Some basic geometry

The context of classical geometry is a domain in which the notions of straight line and parallel do not jar with common intuitions. Some interesting contexts do not follow rules compatible with those intuitions, but a broad class of real-world situations are compatible with them to within close enough tolerances that many honest folk don't care about the difference, in practice. Contexts in which these intuitions are exactly correct are formally described as flat or Euclidean, so I am effectively describing Euclidean geometry. I'll leave it to other pages to address matters more formally, in terms which either address deviations from flatness or formally specify flatness, but I'll here endeavour to set forth some of the more prominent results of those intuitions and some of the consequences that follow for any context in which they hold good enough that you don't care about the difference. Many of the intuitive truths here addressed remain true even in non-Euclidean (or curved) geometries, at least if we add some qualifying clauses, typically confining the points, lines and regions under discussion to some sufficiently small portion of the whole context of the curved geometry.

I'll aim to describe geometries verbally, and I'll also use Scalable Vector Graphics (the W3C specification for an XML-based image format) for diagrams: some browsers shall need a plug-in to help them view these, but those that keep up with the times do now support it. Some implementations are still buggy in places: your mileage may vary !

See also


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