]

We conclude:

\[ \lim_{t \to \infty} x(t) = -1\]

Closing Remarks

More extensive evaluation of the limits on \( x(t) \) and \( y(t) \) for \( t \to \infty \) and \( t \to -\infty \) should reveal that there is indeed a gap at \( (-1, 0) \), although I personally haven't verified this.

This is no surprise given that this parameterization of a circle arose from drawing lines from \( (-1, 0) \) to all points on the unit circle.