Poncelet–Steiner theorem
In the branch of mathematics known as Euclidean geometry, the Poncelet–Steiner theorem is one of several results concerning compass and straightedge constructions having additional restrictions imposed on the traditional rules.
In other words, the compass may be used after all of the key points are determined, in order to "fill-in" the arcs purely for visual or artistic purposes, if it is desirable, and not as a necessary step toward construction.
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.
As a restricted construction paradigm, no stipulations are made about what geometric objects already exist in the plane or their relative placement; any such conditions are postulated ahead of time.
The Poncelet-Steiner theorem, on the other hand, presumes the geometer has no control over the placement of the circle or over the constructions undertaken, making the two independent.
Jean-Victor Poncelet was a major contributor to the subject when he postulated the theorem of this article, which Jakob Steiner later proved.
Though these are the usual meanings, any property the geometer chooses is valid, provided that it takes two elements - no more or less - to establish the underlying set.
In keeping with the intent of the theorem which we aim to prove, the actual circle need not be drawn but for aesthetic reasons.
Many examples of constructability with a straightedge one may find in various references on and offline, will presume that the circle is not placed in general position.
These concerns are in fact intrinsic to the straightedge itself, and have broad ramifications for traditional geometry beyond the scope of straightedge-only constructions.
This article takes a more traditional approach and proves the theorem using pure geometric constructive techniques, which also showcases the practical application.
Once arrived at the arbitrary rotation AB' , which defines the same circle, the radical axis construction can begin anew without issue.
Since all five basic constructions have been shown to be achievable with only a straightedge, provided that a single circle with its center is placed in the plane, this proves the Poncelet-Steiner theorem.
The Poncelet-Steiner theorem is a fundamental result in projective geometry that has significant practical applications for geometers and mathematicians.
For practicing geometers, understanding this theorem is crucial as it demonstrates the power of projective techniques and provides alternative methods for solving classical construction problems, as well as broader insights.
Often coordinates may be calculated using a sequence of linear equations, rather than the square roots associated with a circle, enabling faster, more accurate, and more numerically stable computation.
Mastering the insights of projective geometry enhances a geometer's ability to approach problems from multiple perspectives, fostering creativity and versatility in their work.
It proves that the operation can be achieved in an abstract way, retained in the geometry of circles, rather than as a feature of a physical tool designed for purpose and transcendent the plane.
The requirement placed on the Poncelet-Steiner theorem - that one circle with its center provided exist in the plane - has been since generalized, or strengthened, to include alternative but equally restrictive conditions.
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.
Though a relatively new concept stemming from computational systems, the notion of control flow and its various restrictions in the context of geometry are also the subject of study.
As with the case of constructible numbers, prohibitions against arbitrary point placement is one possible control flow restriction, previously discussed in this article.
In the aforementioned Mohr-Mascheroni Theorem, restrictions on compass radius could be imposed, such as minimums and maximums given a set of starting points.
Some of the traditional unsolved problems such as angle trisection, doubling the cube, squaring the circle, finding cubic roots, etc., since proven to be impossible by straightedge and compass alone, have been resolved using an expanded set of tools.
In general, the objects studied to extend the scope of what is constructible have included: Each of the three above categorical approaches have their own unique angle trisection solutions, as do many of the various tools and curves listed.
They favored thirdly the use of arbitrary smooth curves in the plane (such as the Archimedean spiral), and least of all the use of neuseis (alternative physical handheld tools).
The graduated ruler is unique in that it defines a metric, and also a norm, and gives rise to the algebraic treatment of geometry, Cartesian graphing, and imports a standard unit for proportionality of segments.
It is taken for granted that, from mostly philosophical considerations such as platonic idealism and operationalism,[8] the ancient Greek geometers emphasized finitude and exactness in their constructions.
Although each point, line or circle is a valid construction, what it aims to approximate can never truly be achieved in finite applications of a compass and/or straightedge.
