Poncelet–Steiner theorem

thumb|Construction of a parallel (h) to a diameter g through any given point P, using only a straightedge

In Euclidean geometry, the Poncelet–Steiner theorem is a result about compass and straightedge constructions with certain restrictions. This result states that whatever can be constructed by straightedge and compass together can be constructed by straightedge alone, provided that a single circle and its centre are given.

This shows that, while a compass can make constructions easier, it is no longer needed once the first circle has been drawn. All constructions thereafter can be performed using only the straightedge, although the arcs of circles themselves cannot be drawn without the compass. This means the compass may be used for aesthetic purposes, but it is not required for the construction itself.

History

thumb|right|350px|"The geometrical constructions, carried out using the straight line and a fixed circle, as a subject of teaching at higher educational institutions and for practical use; by Jacob Steiner, Doctor of philosophy, Royal Prussian professor and distinguished teacher of mathematics at the commercial school in Berlin. With two copper plaques. Berlin, with [[Ferdinand_Dümmler_(publisher)|Ferdinand Dummler. 1833."]]

In the tenth century, the Persian mathematician Abu al-Wafa' Buzjani (940−998) considered geometric constructions using a straightedge and a compass with a fixed opening, a so-called rusty compass. Constructions of this type appeared to have some practical significance as they were used by artists Leonardo da Vinci and Albrecht Dürer in Europe in the late fifteenth century. A new viewpoint developed in the mid sixteenth century when the size of the opening was considered fixed but arbitrary and the question of how many of Euclid's constructions could be obtained was paramount.

Renaissance mathematician Lodovico Ferrari, a student of Gerolamo Cardano in a "mathematical challenge" against Niccolò Fontana Tartaglia was able to show that "all of Euclid" (that is, the straightedge and compass constructions in the first six books of Euclid's Elements) could be accomplished with a straightedge and rusty compass. Within ten years additional sets of solutions were obtained by Cardano, Tartaglia and Tartaglia's student Benedetti. During the next century these solutions were generally forgotten until, in 1673, Georg Mohr published (anonymously and in Dutch) Euclidis Curiosi containing his own solutions. Mohr had only heard about the existence of the earlier results and this led him to work on the problem.

Showing that "all of Euclid" could be performed with straightedge and rusty compass is not the same as proving that all straightedge and compass constructions could be done with a straightedge and just a rusty compass. Such a proof would require the formalization of what a straightedge and compass could construct. This groundwork was provided by Jean Victor Poncelet in 1822, having been motivated by Mohr's work on the Mohr–Mascheroni theorem. He also conjectured and suggested a possible proof that a straightedge and rusty compass would be equivalent to a straightedge and compass, and moreover, the rusty compass need only be used once. The result of this theorem, that a straightedge and single circle with given centre is equivalent to a straightedge and compass was proved by Jakob Steiner in 1833. The Poncelet–Steiner theorem covers a particular subset of Steiner constructions: those in which a fixed circle and its center are present on the plane. In this sense, all constructions adhering to the Poncelet–Steiner theorem are Steiner constructions, though not all Steiner constructions abide by the same restrictions.

Rusty compass

The rusty compass describes a compass whose distance is fixed — its hinge is so rusted that its legs are unable to adjust width. Circles may be drawn centered at any arbitrary point, but the radius is unchangeable. Historically, it was shown that all Euclid constructions can be performed with a rusty compass and straightedge. The Poncelet–Steiner theorem generalizes this further, showing that a single arbitrarily placed circle with its center is sufficient to replace all further use of the compass.

Constructive proof

Outline

To prove the Poncelet–Steiner theorem, it suffices to show that each of the basic constructions of compass and straightedge is possible using a straightedge alone (provided that a circle and its center exist in the plane), as these are the foundations of all other constructions. All constructions can be written as a series of steps involving these five basic constructions:

thumb|right|400px|The basic constructions 1 through 5 illustrated, from left to right. The top row being the information given, the bottom row being the desired construction; red indicating the newer information.

Creating the line through two existing points.
Creating the circle through one point with centre another point.
Creating the point which is the intersection of two existing, non-parallel lines.
Creating the one or two points in the intersection of a line and a circle (if they intersect).
Creating the one or two points in the intersection of two circles (if they intersect).

Constructions (1) and (3) can be done with a straightedge alone. For construction (2), a circle is considered to be given by any two points, one defining the center and one existing on the circumference at radius. It is understood that the arc of a circle cannot be drawn without a compass, so the proof of the theorem lies in showing that constructions (4) and (5) are possible using only a straightedge, along with a fixed given circle and its center. Once this is done, it follows that every compass-straightedge construction can be done under the restrictions of the theorem.

The following proof is based on the one given by Howard Eves in 1963.

As a result, a single circle without its center is not sufficient to perform general straightedge-compass constructions. Consequently, the requirements on the Poncelet–Steiner theorem cannot be weakened with respect to the circle center.

However, the center of a circle may be reconstructed as long as sufficient additional information is given on the plane. In each of the following scenarios, it becomes possible to recover the center of a circle, and therefore making every straightedge-compass construction possible:

one circle and two distinct sets of parallel lines
one circle and three parallel lines equidistant from each other
two intersecting or tangent circles
two concentric circles or on their radical axis
two circles and a single set of parallel lines
two circles that can inscribe and circumscribe a bicentric polygon with an even number of sides
three non-intersecting circles not all in the same coaxial system

Given only two circles without their centers, it is generally not possible to construct their centers using only a straightedge. However, in certain special cases, it is possible, such as when the two circles intersect or are concentric.

Poncelet–Steiner without a complete circular arc

In 1904, Francesco Severi proved that any small arc (of the circle), together with the centre, will suffice. Severi's proof illustrates that any arc of the circle fully characterizes the circumference and allows intersection points (of lines) with it to be found, regardless of the absence of some portion of the completed arc. Consequently, the completeness of the circle is not essential, provided an arc and the center are available.

Further generalizations

The Poncelet–Steiner theorem has been generalized to higher dimensions, such as, for example, a three-dimensional variation where the straightedge is replaced with a plane, and the compass is replaced with a sphere. It has been shown that n-dimensional "straightedge and compass" constructions can still be performed using only n-dimensional planes, provided that a single n-dimensional sphere with its center is given. Many of the properties that apply to the two dimensional case also apply to higher dimensions, as implementations of projective geometry.

Additionally, some research is underway to generalize the Poncelet–Steiner theorem to non-Euclidean geometries.

Notes

References

External links

Jacob Steiner's theorem at cut-the-knot (It is impossible to find the center of a given circle with the straightedge alone)
Two circles and only a straightedge, an article by Arseniy Akopyan and Roman Fedorov
A remark on the construction of the centre of a circle by means of the ruler, by Christian Gram
Hilbert's Error, an article by Alexander Shen