66 18. GEOMETRIC TOOLS AND POINT PICKING METHODS
close to that of the standard cylinder, we have that, provided A 2: 51f,^17 the
Euclidean comparison angle satisfies
- 1f
LrJs (ti) qo( (t2) :S 5·
By applying the Toponogov comparison theorem (hinge version) to the tri-
angle Llqo/ ( s) q and the law of cosines, we haved^2 ( q, / ( s)) :S d^2 ( qo, / ( s)) + d^2 ( qo, q)
- 2d (qo, ')' (s)) d (qo, q) cos ( LrJs (t1) qo( (t2))
(18.56) :S d^2 (qo, / (s)) + d^2 (qo, q) - d (qo, / (s)) d (qo, q).
If s is large enough,^18 then
(18.57)
1 3
d (qo, / (s)) 2: :2d (qo, q) + d^2 (q, / (s)) -
4d^2 (qo, q)1
2: d (q, / (s)) + 4d (qo, q),where the second inequality holds provided d ( q, I ( s)) 2:^1 l d ( qo, q). Hence
1
s-d(1(s) ,q) 2: s-d(qo,1(s)) + 4d(qo,q).By taking the limit s --t oo, we have for q E Nout - '¢ ( sn-l x ( 0, A)) , where
A 2: 57r,(18.58)Third we prove the second inequality in (18.51), i.e., diam (b,;-^1 (bo)) :Sll1fcnR (qo)-^1 /^2. Now choose A = 51f. Note that for any q E b,;-^1 (bo), by
(18.55) and (18.58), we have q E '¢ (sn-l x [-51f, 51fl).^19 That is,
b::;^1 (bo) c '¢ (sn-l x [-51f, 51fl).
(^17) In fact we can make the angle L.71 8 (t1) qo( (t2) as small as we like by choosing A
sufficiently large and c (n) sufficiently small. In the standard unit cylinder sn-l x JR, if
we have points a E sn-l x {O} and b,c E sn-l x {A}, where A;::::: 57r, then
L.bac:::; 2tan-^1 ( 110 ) < ;;.
(^18) If d (qo, 'Y (s )) ;::::: ~d (qo, q), then the first inequality in (18.57) is equivalent to (18.56).
19If.
q E (Alinn -'if; (sn-l X (-57r,0))) U (Nout-'lf; (sn-l X (0,57r)))
= N -'if; (sn-l x [-511", 57rl),
then b 7 ( q) =F bo.