What Is Circle?
Circle is the locus of points equidistant from a given point, the center of the circle. The common distance from the center of the circle to its points is called radius. Thus a circle is completely defined by its center (O) and radius (R):
C(O, R) = O(R) = {x: dist(O, x) = R}.
This definition that appears so natural to a 21^{st} century reasonably educated person is markedly different from the one given by Euclid (Definition I.15). In Sir Heath's translation:
A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.
And in the translation by Robert Simson:
A circle is a plane figure contained by one line, which is called circumference, and is such, that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.
As an extra, comes Definition I.16:
And the point is called the center of the circle.
What is so different between our definition and Euclid's? Strange as it may seem, Euclid did not have a notion of metric or a distance function. Nowhere in the Elements he refers to a distance between two points, but only to the length of the line segment joining the two. The whole of geometry has been developed without the idea of a 2ddistance! And so it does not seem to be necessary. The definition we gave which is the most common nowadays is the outgrowth of the culture permeated by analytic geometry. Ironically, in the modern terminology being Euclidean exactly means being in possession of a suitable metric. Euclid might have objected.
Now, returning to the 21^{st} century, our definition is meaningful in any metric space. For some metrics the circles may look rather square. However, a circle looks like a circle in Euclidean geometry. Although the Euclidean circle has much to recommend it for, long, long ago Piet Hein has discovered that circles in the metric
dist((x_{1}, y_{1}), (x_{2}, y_{2})) = ((x_{1}  x_{2})^{5/2} + (y_{1}  y_{2})^{5/2})^{2/5}
are more pleasant to the eye than the Euclidean ones.
For square circles, it is relatively simple to define their length as the sum of lengths of the constituent sides. The length of the Euclidean (ordinary) circle may be approximately measured to any precision with a variety of means. A rigorous definition, nonetheless, requires a theory of limits  the fundamental notion in Calculus. The length of a circle is usually called its circumference and is equal 2πR, where R is the radius of the circle. The area enclosed by the circle can also be computed with a simple formula: πR².
In any affine space endowed with a metric, all circles are similar. Which does not mean there is similarity between circles in various metric spaces. But which does mean that, however defined, the ratio of the circumference to the radius of circles in a given metric space is constant. It may vary between spaces, though. In Euclidean space, the ratio equals 2p, named so by L. Euler.
The most common tool for constructing (Euclidean) circles is the venerable compass. But there are of course other ways. The inversion can be used to convert circular motion into rectilinear and vice versa. A circle is also traced by the point common to two lighthouse beams rotating at the same angular speed. The midpoint of a ladder sliding down a wall also traces a circle. Circle is also the locus of points having the same ratio of distances to a pair of points. Circle (a part of it actually) is also the locus of points from which a given line segment is seen under a given angle.
The line joining the center of the circle to one of its points is often called the radiusvector of that point, or just one of the radii. Likewise, the word diameter is used in two different, but close meanings. On one hand, diameter is the largest distance between any two points on a circle. As such, it equals twice the radius of the circle. For a given circle, there is exactly one and only one diameter. On the other hand, the word "diameter" designates a line segment joining a point on the circle with its opposite, i.e. the point farthest away from the point at hand. For every point on a circle there is exactly one diameter that contains this point. The length of a diameter, as a line segment, is exactly the diameter, as a number. Any diameter passes through the center of the circle and is divided by the center into two equal halves, each the length of the radius.
In Cartesian coordinates, circle is defined by a second degree equation:
(x  x_{0})^{2} + (y  y_{0})^{2} = R^{2},
where (x_{0}, y_{0}) is the center of the circle and R its radius. The polar coordinates of a circle centered at the origin are particularly simple:
r = R.
A segment of a line joining two points on a circle is called a chord; a piece of a circle between two points is an arc. A chord that passes through the center of the circle is (ambiguously) called a diameter. Diameter is the longest chord in a circle. A shape bounded by an arc and a chord with the same end points is a segment. A central angle is formed by two radiusvectors. A central angle cuts from a circle a sector. A straight line that has only one point in common with a circle is tangent to the circle. Other than tangents, straight lines cross a circle either in no points or in two points. The lines of the latter variety are called secants.
Two secants that meet on a circle define an inscribed angle. Two radii define a central angle.
Two intersecting chord are divided by the point of intersection into two parts each such that the products of their lengths are equal. This is known as the Intersecting Chords Theorem. The latter carries on to the intersecting secants and gives rise to the important notion of the power of a point with respect to a circle.
Triangles always have one inscribed and one circumscribed circle. Three more circles (excircles) touch the triangle's sidelines. Many other special circles are associated with a triangle: 9Point circle, Taylor circle, Tucker circles, Adams' circle, to name a few. Archimedes has discovered his Twin Circles in the wonderful arbelos.
Some quadrilaterals have inscribed circles, and these are called inscriptible. Other quadrilaterals have concyclic vertices, and these are called circumscriptible, or cyclic. Those quadrilaterals that have both inscribed and circumscribed circles are said to be bicentric.
Surprisingly, in many respects circles behave like straight lines. Nowehere it is more apparent than in inversive geometry.
An Aside
The famous French mathematician Alexandre Grothendieck  a 1966 Fields medalist  has an uncommonly tragic life story [Ruelle]: growing up in a concentration camp and later being an outsider in the French mathematics establishment. He was one of the first permanent professors at the Institut des Hautes Études Scientifiques, a position which he resigned in protest of the institute being partially funded by military grants. Years later he applied for a position at Le Centre National de la Recherche Scientifique. As an application, he submitted Esquisse d'un Programme (Sketch of a Program, in English translation), where he summarized his past activity and outlined his plans for future research. He described one of his results, namely (p 15 in the original, p 246 in the translation),
Every "finite" oriented map is canonically realised on a complex algebraic curve.
A few lines later he wrote,
This discovery, which is technically so simple, made a very strong impression on me, and it represents a decisive turning point in the course of my reflections, a shift in particular of my centre of interest in mathematics, which suddenly found itself strongly focused. I do not believe that a mathematical fact has ever struck me quite so strongly as this one, nor had a comparable psychological impact.
This was accompanied with a foot note (p 44 in the original, p 274 in the translation):
With the exception of another "fact", at the time when, around the age of twelve, I was interned in the concentration camp of Rieucros (near Mende). It is there that I learnt, from another prisoner, Maria, who gave me free private lessons, the definition of the circle. It impressed me by its simplicity and its evidence, whereas the property of "perfect rotundity" of the circle previously had appeared to me as a reality mysterious beyond words. It is at that moment, I believe, that I glimpsed for the first time (without of course formulating it to myself in these terms) the creative power of a "good" mathematical definition, of a formulation which describes the essence. Still today, it seems that the fascination which this power exercised on me has lost nothing of its force.
(My sincere thanks go to V. Gutenmacher who brought the quote to my attention and to Ethan Duckworth for suggesting a more complete reference.)
References
 T. L. Heath, Euclid. The Thirteen Books of the Elements, vol 1, Dover, 1956
 D. Ruelle, The Mathematician's Brain: A Personal Tour Through the Essentials of Mathematics and Some of the Great Minds Behind Them , Princeton University Press, 2007
 Robert Simson, The Elements of Euclid, Elibron Classics, 2005
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