User:Jacobolus/Unit-diameter circle


 * This article is a draft, not yet suitable for inclusion in the Wikipedia main namespace.

A unit-diameter circle is a circle whose diameter is equal to $1$. It is a generalization of the endpoints of a unit interval, and is sometimes more convenient than the circle of unit radius (the unit circle).

Core circle
In the Cartesian plane, the circle whose diameter is the unit interval on the $x$-axis consists of solutions to the implicit equation


 * $$x^2 + y^2 = x.$$

This circle is the inversion of the line $$x=1$$ across the unit circle $$x^2 + y^2 = 1.$$ If the line is parametrized by its vertical coordinate $$t \mapsto (1, t)$$, then the same $t$ can be used to parametrize the unit-diameter circle:


 * $$t \mapsto \left(\frac1{1+t^2}, \frac{t}{1+t^2} \right).$$

In polar coordinates this circle can be expressed as


 * $$r^2 = \cos \theta.$$

This is a "clover".

Norman Wildberger calls it the core circle.