Re: Why six and seven?

Robert Billing wrote:

I have an unusual personal take on this. In the early days of digital
television, when we were trying to get the equipment to work at all, I ran
into the problem that a lot of effects involved flat things turning into
curved things. A flat thing is effectively a sphere of infinite radius,
but infinities and divide by zeroes are difficult to handle correctly.

I invented an alternative trigonometry, which used a different set of
functions to achieve the same result, but could never produce a divide by
zero, or a zero over zero.

Projective geometry is the usual approach to this sort of thing,
I think. So instead of working with a ratio x/y that might be
infinite or indeterminate, you work with the pair (x,y) and avoid
doing the division until the last possible moment. And instead of
working with an angle A, you might work with (cos A, sin A), or
with (k cos A, k sin A) for any choice of k. Then all the usual
trigonometrical formulae tend to turn into polynomial identities.

(Another trick which has similar effects is to represent an angle A
by t = tan(A/2). Then, e.g., cos(A) = (1-t^2)/(1+t^2) etc., and the
things you're dividing by are all kept safely away from 0. Again,
trigonometry gets turned into algebra.)

Was your approach something like that, or did it involve a more
radical uprooting of trigonometry?

--
Gareth McCaughan
sig under construc


.

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