Re: RAIB publish report on 2006 Ravenglass & Eskdale derailments

In article <pan.2007.03.29.19.46.43@xxxxxxxxxxxxxx>, Ian Johnston <ian.groups@xxxxxxxxxxxxxx> writes
Oh? Why aren't concentric circles parallel?

First book of Euclid: Definition 23:

"Parallel straight lines are
[...]

Since that says "straight", it can't be used as evidence against non-straight lines.

It's really difficult to come up with an alternative which works for the
sort of curved lines we all thing of intuitively as parallel:

"Two lines are parallel if the shortest distance between them is a
constant value for all starting points on both lines"

is nice, but doesn't allow the pairs to cross.

Try:

"Two curves are parallel if there is a bijection between the points on the two curves [*], the bijection preserves the relative ordering of the points [+], and, for points P1 on C1 and P2 on C2 related by the bijection, the normal to C1 at P1 is also normal to C2 at P2 and the distance between P1 and P2 along the normal is the same for all points on the curves."

[*] That is, you can put the points on the two curves in a one-to-one correspondence. For those unfamiliar with the mathematics of infinities, that doesn't mean that the corresponding points have to be at equal distances from other corresponding points. So if P1 and Q1 on C1 correspond to P2 and Q2 on C2, there's no requirement that P1 to Q1 equals P2 to Q2. This allows concentric circles to be "parallel".

[+] That is, if on C1, P1 is between Q1 and R1, and P1 Q1 R1 correspond to P2 Q2 R2 on C2, then P2 is between Q2 and R2.

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Clive D.W. Feather | Home: <clive@xxxxxxxxxx>
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