乐胖代购免代理版

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283.146 366.588 283.764 335.476 313.48 337.626 0.819608 0.819608 0.819608 epoly pop grestore showpage %%EndDocument @endspecial 1256 w @beginspecial 18 @llx 108 @lly 594 @urx 684 @ury 1267 @rwi @setspecial %%BeginDocument: pix/box_complete2.eps %!PS-Adobe-2.0 EPSF-1.2 %%Title: Geomview Snapshot %%Creator: Geomview %%CreationDate: Thu Apr 25 15:30:37 2002 %%For: pnijjar %%BoundingBox: 18 108 594 684 %%EndComments gsave 1 setlinecap 1 setlinejoin 18.000000 108.000000 translate 1.280000 1.280000 scale [ % stack mark /poly { setrgbcolor newpath moveto counttomark 2 idiv { lineto } repeat closepath fill } bind def /epoly { setrgbcolor newpath moveto counttomark 4 sub 2 idiv { lineto } repeat closepath gsave fill grestore setrgbcolor setlinewidth stroke } bind def /lines { setlinewidth setrgbcolor newpath moveto counttomark 2 idiv { lineto } repeat stroke } bind def /clines { setlinewidth setrgbcolor newpath moveto counttomark 2 idiv { lineto } repeat closepath stroke } bind def /circ { 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4814 l gs col0 s gr % Polyline n 3310 3284 m 3131 3284 l 3131 4814 l 3310 4814 l gs col0 s gr % Polyline n 4589 4819 m 4589 4998 l gs col0 s gr % Polyline n 4580 3106 m 4588 3106 l gs col0 s gr % Polyline n 3304 3106 m 3312 3106 l gs col0 s gr % Polyline n 3306 3106 m 3314 3106 l gs col0 s gr % Polyline n 3304 1829 m 3312 1829 l gs col0 s gr % Polyline n 4582 1829 m 4582 3104 l gs col0 s gr % Polyline n 4579 1831 m 4587 1831 l gs col0 s gr % Polyline n 4583 1653 m 3309 1653 l 3309 1831 l gs col0 s gr % Polyline n 1597 3106 m 1605 3106 l gs col0 s gr % Polyline n 1601 1831 m 1424 1831 l 1424 3106 l 1601 3106 l gs col0 s gr % Polyline n 1597 1832 m 1605 1832 l gs col0 s gr % Polyline n 3131 1831 m 3131 1653 l 1601 1653 l 1601 1831 l gs col0 s gr % Polyline n 3123 1825 m 3131 1825 l gs col0 s gr % Polyline n 3127 3100 m 1597 3100 l gs col0 s gr % Polyline n 3306 3099 m 3127 3099 l gs col0 s gr % Polyline n 3127 1824 m 3306 1824 l gs col0 s gr % Polyline n 5682 1474 m 4585 1474 l 4585 1295 l 5682 1295 l gs col0 s gr % Polyline n 5682 1294 m 6779 1294 l 6779 1473 l 5682 1473 l gs col0 s gr % Polyline n 2387 7339 m 2395 7339 l gs col0 s gr % Polyline n 2387 7517 m 2395 7517 l gs col0 s gr % Polyline n 2391 7517 m 1296 7517 l 1296 7339 l 2391 7339 l gs col0 s gr % Polyline n 2391 7339 m 3488 7339 l 3488 7517 l 2391 7517 l gs col0 s gr % Polyline n 2210 6625 m 2218 6625 l gs col0 s gr % Polyline n 2210 6803 m 2218 6803 l gs col0 s gr % Polyline n 2214 6625 m 3310 6625 l 3310 6803 l 2214 6803 l gs col0 s gr % Polyline n 2214 6803 m 1117 6803 l 1117 6625 l 2214 6625 l gs col0 s gr % Polyline n 5502 761 m 4407 761 l 4407 582 l 5502 582 l gs col0 s gr % Polyline n 5502 582 m 6599 582 l 6599 761 l 5502 761 l gs col0 s gr % Polyline n 4580 6268 m 4588 6268 l gs col0 s gr % Polyline n 4580 4993 m 4588 4993 l gs col0 s gr % Polyline n 3331 6268 m 3339 6268 l gs col0 s gr % Polyline n 4580 6268 m 4588 6268 l gs col0 s gr % Polyline n 4508 8232 m 4508 6701 l 6547 6701 l 6547 8232 l gs col0 s gr % Polyline n 4504 8232 m 4512 8232 l gs col0 s gr % Polyline n 6543 8232 m 6551 8232 l gs col0 s gr % Polyline n 6547 8232 m 6547 8358 l 4508 8358 l 4508 8232 l gs col0 s gr % Polyline n 6675 10144 m 6547 10144 l 6547 8358 l 6675 8358 l gs col0 s gr % Polyline n 6547 10268 m 6547 11798 l 4508 11798 l 4508 10268 l gs col0 s gr % Polyline n 4504 10271 m 4512 10271 l gs col0 s gr % Polyline n 6543 10271 m 6551 10271 l gs col0 s gr % Polyline n 4508 10271 m 4508 10144 l 6547 10144 l 6547 10271 l gs col0 s gr % Polyline n 4380 8358 m 2852 8358 l 2852 10144 l 4380 10144 l gs col0 s gr % Polyline n 4376 8358 m 4384 8358 l gs col0 s gr % Polyline n 4376 10144 m 4384 10144 l gs col0 s gr % Polyline n 4380 10144 m 4508 10144 l 4508 8358 l 4380 8358 l gs col0 s gr % Polyline n 4639 11805 m 4639 12057 l 6424 12057 l 6424 11802 l gs col0 s gr % Polyline n 6289 6268 m 6297 6268 l gs col0 s gr % Polyline n 4769 6263 m 4591 6263 l gs col0 s gr % Polyline n 4764 4995 m 6295 4995 l gs col0 s gr % Polyline n 4583 4995 m 4761 4995 l gs col0 s gr % Polyline n 3311 5000 m 3311 4821 l gs col0 s gr % Polyline n 3311 6290 m 3311 5015 l gs col0 s gr % Polyline n 3310 6446 m 2214 6446 l 2214 6268 l 3310 6268 l gs col0 s gr % Polyline n 4584 6268 m 4584 6446 l 3310 6446 l gs col0 s gr % Polyline n 3306 6268 m 3314 6268 l gs col0 s gr % Polyline n 3307 6270 m 3315 6270 l gs col0 s gr % Polyline n 3307 6447 m 3315 6447 l gs col0 s gr % Polyline n 3306 6268 m 3314 6268 l gs col0 s gr % Polyline n 3306 6446 m 3314 6446 l gs col0 s gr % Polyline n 6293 6268 m 6293 6446 l 4762 6446 l 4762 6268 l gs col0 s gr % Polyline n 8206 8357 m 8340 8357 l gs col0 s gr % Polyline n 8328 8617 m 8336 8617 l gs col0 s gr % Polyline n 8328 9892 m 8336 9892 l gs col0 s gr % Polyline n 8332 8617 m 8460 8617 l 8460 9892 l 8332 9892 l gs col0 s gr % Polyline n 8329 9891 m 8329 10144 l gs col0 s gr % Polyline n 8332 8359 m 8332 8616 l gs col0 s gr % Polyline n 6680 8357 m 8204 8358 l 8204 10146 l 6673 10146 l gs col0 s gr % Polyline n 8332 10146 m 8206 10146 l gs col0 s gr % Polyline n 4576 3101 m 4576 3279 l gs col0 s gr % Polyline n 3302 3279 m 3302 3101 l gs col0 s gr % Polyline n 4579 1656 m 4587 1656 l gs col0 s gr % Polyline n 5680 1656 m 5680 1834 l 4583 1834 l gs col0 s gr % Polyline [45] 0 sd n 3316 6268 m 4583 6268 l gs col0 s gr [] 0 sd % Polyline [45] 0 sd n 4582 4993 m 4582 6265 l gs col0 s gr [] 0 sd % Polyline [45] 0 sd n 4758 6270 m 4758 4988 l gs col0 s gr [] 0 sd % Polyline [45] 0 sd n 6299 6267 m 4764 6267 l gs col0 s gr [] 0 sd % Polyline [45] 0 sd n 6298 4998 m 6298 6270 l gs col0 s gr [] 0 sd % Polyline [45] 0 sd n 4583 3282 m 4583 4807 l gs col0 s gr [] 0 sd % Polyline [45] 0 sd n 3316 4814 m 4588 4814 l gs col0 s gr [] 0 sd % Polyline [45] 0 sd n 3313 3287 m 3313 4812 l gs col0 s gr [] 0 sd % Polyline [45] 0 sd n 3317 3286 m 4584 3286 l gs col0 s gr [] 0 sd % Polyline [45] 0 sd n 3293 3099 m 4570 3099 l gs col0 s gr [] 0 sd % Polyline [45] 0 sd n 3308 3101 m 3308 1829 l gs col0 s gr [] 0 sd % Polyline [45] 0 sd n 3127 1833 m 3127 3100 l gs col0 s gr [] 0 sd % Polyline [45] 0 sd n 1598 1832 m 3128 1832 l gs col0 s gr [] 0 sd % Polyline [45] 0 sd n 1599 1834 m 1599 3106 l gs col0 s gr [] 0 sd % Polyline [45] 0 sd n 3306 1829 m 4583 1829 l gs col0 s gr [] 0 sd % Polyline [45] 0 sd n 5682 1297 m 5682 1469 l gs col0 s gr [] 0 sd % Polyline [45] 0 sd n 4583 1652 m 4583 1829 l gs col0 s gr [] 0 sd % Polyline [45] 0 sd n 5502 563 m 5502 778 l gs col0 s gr [] 0 sd % Polyline [45] 0 sd n 3321 5000 m 4588 5000 l gs col0 s gr [] 0 sd % Polyline [45] 0 sd n 3310 6276 m 3310 6448 l gs col0 s gr [] 0 sd % Polyline [45] 0 sd n 2212 6628 m 2212 6809 l gs col0 s gr [] 0 sd % Polyline [45] 0 sd n 2392 7338 m 2392 7514 l gs col0 s gr [] 0 sd $F2psEnd rs %%EndDocument @endspecial Black 661 3949 a Fp(Figure)31 b(4:)42 b Fj(Net)26 b(fo)n(r)g(Figure)f(2.)p Black Black -240 4515 a Fs(4)135 b(Gen)l(us)44 b(Must)h(Be)g(A)l(t)g(Least)g(3)-240 4753 y Fr(Our)37 b(pro)r(of)g(that)g(a)h(nonorthogonal)d(p)r(olyhedron) h(with)i(or-)-240 4853 y(thogonal)k(faces)g(m)n(ust)h(ha)n(v)n(e)e(gen) n(us)h(at)h(least)f(3)h(w)n(orks)e(as)-240 4952 y(follo)n(ws.)70 b(First)39 b(w)n(e)g(dev)n(elop)f(a)g(general)g(upp)r(er)h(b)r(ound)g (on)-240 5052 y(the)45 b(n)n(um)n(b)r(er)f(of)g(v)n(ertices)g(of)g(the) h(\\nonorthogonal)c(part")-240 5151 y(of)c(a)f(p)r(olyhedron)g(in)h (terms)f(of)h(its)g(gen)n(us.)63 b(In)37 b(particular,)-240 5251 y(for)i(gen)n(us)h(2,)i(the)f(b)r(ound)f(is)g(8.)73 b(Then)40 b(w)n(e)g(pro)n(v)n(e)e(that)i(a)-240 5351 y(nonorthogonal)28 b(p)r(olyhedron)h(with)i(orthogonal)d(faces)i(m)n (ust)-240 5450 y(ha)n(v)n(e)f(at)h(least)g(9)g(v)n(ertices)f(in)i(its)f (nonorthogonal)e(part,)i(and)-240 5550 y(hence)e(m)n(ust)f(ha)n(v)n(e)g (gen)n(us)g(more)f(than)i(2.)1920 83 y Fi(4.1)112 b(Basic)37 b(De\014nitions)f(and)j(Coun)m(ts)1920 236 y Fr(F)-7 b(ollo)n(wing)32 b([4],)i(w)n(e)f(think)h(of)f(an)g(edge)f(of)i(a)e(p)r (olyhedron)h(as)1920 336 y(colored)18 b Fo(gr)l(e)l(en)25 b Fr($go)r(o)r(d$)19 b(if)g(its)g(dihedral)g(angle)f($angle)g(b)r (et)n(w)n(een)1920 436 y(the)39 b(t)n(w)n(o)g(inciden)n(t)g(faces$)g (is)g(a)g(m)n(ultiple)h(of)f(90)3518 405 y Fq(\016)3555 436 y Fr(,)j(and)d Fo(r)l(e)l(d)1920 535 y Fr($bad$)30 b(otherwise.)44 b(De\014ne)30 b(the)h(graph)e Fh(G)3266 547 y Fg(r)3333 535 y Fr($the)i Fo(nonortho)l(g-)1920 635 y(onal)37 b(p)l(art)8 b Fr($)35 b(b)n(y)f(starting)g(with)g(all)h (of)f(the)h(red)f(edges,)i(then)1920 734 y(remo)n(ving)c(an)n(y)h (degree-t)n(w)n(o)e(v)n(ertices,)j(coalescing)d(the)j(inci-)1920 834 y(den)n(t)27 b(edges,)f(and)g(\014nally)g(fo)r(cusing)h(atten)n (tion)f(to)g(just)h(a)f(sin-)1920 934 y(gle)37 b(connected)g(comp)r (onen)n(t.)66 b(An)n(y)38 b(edges)e(that)i(w)n(ere)e(coa-)1920 1033 y(lesced)31 b(w)n(ere)e(already)h(collinear)f([4,)j(Lem.)f(7],)g (so)f(the)h(graph)1920 1133 y Fh(G)1985 1145 y Fg(r)2050 1133 y Fr(remains)d(em)n(b)r(edded)h(in)f Ff(R)2905 1103 y Fe(3)2948 1133 y Fr(,)h(de\014ning)f(angles)f(and)i(faces)1920 1233 y($com)n(binatorial)h(faces,)j(whic)n(h)f(do)g(not)g(necessarily) e(lie)i(in)g(a)1920 1332 y(plane$.)2003 1432 y(The)h(follo)n(wing)f(t) n(w)n(o)g(lemmas)g(relate)g Fh(G)3303 1444 y Fg(r)3373 1432 y Fr(to)g(the)i(original)1920 1531 y(p)r(olyhedron.)68 b(F)-7 b(or)38 b(their)g(pro)r(ofs,)i(w)n(e)e(need)g(the)h(notion)f(of) 1920 1631 y(an)h Fo(ortho)l(gonal)i(p)l(ath)46 b Fr(around)38 b(a)h(v)n(ertex)e Fh(v)43 b Fr([4)o(]:)60 b(a)38 b(path)h(of)1920 1731 y(circular)23 b(arcs)g(on)g(the)i(in)n(tersection)e(of)h(the)g(p)r (olyhedron)g(with)1920 1830 y(a)d(small)h(sphere)f(cen)n(tered)h(at)f Fh(v)s Fr(,)j(suc)n(h)d(that)i(ev)n(ery)d(turn)i(along)1920 1930 y(the)29 b(path)g(is)g(b)n(y)f Fd(\006)p Fr(90)2609 1900 y Fq(\016)2646 1930 y Fr(.)41 b(Let)29 b Fh(p)2902 1900 y Fq(0)2954 1930 y Fr(denote)g(the)g(pro)5 b(jection)28 b(of)g(a)1920 2030 y(p)r(oin)n(t)g Fh(p)f Fr(on)n(to)g(this)h(sphere.)p Black 1920 2181 a Fp(Lemma)i(1)p Black 41 w Fo(The)g(fac)l(e)g(angles)g (of)g Fh(G)3098 2193 y Fg(r)3164 2181 y Fo(ar)l(e)g(multiples)f(of)h Fr(90)3837 2151 y Fq(\016)3875 2181 y Fo(.)1920 2332 y Fp(Pro)s(of:)46 b Fr(Consider)28 b(a)h(face)g(angle)f(at)h Fh(v)j Fr(made)d(b)n(y)g(t)n(w)n(o)g(edges)1920 2432 y Fh(v)s(;)14 b(v)2040 2444 y Fe(0)2115 2432 y Fr(and)36 b Fh(v)s(;)14 b(v)2405 2444 y Fe(1)2480 2432 y Fr(in)37 b Fh(G)2651 2444 y Fg(r)2688 2432 y Fr(.)65 b(By)37 b(the)g (de\014nition)h(of)f Fh(G)3616 2444 y Fg(r)3653 2432 y Fr(,)i(there)1920 2531 y(is)32 b(an)g(orthogonal)e(path)j(around)e Fh(v)36 b Fr(from)c Fh(v)3353 2501 y Fq(0)3350 2552 y Fe(0)3420 2531 y Fr(to)g Fh(v)3569 2501 y Fq(0)3566 2552 y Fe(1)3603 2531 y Fr(.)51 b(By)32 b([4,)1920 2631 y(Lem.)k(4],)h(the)f (great)f(arc)f(length)i(b)r(et)n(w)n(een)g Fh(v)3414 2601 y Fq(0)3411 2651 y Fe(0)3484 2631 y Fr(and)g Fh(v)3697 2601 y Fq(0)3694 2651 y Fe(1)3767 2631 y Fr(is)f(a)1920 2730 y(m)n(ultiple)25 b(of)f(90)2417 2700 y Fq(\016)2454 2730 y Fr(,)h(and)f(this)g(arc)f(length)i(is)f(precisely)f(the)h(face) 1920 2830 y(angle.)1705 b Fc(2)p Black 1920 3019 a Fp(Lemma)30 b(2)p Black 41 w Fo(L)l(et)g Fh(e)2531 3031 y Fe(0)2568 3019 y Fh(;)14 b(e)2644 3031 y Fe(1)2681 3019 y Fh(;)g(e)2757 3031 y Fe(2)2824 3019 y Fo(b)l(e)31 b(c)l(onse)l(cutive)f(e)l(dges)i (ar)l(ound)f(a)1920 3118 y(c)l(ommon)c(vertex)g(in)g Fh(G)2655 3130 y Fg(r)2692 3118 y Fo(.)38 b(Then)28 b(the)f(dihe)l(dr)l (al)j(angle)d(b)l(etwe)l(en)1920 3218 y(the)37 b(plane)g Fh(e)2331 3230 y Fe(0)2368 3218 y Fh(;)14 b(e)2444 3230 y Fe(1)2518 3218 y Fo(and)37 b(the)f(plane)i Fh(e)3097 3230 y Fe(1)3134 3218 y Fh(;)14 b(e)3210 3230 y Fe(2)3283 3218 y Fo(is)37 b(not)f(a)h(multiple)1920 3317 y(of)31 b Fr(90)2102 3287 y Fq(\016)2139 3317 y Fo(.)1920 3469 y Fp(Pro)s(of:)41 b Fr(Let)25 b Fh(v)k Fr(b)r(e)d(the)g(common)f(endp)r (oin)n(t)h(of)g(the)g Fh(e)3634 3481 y Fg(i)3661 3469 y Fr('s,)g(and)1920 3568 y(let)31 b Fh(w)2102 3580 y Fg(i)2161 3568 y Fr(b)r(e)g(the)g(other)f(endp)r(oin)n(t)h(of)g Fh(e)3129 3580 y Fg(i)3156 3568 y Fr(.)46 b(As)31 b(in)g(the)g (previous)1920 3668 y(lemma,)g(there)f(is)g(an)g(orthogonal)e(path)i (around)f Fh(v)k Fr(from)d Fh(w)3864 3638 y Fq(0)3862 3689 y Fe(0)1920 3768 y Fr(to)23 b Fh(w)2078 3737 y Fq(0)2076 3788 y Fe(1)2114 3768 y Fr(.)36 b(By)23 b([4)o(,)i(Lem.)f(5],)g(the)f (great)g(arc)f(b)r(et)n(w)n(een)i Fh(w)3587 3737 y Fq(0)3585 3788 y Fe(0)3646 3768 y Fr(and)f Fh(w)3864 3737 y Fq(0)3862 3788 y Fe(1)1920 3867 y Fr(meets)h(this)h(orthogonal)d(path)i(at)h Fh(w)3073 3837 y Fq(0)3071 3888 y Fe(0)3133 3867 y Fr(and)f Fh(w)3352 3837 y Fq(0)3350 3888 y Fe(1)3412 3867 y Fr(with)h(t)n(w)n(o) e(90)3835 3837 y Fq(\016)3872 3867 y Fr(-)1920 3967 y(m)n(ultiple)29 b(angles.)40 b(Similarly)-7 b(,)29 b(there)f(is)h(an)f(orthogonal)f (path)1920 4066 y(from)32 b Fh(w)2182 4036 y Fq(0)2180 4087 y Fe(1)2250 4066 y Fr(to)g Fh(w)2417 4036 y Fq(0)2415 4087 y Fe(2)2453 4066 y 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