Circles, Semicircles, and Choosing a Branch

Definition Let $(h, k)$ be a point in the plane and $r > 0$ a real number. The circle with center $(h, k)$ and radius $r$ is the set of all points $(x, y)$ whose distance from $(h, k)$ is exactly $r$. The center itself is not on the circle, and the points inside are not on it either.

Definition The distance between points $(x_1, y_1)$ and $(x_2, y_2)$ is

It comes from the Pythagorean theorem applied to the right triangle whose legs are the horizontal distance $\lvert x_2 - x_1 \rvert$ and the vertical distance $\lvert y_2 - y_1 \rvert$, and whose hypotenuse is the segment between the points.

Definition A point $(x, y)$ lies on the circle with center $(h, k)$ and radius $r > 0$ exactly when

This is the distance condition with both sides squared, which is reversible here because both the distance and $r$ are nonnegative. The equation describes the curve implicitly, as a relation between $x$ and $y$ rather than as a formula for $y$.

Definition For a real number $a \ge 0$, the principal square root $\sqrt{a}$ is the unique real number $b \ge 0$ with $b^2 = a$. For $a < 0$ there is no real square root. The equation $u^2 = a$ with $a > 0$ has the two solutions $u = \sqrt{a}$ and $u = -\sqrt{a}$, and in general $\sqrt{u^2} = \lvert u \rvert$.

Definition Solving the circle equation for $y$ gives $(y - k)^2 = r^2 - (x - h)^2$, so

Each choice of sign gives a function, called a branch of the circle. The radical is a nonnegative distance from the horizontal line $y = k$, and the sign in front of it chooses the side. The complete circle with $r > 0$ is not the graph of a function of $x$, since the vertical line $x = h$ meets it at the two points $(h, k + r)$ and $(h, k - r)$.

Definition The upper semicircle of the circle with center $(h, k)$ and radius $r > 0$ is the graph of

It is the part of the circle with $y \ge k$, including the two endpoints $(h - r, k)$ and $(h + r, k)$.

Definition The lower semicircle of the same circle is the graph of

It is the part of the circle with $y \le k$, and it shares the two endpoints $(h \pm r, k)$ with the upper semicircle.

Definition Both branches have the closed domain $[h - r, h + r]$. The radical is real exactly when $r^2 - (x - h)^2 \ge 0$, that is when $\lvert x - h \rvert \le r$. The endpoints are included, because at $x = h \pm r$ the radicand is $0$ and the root is $0$.

Definition As $x$ runs over $[h - r, h + r]$, the radical $\sqrt{r^2 - (x - h)^2}$ takes every value from $0$, at the two endpoints, up to $r$, at $x = h$. Hence the upper branch has range $[k, k + r]$ and the lower branch has range $[k - r, k]$. Both ranges are closed intervals.