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1 0 bop 1673 -75 a Fl(1)554 70 y Fk(Supplemen)n(tary)19
b(PR)n(OJ.4)h(Notes|)521 145 y(Swiss)f(Oblique)g(Mercator)h(Pro)s
(jection)817 203 y Fj(August)d(5,)f(1995)840 261 y(Release)g(4.3.3)749
319 y(b)o(y)g(Gerald)g(I.)f(Ev)o(enden)324 401 y Fl(This)10
b(pro)r(jection)g(is)g(an)g(oblique)f(Mercator)i(that)f(is)g(based)h
(up)q(on)f(a)g(metho)q(d)f(in)o(v)o(olving)262 451 y(three)15
b(steps:)312 526 y(1.)20 b(conformally)11 b(transform)i(ellipsoid)g(co)
q(ordinates)h(to)g(a)g(sphere,)312 605 y(2.)20 b(transform)13
b(the)h(spherical)g(system)f(so)g(that)h(the)g(sp)q(eci\014ed)h(pro)r
(jection)f(origin)e(will)365 655 y(lie)i(on)f(the)i(equator,)e(and)312
734 y(3.)20 b(p)q(erform)13 b(spherical)i(Mercator)g(con)o(v)o(ersion)f
(to)f(cartesian)i(co)q(ordinates.)262 809 y(The)f(pro)r(jection)h(name)
e(used)j(here)f(is)f(ten)o(tativ)o(e)h(and)f(is)g(deriv)o(ed)h(from)e
(it)h(use)h(with)f(the)262 859 y(Swiss)g(co)q(ordinate)g(system)g
(CH1903.)324 909 y(This)e(pro)r(jection)g(is)g(selected)i(in)e(the)h
(PR)o(OJ.4)f(system)g(b)o(y)g Fi(+proj=somerc)d Fl(with)j(pa-)262
959 y(rameters)g Fi(+lon_0=)f Fl(and)h Fi(+lat_0)f Fl(required)i(to)f
(sp)q(ecify)h(pro)r(jection)g(origin.)j(Optionally)m(,)262
1008 y(false)c(easting)i(and)f(northing,)f Fi(+x_0=)g
Fl(and)h Fi(+y_0)p Fl(,)f(ma)o(y)f(b)q(e)j(used)g(as)f(w)o(ell)g(as)g
(origin)f(scale)262 1058 y(factor)18 b Fi(k_0)p Fl(.)32
b(F)m(or)18 b(usage)h(with)f(the)i(Swiss)e(CH1903)g(system,)h
Fi(+init=world:CH1903)262 1108 y Fl(ma)o(y)11 b(b)q(e)k(used.)262
1214 y Fh(Ac)o(kno)o(wledgemen)o(t)262 1291 y Fl(The)23
b(dev)o(elopmen)o(t)f(presen)o(ted)j(here)f(w)o(as)f(due)g(to)g(the)h
(kind)e(assistance)i(of)f(Daniel)262 1341 y(Ebneter)17
b(of)f(the)g(Univ)o(ersit)o(y)h(of)e(Bern)i(who)f(con)o(tributed)h(in)o
(v)n(aluable)d(do)q(cumen)o(tation,)262 1390 y(co)q(de)g(and)g(other)h
(information)c(that)k(allo)o(w)o(ed)d(deriv)n(ation)i(of)f(the)i(basic)
f(pro)r(jection)h(and)262 1440 y(pro)o(vided)e(b)q(enc)o(hmark)h(v)n
(alues.)262 1547 y Fh(F)l(orw)o(ard)h(Pro)s(jection)262
1623 y Fl(The)e(\014rst)h(step)g(is)f(the)g(con)o(v)o(ersion)h(of)e
(geo)q(detic)i(latitude)e(to)h(a)g(sphere)i(b)o(y)e(the)g(follo)o(wing)
262 1673 y(equation:)355 1777 y(ln)6 b(tan)463 1719 y
Fg(\022)498 1749 y Ff(\031)p 498 1768 26 2 v 500 1806
a Fl(4)538 1777 y(+)584 1749 y Ff(\036)609 1734 y Fe(0)p
584 1768 37 2 v 592 1806 a Fl(2)626 1719 y Fg(\023)698
1777 y Fl(=)42 b Ff(c)797 1719 y Fg(\024)818 1777 y Fl(ln)7
b(tan)927 1719 y Fg(\022)962 1749 y Ff(\031)p 962 1768
26 2 v 964 1806 a Fl(4)1002 1777 y(+)1048 1749 y Ff(\036)p
1048 1768 25 2 v 1050 1806 a Fl(2)1078 1719 y Fg(\023)1118
1777 y Fd(\000)1165 1749 y Ff(e)p 1164 1768 21 2 v 1164
1806 a Fl(2)1197 1777 y(ln)1238 1719 y Fg(\022)1274 1749
y Fl(1)i(+)g Ff(e)e Fl(sin)g Ff(\036)p 1274 1768 181
2 v 1274 1806 a Fl(1)i Fd(\000)g Ff(e)e Fl(sin)g Ff(\036)1459
1719 y Fg(\023\025)1521 1777 y Fl(+)i Ff(K)44 b Fl(\(1\))620
1866 y Ff(\025)644 1849 y Fe(0)698 1866 y Fl(=)e Ff(c)p
Fl(\()p Ff(\025)9 b Fd(\000)h Ff(\025)905 1872 y Fc(0)923
1866 y Fl(\))702 b(\(2\))262 1948 y(where)12 b Ff(\036)g
Fl(is)f(latitude)h(on)f(the)i(ellipsoid,)d Ff(\036)929
1933 y Fe(0)952 1948 y Fl(is)i(latitude)f(on)h(the)g(sphere)i(and)d
Ff(e)h Fl(is)g(ellipsoid)262 1998 y(eccen)o(tricit)o(y)m(.)21
b(The)16 b(factors)f Ff(c)f Fl(and)h Ff(K)j Fl(are)d(determined)g(from)
e(the)j(designated)f(latitude)262 2048 y(origin)d(of)i(the)g(pro)r
(jection:)k Ff(\036)736 2054 y Fc(0)755 2048 y Fl(.)g(First,)c
Ff(c)f Fl(is)h(obtained)g(from:)759 2165 y Ff(c)d Fl(=)832
2106 y Fg(\022)863 2165 y Fl(1)e(+)939 2137 y Ff(e)958
2122 y Fc(2)984 2137 y Fl(cos)1040 2122 y Fc(4)1065 2137
y Ff(\036)1090 2143 y Fc(0)p 939 2155 170 2 v 969 2193
a Fl(1)g Fd(\000)h Ff(e)1060 2181 y Fc(2)1114 2106 y
Fg(\023)1144 2113 y Fc(1)p Fb(=)p Fc(2)262 2269 y Fl(Next)k(the)g
(equiv)n(alen)o(t)g(origin)e(latitude)i(on)g(the)g(sphere)i(is)d
(obtained)h(b)o(y:)844 2370 y(sin)7 b Ff(\036)927 2353
y Fe(0)927 2380 y Fc(0)956 2370 y Fl(=)1005 2342 y(sin)g
Ff(\036)1088 2348 y Fc(0)p 1005 2360 102 2 v 1047 2398
a Ff(c)262 2466 y Fl(Substitute)14 b Ff(\036)487 2472
y Fc(0)520 2466 y Fl(and)f Ff(\036)625 2450 y Fe(0)625
2476 y Fc(0)657 2466 y Fl(in)o(to)h(Eqn.)f(1)h(and)g(solv)o(e)f(for)h
Ff(K)s Fl(.)324 2515 y(Next)i(the)g(spherical)g(co)q(ordinates,)g
Ff(\036)934 2500 y Fe(0)961 2515 y Fl(and)g Ff(\025)1068
2500 y Fe(0)1080 2515 y Fl(,)f(are)h(rotated)g(b)o(y)f
Ff(\036)1409 2521 y Fc(0)1443 2515 y Fl(ab)q(out)h(an)f(axis)262
2565 y(through)k(the)h(equator)g(and)f(p)q(erp)q(endicular)h(to)g(the)g
(plane)f(of)g(the)h(cen)o(tral)f(meridian)262 2615 y(\(W)m(ra)o(y's)12
b Fa(simple)j(oblique)p Fl(\))f(so)g(that)g(the)g(latitude)g
Ff(\036)1092 2621 y Fc(0)1124 2615 y Fl(b)q(ecomes)g
Ff(\036)1314 2600 y Fe(00)1347 2615 y Fl(=)d(0)j(at)g
Ff(\025)1500 2621 y Fc(0)1519 2615 y Fl(:)583 2697 y(sin)7
b Ff(\036)666 2679 y Fe(00)728 2697 y Fl(=)42 b(cos)8
b Ff(\036)890 2679 y Fe(0)890 2707 y Fc(0)915 2697 y
Fl(sin)f Ff(\036)998 2679 y Fe(0)1018 2697 y Fd(\000)j
Fl(sin)d Ff(\036)1143 2679 y Fe(0)1143 2707 y Fc(0)1168
2697 y Fl(cos)h Ff(\036)1256 2679 y Fe(0)1274 2697 y
Fl(cos)f Ff(\025)1360 2679 y Fe(0)1641 2697 y Fl(\(3\))583
2759 y(sin)g Ff(\025)665 2742 y Fe(00)728 2759 y Fl(=)42
b(cos)8 b Ff(\036)890 2742 y Fe(0)908 2759 y Fl(sin)f
Ff(\025)990 2742 y Fe(0)1002 2759 y Ff(=)g Fl(cos)g Ff(\036)1117
2742 y Fe(00)1641 2759 y Fl(\(4\))p eop
%%Page: 2 2
2 1 bop 1673 -75 a Fl(2)324 70 y(Finally)m(,)9 b(transform)i(the)h
(rotated)h(co)q(ordinates)f(to)g(cartesian)g(b)o(y)g(the)g(spherical)g
(form)262 120 y(of)h(the)h(Mercator)h(pro)r(jection:)690
234 y Ff(y)43 b Fl(=)f Ff(R)7 b Fl(ln)g(tan)974 175 y
Fg(\022)1010 206 y Ff(\031)p 1010 224 26 2 v 1012 263
a Fl(4)1049 234 y(+)1096 206 y Ff(\036)1121 191 y Fe(00)p
1096 224 46 2 v 1108 263 a Fl(2)1147 175 y Fg(\023)1186
234 y Fl(+)j Ff(y)1248 240 y Fc(0)1641 234 y Fl(\(5\))688
323 y Ff(x)41 b Fl(=)h Ff(R\025)883 306 y Fe(00)914 323
y Fl(+)9 b Ff(x)979 329 y Fc(0)1641 323 y Fl(\(6\))262
415 y(where)14 b Ff(R)f Fl(is)g(the)h(geometric)e(mean)g(of)h(the)g
(merdinal)f(and)h(parallel)f(radius)h(at)g(the)h(refer-)262
464 y(ence)h(p)q(oin)o(t:)780 543 y Ff(R)c Fl(=)h Ff(k)889
549 y Fc(0)907 543 y Ff(a)980 480 y Fd(p)p 1015 480 110
2 v 35 x Fl(1)d Fd(\000)g Ff(e)1105 503 y Fc(2)p 934
533 237 2 v 934 574 a Fl(1)g Fd(\000)h Ff(e)1025 562
y Fc(2)1051 574 y Fl(sin)1101 557 y Fc(2)1127 574 y Ff(\036)1152
580 y Fc(0)262 640 y Fl(where)17 b Ff(a)g Fl(is)f(the)h(ma)r(jor)e
(axis)h(of)g(the)h(ellipsoid.)25 b(The)17 b(scaling)f(factor,)h
Ff(k)1445 646 y Fc(0)1463 640 y Fl(,)g(is)f(not)h(used)262
690 y(\(=)12 b(1\))j(in)f(the)h(Swiss)g(v)o(ersion)f(but)h(it)f(can)h
(reduce)h(o)o(v)o(erall)d(scale)i(error)g(if)f(appropriately)262
740 y(applied.)262 848 y Fh(In)o(v)o(erse)g(Pro)s(jection)262
924 y Fl(T)m(o)f(get)h(the)g(cartesian)h(co)q(ordinates)g(bac)o(k)e(in)
o(to)h(the)g(rotated)g(spherical)h(system:)632 1038 y
Ff(\036)657 1021 y Fe(00)719 1038 y Fl(=)42 b(2)821 980
y Fg(\024)843 1038 y Ff(tan)905 1021 y Fe(\000)p Fc(1)956
1038 y Fl(exp)1026 980 y Fg(\022)1062 1010 y Ff(y)11
b Fd(\000)e Ff(y)1154 1016 y Fc(0)p 1062 1029 112 2 v
1102 1067 a Ff(R)1178 980 y Fg(\023)1218 1038 y Fd(\000)1265
1010 y Ff(\031)p 1265 1029 26 2 v 1267 1067 a Fl(4)1295
980 y Fg(\025)1641 1038 y Fl(\(7\))632 1147 y Ff(\025)656
1130 y Fe(00)719 1147 y Fl(=)798 1119 y Ff(y)i Fd(\000)e
Ff(y)890 1125 y Fc(0)p 798 1137 112 2 v 838 1175 a Ff(R)1641
1147 y Fl(\(8\))262 1250 y(Rotate)k(the)i(spherical)f(co)q(ordinates)h
(bac)o(k)e(to)h(the)h(original)d(p)q(osition:)573 1341
y(sin)7 b Ff(\036)656 1324 y Fe(0)709 1341 y Fl(=)42
b(cos)7 b Ff(\036)870 1324 y Fe(0)870 1351 y Fc(0)896
1341 y Fl(sin)g Ff(\036)979 1324 y Fe(00)1009 1341 y
Fl(+)i(sin)e Ff(\036)1133 1324 y Fe(0)1133 1351 y Fc(0)1158
1341 y Fl(cos)h Ff(\036)1246 1324 y Fe(00)1274 1341 y
Fl(cos)f Ff(\025)1360 1324 y Fe(00)1641 1341 y Fl(\(9\))574
1403 y(sin)g Ff(\025)656 1386 y Fe(0)709 1403 y Fl(=)42
b(cos)7 b Ff(\036)870 1386 y Fe(00)898 1403 y Fl(sin)g
Ff(\025)980 1386 y Fe(00)1002 1403 y Ff(=)g Fl(cos)g
Ff(\036)1117 1386 y Fe(0)1620 1403 y Fl(\(10\))324 1495
y(The)19 b(ellipsoid)e(longitude)h(is)h(obtained)f(from)f(Eqn.)i(2)f
(but)h(latitude)f(requires)i(ap-)262 1544 y(plication)c(of)h
(Newton-Raphson)g(metho)q(d)g(for)g(a)g(solution)g(of)g(Eqn.)g(1:)25
b Ff(x)1471 1550 y Fb(n)p Fc(+1)1553 1544 y Fl(=)18 b
Ff(x)1627 1550 y Fb(n)1661 1544 y Fd(\000)262 1594 y
Ff(f)t Fl(\()p Ff(x)326 1600 y Fb(n)349 1594 y Fl(\))p
Ff(=f)410 1579 y Fe(0)422 1594 y Fl(\()p Ff(x)462 1600
y Fb(n)485 1594 y Fl(\).)g(The)c(correction)h(ratio)e(is:)461
1656 y Fg(\024)483 1715 y Ff(C)f Fl(+)d(ln)e(tan)675
1656 y Fg(\022)710 1686 y Ff(\031)p 710 1705 26 2 v 712
1743 a Fl(4)750 1715 y(+)796 1686 y Ff(\036)p 796 1705
25 2 v 798 1743 a Fl(2)826 1656 y Fg(\023)866 1715 y
Fd(\000)913 1686 y Ff(e)p 912 1705 21 2 v 912 1743 a
Fl(2)945 1715 y(ln)986 1656 y Fg(\022)1022 1686 y Fl(1)i(+)g
Ff(e)e Fl(sin)g Ff(\036)p 1022 1705 181 2 v 1022 1743
a Fl(1)i Fd(\000)g Ff(e)e Fl(sin)g Ff(\036)1207 1656
y Fg(\023\025)1272 1686 y Fl(1)i Fd(\000)g Ff(e)1362
1671 y Fc(2)1388 1686 y Fl(sin)1439 1669 y Fc(2)1464
1686 y Ff(\036)p 1272 1705 218 2 v 1326 1743 a Fl(1)g
Fd(\000)g Ff(e)1416 1731 y Fc(2)262 1828 y Fl(where:)703
1893 y Ff(C)15 b Fl(=)796 1865 y(1)p 796 1883 21 2 v
797 1921 a Ff(c)829 1834 y Fg(\024)851 1893 y Ff(K)d
Fd(\000)e Fl(ln)c(tan)1048 1834 y Fg(\022)1084 1865 y
Ff(\031)p 1084 1883 26 2 v 1086 1921 a Fl(4)1123 1893
y(+)1170 1865 y Ff(\036)p 1170 1883 25 2 v 1172 1921
a Fl(2)1199 1834 y Fg(\023\025)262 1992 y Fl(Use)19 b
Ff(\036)371 1977 y Fe(0)402 1992 y Fl(as)g(the)h(initial)d(estimate)h
(of)h Ff(\036)f Fl(and)h(iterate)h(un)o(til)e(the)i(correction)g(ratio)
e(has)262 2042 y(reac)o(hed)d(su\016cien)o(t)f(tolerance.)19
b(The)14 b(function)g(is)g(rapidly)e(con)o(v)o(ergen)o(t.)262
2150 y Fh(Standard)h(De\014nition)262 2227 y Fl(The)f(follo)o(wing)d
(en)o(try)j(has)g(b)q(een)h(placed)f(in)f(the)i Fi(world)d
Fl(\014le)i(to)g(facilitate)e(CH1903)h(usage)262 2277
y(with)i(the)i(PR)o(OJ.4)e(system)h(b)o(y)f(sp)q(ecifying)h
Fi(+init=world:CH1903)o Fl(:)262 2360 y Fi(<CH1903>)20
b(#)h(Swiss)g(Coordinate)f(System)349 2409 y(+proj=somerc)f
(+lat_0=46d57'8.66)o(0"N)g(+lon_0=7d26'22.5)o(00"E)349
2459 y(+ellps=bessel)g(+x_0=600000)g(+y_0=200000)349
2509 y(+k_0=1.)42 b(no_defs)20 b(<>)262 2592 y Fl(Note,)g(the)f
(original)e(are)j("rev)o(erse)g(engineered")h(from)c(decimal)g(v)n
(alues)i(as)g(I)g(susp)q(ect)262 2642 y(original)12 b(sp)q
(eci\014cations)j(w)o(ere)f(in)g(DMS)f(format|needs)g(v)o
(eri\014cation.)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF