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*To*: Steve VanDevender <stevev@efn.org>*Subject*: Re: C-ship*From*: Steve VanDevender <stevev@efn.org>*Date*: Tue, 2 Jan 1996 02:14:26 -0800*Cc*: T.L.G.vanderLinden@student.utwente.nl (Timothy van der Linden)*In-Reply-To*: <199601020930.BAA25497@tzadkiel.efn.org>*References*: <199601012207.AA10231@student.utwente.nl><199601020930.BAA25497@tzadkiel.efn.org>

Steve VanDevender writes: > Things are substantially more complicated when dealing with > accelerated worldlines. I've got a preliminary solution stated > in similar terms as the above discussion; perhaps you'd like to > check my math :-). > > You are at the point S = [ t x y z ] attempting to view an object > whose coordinates are P(t') = > > 1/a^2 * [ a * sinh(a * t') > ax * (cosh(a * t') - 1) > ay * (cosh(a * t') - 1) > az * (cosh(a * t') - 1) ] > > where the acceleration is represented by A = [ 0 ax ay az ] (in > the object local frame) and a = sqrt(ax^2 + ay^2 + az^2) (the > magnitude of the acceleration). > > So again, we want to solve the equation (S - P(t'))^2 = 0. Of > course, the components of P(t') are much more complicated. I > won't bore you with the full derivation, other than to note that > it becomes easier to isolate t' by writing the sinh and cosh > terms in terms of their definitions using exp (e^x). > > Eventually, you get: > > exp(a * t')^2 * (1 - A|S - a * t') + > exp(a * t') * (a^2 * S^2 - 2 * (1 - A|S)) + > (1 - A|S + a * t') > = 0 Sigh. That should actually be: exp(a * t')^2 * (1 - A|S - a * t) + exp(a * t') * (a^2 * S^2 - 2 * (1 - A|S)) + (1 - A|S + a * t) = 0 I wrote t' rather than the t component of S in a couple wrong places. > It's convenient to make some substitutions for common > subexpressions: > > k = 1 - A|S > p = k - a * t' > q = a^2 * S^2 - 2 * k > r = k + a * t' And these should be p = k - a * t q = a^2 * S^2 - 2 * k r = k + a * t > So then applying the quadratic formula and isolating t' gives: > > t' = 1/a * ln((-q - sqrt(q^2 - 4 * p * r)) / (2 * p)) Fortunately I still copied that correctly.

**References**:**Re: C-ship***From:*T.L.G.vanderLinden@student.utwente.nl (Timothy van der Linden)

**Re: C-ship***From:*Steve VanDevender <stevev@efn.org>

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