Editing Poncelet–Steiner theorem (section)

== Practical Application ==
The Poncelet-Steiner theorem is a fundamental result in projective geometry that has significant practical applications for geometers and mathematicians. It serves greater purpose than merely being purely academic, [[recreational mathematics|recreational]], or a novelty. 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.

The practical applications of the Poncelet-Steiner theorem extend beyond pure mathematics. In fields such as [[computer graphics]] and [[computational geometry]], it offers efficient algorithms for geometric constructions without the need for more direct and lengthy approaches. This can lead to more streamlined and robust software implementations. Additionally, the theorem has implications in [[architectural design]] and [[engineering]], where it can simplify certain [[technical drawing|drafting]] and [[modeling]] processes, and lending itself to improved design and modeling techniques. Often coordinates may be calculated using a sequence of [[linear equation]]s, rather than the [[square root]]s associated with a circle, enabling faster, more accurate, and more [[numerical stability|numerically stable]] computation. In fact, [[Paul Dirac]] applied projective geometry in his contribution to the development of [[quantum mechanics]].<ref>{{cite journal |last=Farmelo |first=Graham |date=15 September 2005 |title=Dirac's hidden geometry |url=https://www.nature.com/articles/437323a.pdf |department=Essay |journal=[[Nature (journal)|Nature]] |publisher=Nature Publishing Group |volume=437 |issue=7057 |page=323|doi=10.1038/437323a |pmid=16163331 |bibcode=2005Natur.437..323F |s2cid=34940597 }}</ref>

Not only does the theorem have historical significance, but it, as well as projective geometry more broadly, also offers deeper insight into the nature of geometric constructions and the relationships between different geometric tools and geometric structures. It challenges conventional thinking about what is necessary for geometric constructions and opens the geometer up to new avenues for problem-solving. Mastering the insights of projective geometry enhances a geometer's ability to approach problems from multiple perspectives, fostering creativity and versatility in their work.