Re: [math-fun] Homotheties of the hyperbolic plane
Fred writes: << On 2/14/08, [I] wrote: << ... In any case, the impossibility of a bijection H^2 -> H^2 that increases all distances by a factor of (say) 2 brings up an interesting question: Can one quantify just how close one can get to such a map?
In particular, can one choose constants c1, c2 both > 1 such that c1 <= |f(x) - f(y)| / |x - y| <= c2 ?>>
Good question. (My guess is no, because lengths pushed a distance R away from a point increase in an exponential manner as a function of R.) --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
On 2/14/08, Dan Asimov <dasimov@earthlink.net> wrote:
Fred writes:
<< On 2/14/08, [I] wrote:
<< ... In any case, the impossibility of a bijection H^2 -> H^2 that increases all distances by a factor of (say) 2 brings up an interesting question: Can one quantify just how close one can get to such a map?
In particular, can one choose constants c1, c2 both > 1 such that
c1 <= |f(x) - f(y)| / |x - y| <= c2 ?>>
Good question. (My guess is no, because lengths pushed a distance R away from a point increase in an exponential manner as a function of R.)
The construction I suggested earlier --- remapping the radius to make the tangential factor exactly c --- has a radial factor which seems to approach c + O(r^2) as r -> 0 (fixed Point), and 1 + O(log(1-r)) as r -> 1 (Infinity). Presumably allowing the tangential factor to vary between fixed bounds c1 and c2 can make no qualitative difference to the second limit. So it doesn't look good. Maybe explore bounding the Jacobian (area differential) away from unity? WFL
participants (2)
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Dan Asimov -
Fred lunnon