A circle with infinite radius is a line

Question

I am curious about the following diagram:

The image implies a circle of infinite radius is a line. Intuitively, I understand this, but I was wondering whether this problem could be stated and proven formally? Under what definition of 'circle' and 'line' does this hold?

Thanks!

Answer 1

A circle of radius $r$ whose center is at $(r,0)$ has the parametric form $$ \begin{array}{}x=r(1-\cos(\theta/r))&y=r\sin(\theta/r)\end{array}\tag{1} $$ the limit of the curve in $(1)$ as $r\to\infty$ is $$ \begin{array}{}x=0&y=\theta\end{array}\tag{1} $$ which is the vertical line in your image.

Addendum:

In Inversive Geometry, circles and lines are considered the same. The inverse of a circle which passes through the center of the inversion is a line which doesn't pass through the center and vice-versa. The inverse of a line which passes through the origin is the line itself.

In the following image, the red and green circles are inverses with respect to the grey circle. Notice that when the red circle passes through the center of the inversion, the green circle becomes a line.

Answer 2

If you take the equation of a circle tangent to the vertical axis to be

$$(x-r)^2+y^2=r^2$$

or

$$\frac{x^2+y^2}{r}-2x=0$$

and let $r\to \infty$, you find that you obtain the equation $x=0$, which is precisely the vertical axis...

Answer 3

Well, this is true in the ordinary Euclidean plane. It is false in the ordinary hyperbolic plane. The "limit" of circles going through a common point, as radius grows without bound, is a horocycle. Sometimes this is spelled horocircle.

In your image above, there would be two horocycles tangent to your line at the point $C$, one on either side of the line.

Anyway, there are pages of stuff to be described about this. Maybe I should just say that there is an "intrinsic" axiomatic definition of a hyperbolic plane, modernized by David Hilbert. You are likely to first come across some of the models, the most popular being the Poincaré disk, the Poincaré upper half plane, the Beltrami-Klein model, the one-sheet hyperboloid model in Minkowski 3-space. The first two and the fourth are linked to in the wikipedia link.

Answer 4

There is no such thing as a circle of infinite radius. One might find it useful to use the phrase "circle of infinite radius" as shorthand for some limiting case of a family of circles of increasing radius, and (as the other answers show) that limit might give you a straight line.

Answer 5

Algebraic description

A generic algebraic curve of degree $2$ is the set of points satisfying $$ax^2 + by^2 + cxy + dx + ey + f = 0$$ You can define circles to be those curves with $a=b$ and $c=0$. You can compute the radius as $$r=\sqrt{\frac{d^2 + e^2}{4a^2} - f}$$ For the case of $a=b=0$ your equation describes a line, and your radius becomes infinite due to the division by zero. Note that the definition also includes circles with imaginary radius, which have no points in the real plane. You may exclude these by restricting the range of parameters.

Complex plane and cross ratios

Interpreting the points of your plane as complex numbers, you can define that four points lie on a common circle or line if their cross ratio is a real number. Using this definition, a line is just a circle, and the only reasonable value to assign to its radius is $\infty$.

Möbius Geometry

A Möbius transformation will map circles and lines in the complex plane (or complex line, depending on your use of the word) $\mathbb C$ to other circles and lines. So the above definition of a circle will suit these transformations as well. There is a scientific topic called Möbius Geometry which uses 4-dimenstional vectors to describe both lines and circles. In this setup again a line is a special case of circle, and performing any radius computation will lead to infinity.

You may extend Möbius geometry a bit further to obtain Lie geometry, where even points are special cases of circles, namely those with zero radius. Lie transformations may transform generic circles to lines or points and vice versa.

Constant curvature

You may define a circle as a line of constant curvature, and without endpoints. You may imagine curvature as the inverse of the radius, but you can define it in other ways as well. Obviously, for the case of zero curvature and therefore infinite radius, you will obtain a line.

Projective geometry

You can define a circle to be a conic incident with the two complex circle points $$\begin{pmatrix}1\\i\\0\end{pmatrix} \text{ and } \begin{pmatrix}1\\-i\\0\end{pmatrix}$$ where the points are given in homogenous coordinates in $\mathbb{CP}^2$. There are conics with real coefficients which degenerate into a pair of lines. One of them is the line at infinity incident with the two points above, and the other may be a finite line. So unless your conic degenerates to a double line at infinity, you get a single line as the finite portion of a conic which by definition is a circle. You can compute the center of that circle using the points above, and the result will be an infinite point, indicating the infinite radius.