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 !
Written by Eddy.