Einen Torus mit √2 wickeln

[size=50][right]Diese Aktivität ist eine Seite des [color=#980000]geogebra-books[/color] [url=https://www.geogebra.org/m/ggvbukux]"Loxodrome" ? Oder nicht ?[/url] [color=#ff7700](03.01.2020)[/color][/right][/size][br][size=85]Ein [color=#1155Cc][i][b]Torus[/b][/i][color=#000000] e[color=#1155Cc][color=#000000]ntsteht [/color][/color][/color][/color]durch [i][b]Rotation[/b][/i] eines kleinen [color=#0000ff][i][b]Kreises[/b][/i][/color] mit Radius [math]r[/math] um eine [i][b]Achse[/b][/i], welche in der Kreisebene liegt. [br]Der Mittelpunkt des kleinen [color=#0000ff][i][b]Kreises[/b][/i][/color] bewegt sich dabei auf einem Kreis um die [i][b]Achse[/b][/i] mit Radius [math]R[/math].[br]Ist [math]R>r[/math], so liegt ein [color=#1155Cc][i][b]Ring-Torus[/b][/i][/color] vor: die [i][b]Achse[/b][/i] hat mit dem rotierenden [color=#0000ff][i][b]Kreis[/b][/i][/color] keinen Punkt gemeinsam.[br][table][tr][td]Ein [color=#1155Cc][i][b]Ring-Torus[/b][/i][/color] ist das [i][b]konforme Bild[/b][/i] eines [color=#980000][i][b]Rechtecks[/b][/i][/color] mit den Seiten [math]a,b[/math], wenn [br]die Verhältnisse [math]a:b=r:\sqrt{R^2-r^2}[/math] übereinstimmen.[br][br]Aus der Zeichnung:[br][br] [math]\frac{h}{p}=\frac{r}{\sqrt{R^2-r^2}}=\frac{a}{b}[/math][/td][td][br][center][img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAc0AAAEWCAIAAACL3lumAAAABmJLR0QAAAAAAAD5Q7t/AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nOydBVQcWdbHs7vfmVmfndnd8ZnsrIxnNMlE0UYD8RB3d524EHcjRlwn7h4SSHCC0+5e1VbVLjjfqy6GEII0UN3VDfXO73A6pKl+XfXqX/fdd+99HaqoRjWqUY1q7mwdyO4A1ahGNaq18UbpLNWoRjWqubdROks132hlZWU6nY7sXlCNao01c4l5wrMJpRWldX7/QmenpEzR2rWe7RXVqOZqo3SWat7cKiorFj9fzNazK6sqX/3fl+xZvpE/NmlsWUWZp/pGNaq52iidpZrXtjO8MxeEFxpRzrp+A2Dx3hDfiGPE1avKVKMaWY3SWap5YZOYJStyVphKTI2/rX7/LOJA1uauTVOluaFjVKNaSxqls1TzqmYptSx7viwfyXflzY2tg9FR+vyM+U1KNdWo5oFG6SzVvKfFs+MfyB9UVro66W863uCa6Nqmgk2vrqC52CCoZX9HNaq91CidpZo3tEQoMY4RB2b8zforl+K67GX2nUU7b0pvun5clapq2zbshUDQrP5QjWr1N0pnqeaBlpJSdehQ/f8ls8jW5q1loawWHLYZ8bMcI2dNzhqxWezKm9etqwI29cqVLegS1ahWT6N0lmoeaAUFVeXlVWkvr0yVlJcc5Ry9ILjQ4sM2O0/hovDiAeYBS6ml8bd17lw1Z07VyZMt7BbVqFanUTpLNQ+0YcOq5s6tYtWyWR/JH8XmxTrKHK05bAvzwdblrXusfFxeWd7QGy5dqrpzpyourqX9ohrVXm6UzlLNA+3evapTpzCrFjSekQeETm6Rt/6wLc+7RYqRn7N+Fhgp/yvVPNEonaWaxxqYr28t3NqsFanGW2vrG6Sr02NzY21lNvyfCKKzWMyt7hXVqFa3UTpLNTe18vJyjUZdWoqFVJVVlN2W3t6Yv5HYjyCmjsyWwi1J6sTgoOj339/x5z8PPXXqNCGHpRrVahqls1RzU/v737/+4IPjHTp0TGGnjH823lxKvKVIWL2ui5cufvEF/8cfqwD/6rhNX/RMln6fgoIoxCl3GA8ukN4NijbGd198/sMPZUC1Oneu6vLTMKL0sE4jTGcPHdr7/ffluM5++PEvKQsi7s0MoaAgirszaDcm+5PeDYo2xnt//Q2uWkBn33ormCg9rNMI01mz2fzb3wZ16VL13XfGd7/8x6bnq8tQnVULUVAQghGWybkM0rtB0Waw6+Ak7u2R+wd/8q8rQGQ/+yz9zBl3OTyJrPPtKCld3Lfn2Y1LLGqlTi1amboogXO9GNHYdDAFRSsxqeRyHpP0blC0DcwaxcSE0WzZczuiUoh4k3t+ych9XuW2IoUE76fwdPVo9tV4/JvYdSq+PHfqk3EwzCH9tFL4OpTOUhBCJaqfnzT9Efu6TVv9mxKz4f7MEDuqJlYMazeCdTZp9WjWrzr7K6qHnGtzE6dW6FHSTzGF70LpLEUrKUN1l+gnD+bsUMN8YAXW/B7o7D3f11kMvVq6P3v76YJD5SilthQtgdJZihbj0KlFivzYtGUs6fNiRF3nf9uOzgLA1+PIstelLcsWJlJOW4rmQuksRcso1mlj05beY10pRXX1vqFN6SwOMN0fsK9uSF9p0sjtZF8ACh+C0lmK5lKGIsfy4/bnbCvWNWbYtUGdxSlHkZ3PN54tPNzQE4aCog6UzlK4TgmizRImbspYDUFcB6Jq4s1tVWdtzmgEnUq4LHV+puAxOCmkXxgKL4fSWQpXsOvUEMzdkbnhEedGmWtmXFvW2epviGhTePc3Z8TKIVbtFUAKijpQOkvRJFYtfIV+emfWhiqD0fW/avs6i1OBotsz116mn7ZolaRfKgrvhNJZikYAVlq2KGn+0xkWHWTVQc362/aiszgOnWrC41EMaQa1PkbxKpTOUjSACoa5S5Pns6TPm3TF1kv70lkcjix7wdOZKhWP7ItH4V1QOkvxKmaN4kT+/tMF8a1Z42mPOmtzBhWfLzx+KGcX5UagqIHSWYo6ZAofA5usDGltzFI71dnqL49oFz2bncp/QK2PUdgonaWohUYlmPpkvFYlIORo7Vpnq08oLByfMFIFU26EtgiispsQQ16m+swJ+eqVknlzwD8tEj47wM+iENtQtXDcaE5IMIcWBGAEBebPmA7+yswuAv+0aZQ2vZoXFcmNDOf1ieBFR/Ki+xjzs8Eb9BkpFkhK/rejcAPAel2WPD+Zd6+5i12NQOkshlULZQkfz0uaTmXr+jZ6jQ1VAW3VXDynOhoPZBR5dJ/t35vj78cO8BcMHypbvQLorFWjNLKKrGoF+BOLTGiRiaxKCcAoFciYBdhxtJBFLsKPaRHzzAK2ITtdd+cGtG8PeA2OL5wwVrlrOzi+5upFXv++sqWLkAd3bVqlzUglxfgwpYj2Nuv8gewdZq2C2CNTOvsCg1p6PH//qfyDLsYeU3gDQDdtqEZ94Zxkzmz+4IGam9fAL6GD+5Q7ttr0WissMwu5FkiCvRlVYyrc8KGa8htgCo6/tqrkAPBC/zwd2r0dKDi7Vw8g5aKpkzAdh6TGvOeknxkK17HrVFx59tLkeQqI5XCDF5HS2ZdwIGqZkrE2bVmhOLXxhGUKMtHCDrsRWJ3AgBVNnsD268Xq/pNk3mzd7esmIRt7A9BTtG7NpCZpoX8WiK9BC2TXxGGY6Hngn8jDe+zu3RxWAzBv4WPxWF19ys71YipQdGXKIreu01A6W99JQbUZgoS16ctNWrmNWiLzGjC1QsALTLa4keH69GTwS31SgrEgGxM7fWtTqwlbB0PVmPNXB5uFHLZ/b/AYEI4bo9i22Szikn4OKWpTZTBuy1x3uiAe3OlujamndLZBrDrobOGRuOxt5ShKoEecotkYNGAyrrlyUTx9Crv7TxYJtgSsT3uGz9wJhPh4A/BUMGoNORmqo/H8oYNZPbrZTZhha2QU2FoU7k5BFKWoLk3wcHPGGhnEfLVcLOFQOtsYDp1KDfO3Za59wr1ZjiKkD472iBaSLl7ICfDjBAUoNm/QZ6Zi7gL3fJYb47qcYQ8mcHBEZZHwmZ1/sEqF2O8xl7Hbb3KK2tgRFQRx1qUtf8K9VdrqwFgXoXS2acqcj75tWeskSrrD/Y8+CgyNUnPlAm70yRYv1F67ZDej7pYkD8XPIipDYS74LlZYxunZQ7lru4lVZDdQ9eQ8QQmiPZq7Jz53j8cUtvpzKZ11BavTWX4kd8/ZgiNWrds/rv2i15gFXExPtRCvf1/00T3nLz2kQZ7OU9Aq4eNHBeNGs3t0l/68QJ+RQv75b7uUorok7u0VqYvMWqXny5tQOttcoJWpi1IFDzz8PGz7AIXlMRWbN7C6dcWDouwWvYedmKTkg9lNiDE3SzJnJtuvN5YZocP80eRfjjYEmIOKFAVrUpcWiFMcOnLmo5TONhu7TiVTMpYkzxMriig3AiGYeSx57Gp27x7C0SORpASyJtFk5t0CE16PjSUzn8nvF034El+7Ra+RnCk4fDL/ILn7qlA629ITh2juMC8ey4szqKn8y5YCzFVgwaFq+bbN/OEx+pREO6mmnDfUN7AisHz9WixQAVEZ6XlUWGFryBYmzUmcVo6Qv4JN6WyrKNZp1qWtSBM8sLptEbzNgmXH/iKZOwtTW6QlaQWE4w06W4Ny/14OLVg0eaJFJiS9M76FXQerYN64RyNQlZj0zuBQOksAZo1i6IOBsIqKQncNLYQXYdGnPkXu3CC/P7/iVToLsEIy4bAYoLbQ7p3ui2ZrY5g08vXpK5L5972qCB+ls4SRJ3o2K3ESldHQOBa5SDAshhcVYbegpHemDt6msziG5+nciDDd7WtUakOT3GD+siNzvVcpLA6ls0Ri1iqO5sUdyt1NrY/Vg0Grij/ACaUJRw6zyETk9+cVvFNnAVaNEtizdoNWtmyxFZaR3h8vRA6xJiaM1mu8dLGE0lmCsetgnUo4I2linviZFz5XSTstFj1/0AC2X2/kwR3SO9MQXquz1aBqbp9wi5hn08KUbVsDuMumJ05ky7y6QBqls27BoVMzpVljHg4jvJCl72HU4SaYPiXJCsmchWC8FG/X2Zp+sumCUcOxs9q+nbZliC4ue+t15jlnvSfy+9MIlM66EasWukI/vSp1kTdElpADqpbHrhKMGYlViSW9M03hKzprEfMFw2K4EWFIwn27vj0mNZQg2izhk43pq5Qwx05S6kHzOkzprFtx6FRKiLUpY/VVxpmy9qS2doPWzMe2HjALOdgLsvvjCr6is5jfQAtDB/aye3aXb1zvE88worA7SzstS1mQI0oqbcUGtB6G0llPUIrq8kTPliTP48pz28PWOHarQb4+lvVTF4uUmG3sPIPP6Cx+kvUafUYKJ8hfn/aM9M54AKsOKkbUcTlbr9BPtWaLb1KgdNZzVKDoL4VH1qYts5BRycJDaCGgrcKxo8GsFiuM4gXZB67jWzqLg2XogpOsVmILjG13cawcRe+wLu7M2qCCeb64vEzprKcxaKSbMlZdY5wBsx4r2Z0hHGNuFi8yXDhmlEUpIb0zzcUXdRZHd/cGN4TmW081FwHzP54sZ336iufCxDJfM2NroHSWjKGj03Bk2avTFhdK0nxuBtQgzptc9/CudNliHzWsfFdnsY1+sVpfkJnH8tGT/yrAbjVoZPG5u08UHKhA9aT3pzVQOksaFXr9VfrpnVkbdSqRL06FXgJRqU8fx+awWsiG+qoD2od11ondoOUN6AsmE/iW6b6L1WnGPmRfW5u+zNYmyj1TOksmVh1UgSKxacvusy/7dCUauwnhD+yHJiWQ3pPW4Os6C7DCUsGQwbz+/Xy3smIJomHJns9OmirHtvhuI54QSme9Ap1KOCtxMlOa6YuGrVnEszmllvSetJI2oLMYeo1g+FBeVCS+564PAQY/ohKvTlucwL3RdvxpTiid9RbAoztT8HjBs5kGjdSH1FYwbjQn0L9tJN23EZ11OhAks2cod2wjvSeuY9EqrzHPrk1b1sYUFofSWe+iFNHufb7lZP4Bn6j7xR86hBNC8zm7qSHajM6+wBlmR343GgUMdbo0feLj0Xq178WouAils96IWauYlTiFLskgvScNgqgEw2O4obS2YcnitDWdRVSa82d5fSKxrYJJ70wDGLXyYfcHsqVeXQWm9VA666VYtZBIUTghYZRZ541rx8idG+wAP7xcd5uhremsDrNnzUIO9sL7gr1KEO3a9GXX6GctvrwC7OqXpXTWq9HCt1kXVqQsqtIbyO8MDr7NjLPaP/mdIZQ2qLNOrCq5YOAA75l5FCOa26yLK1IX6VTeWIbYHVA66+3YdSqtSrg9c/0V+mly9+zEEU+eKJ42hfRuuIO2qrPAqhXPmMHrG0W6VWvXwTDEBbM0gTzf4TuLva2H0lnfwIGomZLMeU+nc2TZdlLvFjOPaeazSD8h7qDN6iy2ja6KP3wof0A/m5m08LsKPTovafod1kVyBzApUDrrSxTr1A9Z1xYnzy1DSbhbVCeOkn4G3Eob1lmb0+PP7xctHDvK81ZtOYoczYvb+3yLTi30oZhFAqF01vcwa5X7c7YdydtTqfdc0jf6NIETFKB7eI/0r+8+2rbO2jCXukS2fKkndbYE0eSLkpelLGBJs9pDRdAGzwOls76IA1HzZDmx6csyBQkeiOs2CzicwAA4/gDpDj630uZ11uZMYQAXUXf/tgc+C1WLV6Qsus44V456b2CZZ6B01ocpQ3V3mRe3Za1Tu7UoJ6LihoeK585sk2X3atMedNaGPTXZ7N493Vq4Ejz7zxYc3pIeW673+WxsYk4IpbO+TqXesDkz9nRBvFukFlFB+/bwhwyymcgPdXA37URnAYacTBsYLQbiZ0LFiCaV/2Bx8jyxsqgdrnc1BKWzbQGgsGatYuHTmZnCx4R6wZyhsqja1+vsuUj70VkMtVK2YimBB3To1EqItTljVQLnepkXBCB6FZTOth0ciLpAnLo+fblUQSckOBFNT4YOxLVtn2xt2pfO6mBenwjZ6hWEXF+zRnEmP373801elFDjTVA629YApsTBnJ1nCw+bNK2rQIqokLu3iDV5vJz2prNmHosbSkMTH7XmIFYtVCBOmZ001Y7AbW8fJqKgdNYbAUMW/DSrIb0S0kggtUTJLZAXZsmf3JMk3v+VBxJ6tlzJV4D/1ckhAwRZtdgf4n9bgaIzEyfni1NsLUse10LVvoJ2Y8za2p/OAjRnT3KCAlpccU0D8xc/m8OWPsfXBvCxZ4QhVAGppRAsUjJy5DmpsheD9r4kI0la9FwOCZVg3CJg3MK/jluyT4VboXTWWwCjzaSCYKHyxgXxonmcMBojNJgOfkaEYkSGNQj43/AQBv7+gf2Y2zfxM5/KgOyaNRAMcyY+Hg3D3OZ2Bnl4l9cv2m5sX162dqizNqy4ZYxo6qTmPlCBsG5KX3WNftquU4NxK2YpzhwVzZrKoQWBQUsHA7JZ43bsCNaBPQJmjhwcyqJpg2VlKJ0lHzlXsTGW3yeM2cigbBkxA5nXzkuecR5syFjp+pzOzGdx/P1Ux46QfmY8TPvU2eYCFPYx98bq1J/pmcrFC3iNi2kLAAecOoGdeL9NlYKjdJYcSo1wdrJ04lh2dORLj33weA8Jog/qx9y5WZCeJFVLID0EgYkYeM6b1bBF8xLgN+D34H/1SkhAV9y5Kl68gFtjBdeMWvAzKrKo39a1N4vOlzQZA2vQCMeMEo4Z6bvbKbaY9qyz6NPHTXoPwNQegtgDzo/tOyQrKoIRUSOOoYywEGzcjhnBit8rYGTLdXKlwTluzeq649Zce9xC2Lg9f1I8cwob/HlYMCMi5MW47RPO6NuHeeaYsBit9kj4LpTOehQHCkvZyjnTOP2inBMr55AKpdHBMN2yjl+QKdfJlGAItthdhQ9HMHw1EmXCLcm0SZygAGwS57wZmCERzz9bOXrXkdQyY4Nqq7t/hxsWYkPaeEpCvbRfnUVU/GFD1OdONeQ9AGZB3jP1DxvH9Yq5ij+5wU+gjH0jmYfjBHy6AlFg8/2a5YFmYXeOW6sWBuaCSqS8fFY8chg7OJBe80FAcIcPZT2+Ky03q3x0+1tKZz1EuQlOSZCOGsaqeWIHB4DnP/vJPSksVOJLAcR+IjggkHVgVsi4igO7BaFBQHDpQG17Dbr56ZJxP89n6yHYUedD9RrJogXwof2kny5SaL8666xRi8dKv/R7LVxphc/GK7svXPftuB3h4YVg9Ab60efMYOemycCzvGXC2iRg3CJyiFcoj13BC/KvFlxgLgzqz9y0ll9uJP90NRdKZ91OpUV15Zx4RAwzNNhpvQZjVsDNS2JYpCzWe2hCBD4FmBvcAsXaFdyA3vTw8IIfxu3+YvKquVPEfLq8xIC9p9iZ+d6uAgzq0J51FkOjFE+bbIOkDlQNHvxVNvjADhFtxO3/zZxPi04OCWQOH8J8+kAKFBCMW890CXwQqoDy0uVTJ7KB4GLmbQijTxhjxWIuKod8yJlA6awbqbCoblzEFDaMhiksGCjLFnLzM2Qk+pvAwAVmyK1L4sgQVlCf1C9mLOo68MKCmbx+0VG03j0OzpzqaGcxBrVp7zqr13AGDwr8trNfb//Vy+9GRxf+Z/aMnsMuBfRk7d4q4Bcp8OcxKZQaYSlHceKQyL9XER7JEE6jr13JA5Yv+efNBSiddc9pNcD5mbJxI1m4DRvgR9+8ji9kkDlSa4NP9xJuS0YM5PcceqNT138VP/++it3l2MoPkx7cJb17ZNHOdbbUqAvz+7i8sHMVp8u0gX/9evga/16MYweEWqnSYwZs4wBVNanhS2dE4aHVcWPhIfST8cIqm7dPwiidJR6jClr+MzckiI47YZcu5AroCi8ZqXUoM8I3LkrHRP6hKvdHQEnm9369wivM3j5q3UR71lk7qpKnMrPPfIaPhCpml7kzL4A5uxcajPhc8PQRIS0Ic92CyeLwGGbWU2mx93W1BkpniQSIaWaSLCSo2m0/bjQrJ1XmJTZsQ1TZjf2DP67idqkq6My88vmH3UcvnydB5djSHOl98zDtVmcdCJS7bNcvb3Wd2u/vVYzOVYWdT67+sCAjhfSONQJQW7MGWr+KF+j024Kbbt8ugdfea5TOEoYDVe3ZJggNrr7qB/cIyoy+YBgiqqy43dFf/2P5pH9/992QXoMefjZrgX/0k6wkL7XB3Uc71Fm7UQffvHH13Z5HP/7s2Mefz/tH94khHy4Z/M8Zn/wLvXrJ+xdFwRDl5svHjGDhGWj9o5l8usIL18conSUGBU8xqD8TN2MnjmXrZErSu+QiVljG9u9tvHqpymGuMKuvnZdEhfC+n7j963Fb9+/hFuu9/U4jkPals3qNISezIKTvyXe/O/zvj8+86Xcuaq3ZgCiGxwg6duSG0ljdu7m1FjiBlBrgret5+DwyJJh+/4bY20wESmdbC3h4ZqfIA/3oeDZX3A5BqQ/F96FqKG63YFiM3fJiqzGTGpowmhMRzP73/Inhky+hivYite1FZxGVRcJXrFqR+G2PQ598fPbtLgc6jn50XeBwapMhIwXNSGV8803B3/+u3LHFpveNjBVwG0rYimGDmXjZhJVLuF7l+KJ0trXcvSrBE676RzN4ha0rRehxrFqI3bunFswQ6wwLPbxjsyCcxgiLzP5w4XChhEFIQVsvp13orEapuXSe5d/rl88/O/nB18ff6TM18rFRXUtMnTHUcPx+TnAQ26+3iVVEfp9dxoFCM6ey8byGwf2ZeqW3lKShdLZVbN3Ax3NnYwYxSwzeclGbhZnPrnf/EmAg5KTJIsGQDWV1HXFqVcJag6ZNlfZ4lTaus4jKyCjg0IKzAn86+u+OZ/9Jm/fd3gXz+C95M1GVYvMG6c8LwJDgDxnEoQVJ5s+16X2s0sWZYyJcamlBdJ3cK+5KSmdbzrRJbDyDdvpktheGvzSJVaNsckMaSKiICKPj5REW3Vn9kHPNqvOKgesO2rDOWhQS4cRxjJCgi13fO/e2/56O48LD2YfjhHXfiaoFw4cCM9YKSZGE+/lvvsnq0lmfnkx6/5tLTqq0z6+FmbResFhC6WwLGTuShRe5iNspIL0zLUN1LJ43oF9tz2y9GGEo/NeaDKlPJKMfxmhUvvqVG6dt6iyiUu7ezgml3erxSW5QzzNvB/UNyQWXsjCzPh9XLZ21G7ScwIDC994XDIvxOZMWoJZANaW/VGKSjQNKZ5uNA4Enj2PjoQWXzopIH08tBtizZhHPlXcaICg6slpqnydL2dLnExNGO3S+sULiOi3RWVTtIVrwjRCVISuFGx761O8bACckaHq3s31Csdh+Ab2BeQzQ2WEx7N69gM5iJ0TIyf/rG6ye3bW3rpF+dVqARqKsKRCqkZIptZTONg8HCk93ugvCQhhXz4l8eLMNo7ZZ0ZEmFTywX7V1wC9SmrWKS/STB7K3lxC5vS7JNFdnlbt2iKZMksyb7QH0ac+a9V0scpFgyKCCoJ43undkBflLVywfEFmIXz52fmOrtWYxzyzkVG93pNfwIsIL/9WRSwv20T2PEQWElakDXzyMboRIcyBQOtsMSo3wskVcPNXv3HGRF4ZDu4jdjAgnTYCPHLI15yuY1XBkeHWMsE6mtOtUKoi3IX1lhiCB0M3MSaNZOuuwG6VLFtlMiLfZs2C+L1u9ghNKu9b9o/ygnoLhw4BuDhnEwS8cp1GRxXi5ZpuJXZT3xz9yggOVe3e10KwmG61MGUajO6vdM8wkbYpD6ayrFOvhQ3sFQGHBNTuyX+htgdDNwliUx+raxcxnNfcPgdTilXGiIhj4Pk5AYRnSzEkJY1Qw3+7jsV/N0ln1mZOGvKwG36CFTHw2tGenHQhxSwM57XaTcutmi0zo4szDbkY1F87y+/V72PPTVP8fgDjqkxMdBvX0KdUim5valMiiavDwEI0fa4VlNb9R7tjGDQlmB/hhoSlkX6OWAQuVuFU7dBDTSobUUjrrEsB0TX4kCQnCBuvaFTyvTaN2CVSt2LBOMmdmy7ZZ1MogvMzjsMEvJMmBqB+wr8SmLStF6wkR8xVc11lw6qRzZ+En0G7Rs/1788LDwOSaExQoiBkMFNYiFRS887ZFLsx/6y271dCS/ug1+X/7m1nAKnj7nyZmQZPX1MQqEo8dk9br20e9Pqf791CByYpGWaxXxe0QgEELuHXJhRlYrXWwF1/WjHD7RHBDaUCCfXdrTmauAvfVLp7H8fxMlNLZprFqYRlXERJEjwhjTBjDLvOhdK/6ALcKsHR0d260+AjMXDkegbBo3ks76aJq8c6sjZeKTlbom4hh8E5c11n9sycvHKZaCH3yMKtDB0eJ1WE3wQf35fz2t44yO5gumDgM+r8/sRuRFvaHQ3fYjPlvvGEWN7ZcadVC8tUriwJ73u353xy/LrL580xcBlBMLAI6FatqBPRlw2q+SzOw+nTWZtIxvu3E/KkLJ8BPn5VK+mVqMedPiSKcj5xzx0UeDsSkdLZpzBpo6CAsimtgX6ZF7fPRo5rLF9i9erSyRMiVX8SRzgTHBzckta2DEkRbKE5dmjKPKc0Er0n/ss2iEZ21W9AX/9RrhCOHYw6BX38jmjqF1flHMG0HOgUkL//tfwLBtYKjrVllkfJdtwE1Vy5YpEJDdrru7g1wgcAfKrdutukgc7290kK4o4AbEQZs2OTe3wgGDkAeP6zpmEaqDA7AjIOxI1jlJtdOAqrmDxrA+qnLSzqLqKywTDRlEjcsRDh6JPY1yb5SLQM8aVYuxtZXggLo/CKPLutROtvUCTLAOzfz8YBnZo6PpdXWi92kM9LzW6mzQFsXzePgm0SoRHWXcav0xkv0kzuy1mtVQh9y2jaks1ZYzvzh+5oFd4tEgNSqhg50h/ntt7zoPqr4A9KF83iRESYu06pR0j/9lP7JJ8xvO72IUAbSadDWoeY4ToXNyOnQQXPtcu5rr1kkfPHkSYUffpD/5z+Z2HWTX8EfGnKzxGNGp3DzLckAACAASURBVPb69kGvz9ghQXD8fqcv+IUdMGkcdoGiIxjNmiaDsWHMe27VvmxPaCF9Zion0B/MhLQ3rpJ+pVpMhQXGbaaoSKYR9pzNROlsE+SkyoMDsZ26j8QJfTiK61cs+PoGESU2So1wvyjMezBkIPPVmh1WHVSK6tamL7tCP23T+sYkoCGd1T28Kxg0QJ/mTItCVLzoqNpx+44Sa3aHDmhSAnghmTWd3bO73YxZlMXlDoCjxFLz9Y352a8GbFl/fQ4BkTWx6bm//Q34CO2lX8BPcEDsIBXFLz0UEZVZzFesi83r3fl+z0/zA7pJFy3ACmvVeo8dhS+fFWM7DoQwclJkTX7x2tj1Gvsredh2q4HXJ4IXFQlMWvBE8dHAAxxgFuARtRtj+R7zHlA62xhAWPFqh1MmsMvN5A+R1gMf2scNpTWZA+YiElb12sKls+J6jSaHTi1U5C9JnpstSvT+2K9G/AaOYnPhxx9j262jauThS1v7ANs2q0MHm0YJxFcwfCjmQLAasO1jXz2OXuNwmOtQ+yEE5uaiyRMa7SSkuXCW4d/rid+Xqb2/E48YbsjNtL+SrKUWKQN6Y/kI61bym7fmg6qly5aIJk14EW+Ag6iUWzeJpk3hRoSD8aPcu9P7S9M2wi8nhOEh2I6ouWkemqFSOtsgxQZ45yY+HhCDKMgfHMSAqvWZKQTeJKuX8sD9HBxQBAkbDAIvRbRJvLvLkhfAMNeBeK8p1Ng6GKIq+s+/TUW5gmFDas8G9OnJwNQtfP99YKvazSgvNIQT4K+7c0N97nQLOpD72mvGhkILDBpDbhawJVP9v7/f43+C6Gj1mZP1ukqBjTZ+NAufZ1Ramnmh610HcwK+IKtbV+W2zXiMl1UpbvAgiAq82aqFTFymySvzmMuM8IQxbHzzaYvGE59I6WyDMHPlQf70sBDG2WM+nFz7EsDmItoMAWb+wH5Ok388u3HTqVJv2JS+6kLRcZPGS93cjccbIImPBENjtDdfykAFpquj2OoosWBTBC0EfqLpyabC3JZ1oP55BqKyyMXSObMKArtf6/4RK4wmW7IIm7k34I15fFeCb1BIz26exwCjYZ3FXAf9++puXeX1ieCGhUiX/FxvmTcb5gBJ116/kvO736mOHcn7w+9f9UJ4A0YVFOmcip04LPSA94DS2fopNarw/NoxI1iVVh+eItVGe+UCr1807j0kkNQnUnzzUVdmYUBkJz8eky1MJP1s1NO3RnXWbtTlv/GGVePR3E2rWgHt28MK9LvV/V+5gT0EMUPMksaK+AArMjgAC8hfMIfbEvkAOjtiGNvf71WddYZSrJDMn6O7dY1Dw0xaQ25mPUdAVLr7t/UZycDGB51Xbt1E+mVtiK3rsakYsKVETLfHHlA6Wz+PbkvCaPSQIHpeevONAu8EVUsXzJPFriK89hIwY8eNwp5JtECGS+/XqYok6aMfDjVpvcuwbTp+1pNlq7SQITOFE0pL6PVFUu9O3NAQfXJi49MRIKzHDgjxIgblppYaB+A7NmSBOjOA7WZUOGYkMGlFE8fVH7KGqvlDBksWziP9gjZOhUU1dBA2FVsyn1vi5vROSmfrvwCDB2AernkzOb60CU2jYPO+vlHIo7vuOLhZDeELYg9vSlxceAFqe7no5Lastd6zPuY9dRGtGog/oF920E83e/yLExIMH9znisMHFioD/bHlr02xPLfGxhiep2MxXrQgzeXzr/6v3aDN7tDBJ/YWS7gtDaNhXtr6q0QSB6Wz9fDgJmbMggmFlE1+hWCiMBTmsLv9ZGl01tkafp7PwWdh5uakchg0sp+fzk7lP/CGfXG8QmcNWunPC4G1eLnbB0XBvYUTxlkULgkWMGb3bMeWbftHMVuesoiqhWNHcUKD6/EbOAVUunCucsdWLO14yc9YjFffqHp9yt65/PUqpb8uiE2fxHarRUXpbF2qbKqYgdhsYtVSrk8Xi6kDtG+3cNRIu8ld+emlBjgqnBEZyki4I21u/rhYUTjlyVgV7FIxXPdBss7qNeqLvwiio252/1dmYBduCM3IyHf9zyEBlv0VHsI4vK8Vj9KG18FwjPR83EFsFnHZ/r25oTRo7y5yr1oryU/DUpNpgXRGthtNWkpn63L3qjiUxgDzL7H7veOeBNy0yJOH7gt7BNo6byaWgBQSRG/JVqNaOE3wYM7TqZXk1UYgTWcRFTAAhTFDnvbu9LDX5wy/HtobV5qV3AGM2YN7BODk9+vDKHMxxbZemtJZvLcYzp2SgUkLpNYXd1uoodykGj8aM2nnz+S6z6SldPblk26GJzn3Slg8n+vbRbnq4olZuVGlxHd1zs9ooWmAqiV7sjZfZZwuQ0moC0WKzlr1Wsnc2YVBPa93/zg/oJti3Vpsk4tmPg6tWghYZODk798lcMUzW25CKuutIoaqeQMHMLt2aUhn7SZEsmCe7v4dm3M/Dk5IMNBZ2fIlPp22kHRfGhpMD/SjK/ju8hNSOvsSAoYCTL7AkC1ws1/cw5gY+fLY1TY3xyQBq2rKBOwpNXo4q8WmgQNRs6VZ69NXMCSZHl4i87DOAs2CDsbxvvnx1r//mv7T18KBAww5GS2zDR/cwDa3jwh1dSbByM7o0KGDub4tEoDKmwXshqxpu1EnmTNLsXUTbtVqrlwAOssJDjK+UoHBh6iwqIYNxta99+0UuCmWltLZF4BTvDGWD4wCIBMVZh9+Pr+KPjlJPG2q3ej2iHF2vjw0GPN2NbgDlWuUINq7rIuLn83xZFKD53TWoEWSEkShEYn/eefx+3/l/P1DY5f+tvqujlWrBDR+NCATeFzdysU8F2UC2L9AZ8tMSMUrq1hA/RuvyAXHHxCOG1NdFcxZ3wuL8ZowzgOjy32cPCzCM3FN7inIR+nsC8pMMJZlG8q4eKqtJIDV4Cyy54EPKjfDI2Iw0+DQXgJMA4dOHZe99UDOjnKPuBE8obOIyizkyObMzvr3R/c+/Ev+u+/Dg0aXbD9e8U2MPTmjzputOqji1K3KxXvt+sYkjFugCPJv3iTs0a3rf3vjjSP79kRHhIkYhS/+C1VLZs/gxwyuW9+gFuizRLZ/75rqi+jTx5zgQADm/Xf/NXITdh0cRnPW+bzplnA0SmdfcO+6BJzrQD+6TuYb9aVcBVXrM1IaW9kglN1bBGBOAEwDm46A02jXqaQK+uq0JSn8B+4uaOt2nUXV8MF9jE8/v//RX9Lf+6esi7911Y6SXadKdp4oi/m5LGx6ndl6BRjuVx5XatCq9KIGJ/IIfGA3tgIWM4jpeuJi965dpk4cX2LQDugbtWf7lto9bHIdzKqUsLp1NWRnVHfAqBOOHQ1MWqDOL1Xp9SmK9fCCOdgq7pzpnFYtJDYApbO/nggDvHAOF5t8LeGVtqkVMCxDgd2rhz7TQ5XwjTAERBaL/c4ibMpfiugesq9tTF8lVTLcV9DWjTpr0KJJCfzuPZ99+Hdgxoo++QqZsrB071mgsNXsPlnx3TA746VPx3SWL6satbJMJGvooQW0tX8UEzzY4uNcDefCnQbFTkfw//3f/yGKWhVhXNBZuxmVzJ+jT095ccULc9gBfhxasPrkMTddGg/w8KYklEb361VkVBFvZlE6Ww14iEWEYnVmn9zzkN3nMYz5z9ndu2Hb+Xnk40qN1Wm4hK8qVOkN2zLXnS04YiXCUn4Vt+gsqjZx6OJx457/66ObHf/Efrejsm9MWVwthf2VctqUskELa/8tprMMYSVTWJVFb8ieZeViK7dBAXQ519VFTgMk/c1vfoO/BoIrZhVCQk5NbwXDh7H96qtvUJs6i3UGrXz1CixtISqywZxdr6fCDOMbqd29SrzrgNLZavDYjgA/uskNTzNy0V65yA0Jdl+Gwqvs2oq5DiLDmWUtzrJvGIdOPSNx0nPhE/CC2CMTrLPO7V6UO7dlfPT+3Y//kv3uO9IewfYN+19V2Gq2Hi3vNtYGvXCMYjo7YFHVrnPlDH69OmtH4OMHheBUTxjDdn3lVq+UDOwbjb/u2e2nRXPn2GtFZTmsBofN2PgRLBI+mpxUu9q3WcznBPhVx3gRfcU9A5jR/jy/ekZLeEwnpbMYYLyuXsbDYpVnt52CBjXIli0RTZ3kyfJ0cp4yJIgO7Cyhe3I9wPViSDPmJk1XQCwC3QgE6qxVByOPH+R1+uru26/ffvf3zz/4yLRgbenuUw2KrJOKTjHFRy/XHAS3Z8FPWwP5deAxNtUZSLdvRzPreTcCEPSmUiR0924Lhg15KeMWUcFHDznTFkI8XNKMQK5dwFZoQoIYFRaCj0zpLAawBUYOZQG74NLpNhdpgKolM6crd2735F4jFRYsdxmczwc3G64G3WpKEO35ouOHcnYaCYr9IkpnzTwmOzw86R9vXPr49cK/v1fYJ2rroO6WrfGNiyy2GjZtfbnfhBqZs2gUNrSxcFqrBgr0wyINip4TVFWu3n0YXwXYv6/E+VpVCm5woLOO13gfTVtAFRAeuUH4ToCUzmIYVRCenuDWHGfSAArr2XHvQOH5s7DV22WL3J5WV44iPyfPSeYRUIeMAJ3VwpJ5s9Pfe+fiR69nv/EWr4efbuNedFPcmdFRF8cNLN15sgmp3X2yossom9xV/2Ci09lFC6JXWQk6n67k3TrBNuZ5xexFnj7G0hZoQYa852696G4C2Ae4vXUynuDFDEpnMe7fqI7oImzy1e45exwL/I6KYJR7JOMDUYsmPR6jgFitOUirdBZVq44fyfv88+sf/j7prT8X/e8z9bL1uo17cHgr1uwa2KOkSZ3deaLim5jic7dc+UQwVvdswyK65swgztnlms7aLXqgp+irAbN6Da9/XyzGa/BAXyx6AOyDDWuwmmcrFhNsH1A6iztnMf/3jMmclheU81rMek5QgIlV6PYPehk5VxEShNWWVDa8bxixWLVQpuDxkuSWl5duqc6qDPnP6X49H73zl7vv/D7/nQ8U42doN+yuEVkAsilu18Duz2ZNa1JnywfOKxuwoCGHbG3KTdVZzkf3E+fsQtX8mEGsHt2a0FkTIhw5XH3mxKv/ZSzKA/YsUGHN1Uueue7EcusyVhO1f1QzgpFdgdJZrKDf9MnYeD2wy3P7DHsS5MnD+rdfdSdVVhUwZkNp9KxnHt2QwqxRnC44cLowvgxp9vY8zdZZLUApGDsq7d23L378Wtbf3hIER2rX76qtsDVkzJ99bERY+a7TTUjtuoPlPV+KOmjwDNtU+E6CyY+IjES0wrJGksFwsEK0Py9QbF5fz/+iasmcmfj2443n73onPLoiOBCrKYMqiYw7onQWi/QePABbtHnonpS79omz3ByWgHv2qMjD3hi7Tq2G+atTFxeKU5sV+9UsnbVbDYr1a3I/+c/lj36f8rc/szv9oFq+vl6FxZGu2bR9QFfDlgNNu2h/GGHnN12KVyVWgulCcABd5nLkrEvfy6iryaltEFSt2LxBsmh+vc4Bi5jHdsZ4ydet9uR1JwQwSwBnFUzFclKJtA8oncU2GaYFYme26HkbXATT3bkhW7PC884yMDNYPB9bCotdQXw0oisUI5p0waMFT2fqVE1Mq2tqtbios0CJkNvXC3t0u/HhHxPe+kPBux8qpszVNqywv7oO9u4a2D1x1pQmdHbXiYpOQxznb9d8XKlBV2nRV1jQSquhdjmujCQZGLRg6FYStQimwwRUNGk8r094k+tg8LHDwonj6w/KRlTwwX1YjFd4qM+lLWChR8NY4SGMK78QGSpD6SyckyLHMxQQeVvLUABAe3fxhw0lZW/n+L1YtsLkCWx3JIy7iha+VHRyW+baClRfXxYZVGxG16/fcOjggVIz2rTOomqLkMMbPeLJB2/defv17Lf+Luk/VLtuZ+MKW62zG/femTLqxKjw0l1NrIaVh04rG7UMDxGxI6qt69d2/fH7h7euRYaGSGqVHzzmzFAAokBkbTmX4w3MPKYhK62hOBYrLOP07gmkVjxtSt3/0igqbCZwI5caEfCTtIHRACV6eOViLJR+20aBg7h5GKWz8O3L4jAaPdCviPDgZG9AumyxZP5cUhZ/H93GEsbDQ+gVFjKjKR06tRJi78ja8IhzrQJ9yWPo0GtOnDhZbLVAcumtWzeb0Fm9RrY+NutfH11/77Vn/3iD26WXevl63aY4V0QWh71s1YEhARW7zzQRRTtpTXmv8dU9RNVmlfytN9+ssOh7/NQ1+dE9/Pd2BN66no8XpCeyHIfLOovRSLAgAuvu38ZM2rAQM59d+7/ALbx7z559+/ZdvHhx7969WrV3RdqCExu3E4viWLKAyJADSmfhfTsws2vsSJY7kkRJRzxzmmJDrCeTFGpg5crBxNavl1dEy5UhujT+w7Vpy2VKZs2ej6Vm/aDBg61ahV2viYqKMsKyenXWbkZ1Vy/Rv//h9ru/f/CPP9A//o9i0izd5n2uKyyOLHbzjgE/NRlyULpsN6azcLUXS0DPmzZxfFVV5Vtv/s346woVkAAgBEAOtqwndPEW6OyIYWz/puobOOMKFOti7eaGPbkaiNe3DxbjNbBf7eEHbmG73Q5+6g0G8FOuUDRrhx4PcOkMFpI4aRy75XuzvwKls/DyRdh4XeGGpGbyQVRg4qbc5dFksBoMMJb9EehPhzwV2tUkJYjuRP7+43n7TFpMxUpMyMyZM0tMaLFZP3XqVNOrOqtXG7MzuEMGP3zvrzfefa3grXfF0YO1m/Y2V2FxNBt2b+nXWbRqfRNSu/dUebcxNdkKUyeNXzhn1s4tG19Ue3GWPZo6EQuSOXmY4Ih6h1XvsNW3pc3LmBgFYGjVu9nti/cU5nCdG9sgj19E2la9orNNFjL3MHipkwF9iQztau86C2yBhc66kzs389vS7rbVoGrJgjmaC+dIyYMEwxToLIBX6FU7WqrAjb08eeF91uVyFBEKBBcuXDh+/LjZoH/Zb6CyKMSytWueffQuUNj0t/7G7R2oid3eMoXFQTfF7R7UI2XOjCZ0ds+piq6jbCLntrIq+T//8Y/kR3Wz3SrMqtHDseWaq4Qu12C4UN8AA4wog9bW1DY5gjGj8aIHNZ6ritJim8VUVV5i0qNV5aUaFZSWlqrXkD4qXpCTKscXGKtsntbZ3Nzcbdu2HT58GIIgHo/XLGH0ap0t0cMzp2A6e/ygsK0Fz4I7gc8ufnTPkfSYFJ0Fw7SPc/vb3FSvC+Qo1ml48twVqQsF8lwwqCorK4C4vNBZg1Z750beF5/d++fvH777R+ZnX0OzFjXLFdtQyMGZ0VG3Jo9wIft2JB7aZdepqsrspca60/MqqwoYXGE0xqPbhAYj6tW8ftHMH7+3Qo0d1mrU2nMy7VcuONKTrY3WgTMrpRxn0QMo/kD9h9IohwwZUmombZPjV+EXYqUmA/3oFuLU3xWdnT9/fseOHcELq9XaoUOHn376qVnC6NU6W2qoTqq5fE7sDW5EAik+f6aqQ4fKP/y+6re/rXz3HZvS09HBQGf7RWGB9B5OVXCdUlR3iX7ycN5enVoEFA3orELANtHz2LTghH/86doHvy98+33p8HHIlma7YhsKObg9ZeTJ0RFN62znEXYet5Gel1vgbl3Pff9tfNJ9DiGnAsdh0F4NpR37+ktE3HAAr15bOmk8GFoVv/0t+FkeHNTYKiuiUm7fzAsLudS1y73rl41w3ScumFv0HzCgrFH/g4fhF4m//+bEN18d0UgIu2Wa1Nnbt28DbU1LSwOvy8rKwOsFCxY0Sxi9WmfLTPC4UVg4/YMbbqws5XmsZrTqN78pXbOKP26MNuFe5VtvlQ4b4uFgRiwBpD9mcxGbsEQ4pYh2Rcqip9y7nOfpVwb0TX3nH5c+ei33z2+yv+usXrFRvXwDUWiWb0iaOOr04GDb2r2NsS6upFN/872HJg69XhwCTmhwUK9ePcPCwt555227iNvQO5uFhcvo9MUXfn5+4eHhr7/+OsoqrPdttmPxQF7Ne3bSu3Y2/HK6skMHx5qVjR2ZWdjxz38ODAwMDQ19/bXXLa9UTaysKLd6zVJYuRnt8uMPAQH+NFrwl198XkHQTjxN6myvXr2AtuKv09PTwWsul9ssYXyhs45Wt5LyisfLRxReiEOVYkKwaMWjhueHheTevcYl6pjegOXQfnAzoHKRJitFx2NaNq0H/0T0Kk/2wY5K+kfn0YJyH9/hkX5CmkTJY37+0Ue0kJBjv/td/u9eZ//1Xc5fiIf+x7/n/+FNzl/fa4I/v8vq3oNFC6qX/ED//37+Oc3ZunbvvvWrLxt6Z7OQhoV89/33+GGBLJ7r/MOr72GGBOs/eN/65pvMoIDcN/7GDPDTv/++7W9vgN83dNi8QP8ffvyR9mvjF+WSfq0b4dnDO506dcK7+s033yTcvk7IYY061e1pQQZIViNl5eXltVXy/fffB1KLv96zZ8/vfvc78OLatWu131NZWfmqJNajs4S0p2tGc64fdqBqQqiwqMeN4kSGsR7flRF1TG+g5PIFIKwOuPpLla1cBv5pt5s82Ycqu2Zwf3YYjZX6WE76CWkK1UPmlX/97a0uXbocefe1jE8+Zn3Rif1lJxbRZHzSMenDd/hff98kvNBQXmR4vciiIz/536e4FnzXufPxLj829M5moYju0+mrr/DDdu/Z83rP7q++h9snwtDxY8cbf+VEhBW8+SYnlGZ+523b398Cv2/osPTw0K++/BI/rJ+/n14pIftaN4aEVfTFF1/gvf3yy69YOZmEHLbMaro/qzF7tkePHlFRUfjrzz///IcffhCJRKtXr3ZdGL3bb2CEJ4zB/LO3r7Qtv4HVUPmXv1T07C5fvdJw4igQ2dJVKzwc3eUrfgMpxBh+f6BRK8fXwTLoCd/Eff3trq8Tc26KxAUEIpEUrr26bPyRka3ssB1RLZw763//+/zHH3/s1aNnBXHlWmZNnfzlF190Bu3H7x36+keLPe0ZGE4lUyayevVwrFuDPc6vXW7isNOmACMRaMe3nb4mZBdk91Gs14wcOrTT1998++33A/pGlxqJcbU16TcoKyv79NNPgUn72WefcTic//u///Pz82uWMHq1zpYa4KltdB3MqpRU/ulP4DbARHbyBM+H0FbZsJJdzpIcXhdvUH2KtMrJj8ewpFl4Sm5NvIFZo9h3b9v7+96N2hfBF+YTp7NFC87OnH16KiHnNjwkNyggO/Uxwc8wVCHWyYSNO0wdt2/g4wpQfCTelWgW7LBy39isREBXBPhl+/fKNqsJO6aLcV01zoRKZ2uWMHq1zhbr4TnTsbiu+Lg2F9cFTA9ULR4xTHvmBFlxXZHOuK6CTK/T2VJEdyh35y9FR02aF7G9teNn7ToVAouGHOr/Qdz7yy/Ml0rorddZqaRoxKEhG66tbH3/wVxhyAAwV6Dfu05kGAmQV5wm32mi53G6/GiTC310i5pG4BZUl0YkMKC+3ecpIPCiuVg+2LYNbTFPQa8RTRynPnWUlJuhxAAHOfMUhAwvSvgpQTTZgsRVaUu0KmGdPRxfrW/gQNRs/vPvd3/Taf+XF54db6XaSqX0Xpu6XU093fpvUWFWjRnBighhXDlHmL8LyOvSJUuGDBkSERFRZmkiJcxYmMvq1tVO0LTaq0hPlALjIDyEhDyF1jSv1lnAqqVYeR6gtm0z73bqJGjfblLybnUyCLcLNBKv0Fmgqjq1cOGzWQxpZglSj0A0VEfGolGeTj70yc6PQ+ODitipYklhy3SWI8juseEnFi+z9d+lzATPcBanPxwHnhbEnB+gs4MGDaoqtpbbTE2atFalBE1J8sXNaZok4Y4UTBSGDGRWElf/iNJZ+Mh+QXgIY+hgZpusIwPsWdnaNaTobH4Glr/Yqwe93Ez+eQDCuitr4y3meUfD+5A3Uq/LjqghGXv2hSnv7PjnnHNTxeKWSG0OMyl4W0C5gYBlKzD3WrkEm4etX80jah4GtHXkqFHz58+Xy2RevlrlVs4eE0WEMqZPJrKeJ6Wz8OM72I5AvXsWeWbHQA+D1cYnqdbyrUtYwcl+UUwiC6Q2n0q9/hr99OHc3Vq1wN6wyNpcqPNdqtflM1ICDvX8bM9/4x/slMtYInEzVsmup50bHNevTN9Yoqrr7NmO1UWcP4uwTRir7dmqCrsLT2V9RjK8by8pdY3dih2Bd2zCTuzqZYQ9wGyUztqwNDtsx8BAf7pS4BXTW4LHjVFnVZNTxmX7Jj6wC+YSuBtrMylBNFxZztbM2CJJenF9joI6uLifQoUBvZp6+vPd/wOC+zDrqkzGcDHYYO7paZOPjrET5Cu/dBor3xcziNVIeV9HqbW41AZ+Okqs4Gfjsgh0Njw8vNTkkrmNJDyQLlnU9A43vgYYq3hhqb3bBQQGIFE6C1faVGE0LA2f2B2BvAT1mZPMH77z/P0AbIF5szh4gVTPLzBixQo08rjsrRcKjwN71sW/atb+YEa1bPG1ue9tf3fa+QlMTiaQ0cZ1li/M7bGh69XUM0R9x4IMGc3p/m5wXQHVCIbFZHXowOvfVzRxPJcWzB8+tPEV0cqKCldTYIHNa9Q1Wa/L56gJ+r5zldq3hlCdBbPaETGsiDDG9Qu+Ed/XLCxSvuF5g/uLuA9sn6WhWOGIGxc8nQBSgmjvsS5vyYytd7GrEZq7320xsJd5OX1PRry/7b3ddzc1rrOphQ9CtgYS4pyt7q0aBiILpJZb0OB8xViQndOhg6PUBl5bJALw2k5Iwj6iMrGLzI3UmvFZLBoIP6vsfCKDESmdfWF5bYxti6FdOpiUReEKCxxKY9CC6MwczwXPAmGlizMWJ8+VQ6xm7XSL0+x9xZ1UGvV30i92OvjVt7s63cu6LJPW40YA1u6qSz9POjrWgRC2IIlbXuE0xt2GLC9EpfnldOF77zlsRrsZla9azuz0NSEronajTjR1snxTffuK+zj56djultgsgVApoHQWY/M6zJM4YiirnMQdA90DuMGkC+eZWEXu/qA6xZK/GgAAIABJREFUMLLlIc7dLXUe2d3SjqjUMG9rxtoE7vUypIULTS3TWSdQhQFde335h3s/iDk6oICVKnk59osnzPlpfee0ggcEfuVSIzxvJmYfbN/YwNY1Bi2r20/0//5XPHlCwbvviCZNsBOUoQuOI4gZrD51zANX1sP8ckocHsIYPqQxr3cLoHQWIyNRFhpM9+9FRxVtLZzFbtHzovvoM1I8/Ln7dmK7ro2IYRG4yVJDWLTK64xzu59vLENbtZTfCp2tBoFEA49Ev73zn1vvxPIEOaJfw78uPDs+cG80sHyJ/eJHDwgjQxnjRrHqD5VBVFkdOmgu/WJzphXkvf4aUW56u9XA7t4NSbjv7ivrYUoM8DLnLlab1hI8taV0FgM8u8AMF0htemJbWwoDUzzhqBG6G1c8+aFgvE6fgkXRx+0QuDubWSjPm/lkollLQExF63XW5kwhy2Mkf7uv0/92/Odu5gWROF8iKaJtCzj4YBfh3z0vFZvkBvSmW+pbvDLmY87Z4opiILjyVcsL3virzaBBnz1p/eeahVxWt65mAbv1h/IqKi2q/n0xVwyxi2A2SmdxykwqvGrXob1u1wVPg6rFUydD27d4MlWhyqqKjmCA+VfaEzdW6rLqoBlPJrGkWY1HxboOITqLU4pqD97f2XHfh8FxfuefHg3fQSPcmLU5XQdB/vQwGv3BjZerHCAq9blTBW+9VfDmm4wvvwDPWkN2Rv6f/qRYH2tREFAPwZCVxvbrZbd60SYIhMArxCobgFMq4xIc4knpbDWbYrFN24cMIDmo3h0otmwUTZvsyZDy58mYHybQj64WuyUk2a5Tx2Vvu0Q/SexhCdRZHLNGMfuXKe/FvbP/4Q53nIcyEzx7GuaiXbmE2zaXcD3L4f1YJtjg/kxinbM2SmdryE7BpmDBAXR+kVdtzkoAmhtXJPPnelJnVyzGSkZMHMsiPEMBmK5PuDdXpi42aIj38BCusziQnPOAdWXRs9nlKGFBXTWcOy4G0gBM2iorwUdujDZXo8vmnNROc27VvmdbA+uKrYDS2Woqraq+fRhgyF451wajaD15Y5SbVcOGMMF4vX5eRGxJX71GMi5hhFBWQJSjoA5u0tnqzqsl69JXXGOcbW5Ub+NI2Qowz6UF0fPSPRQ/ZzfqZD8vgPYS724mF0QG0QLpwNgCszHCD07p7K+jB4HXrsSssPGjPLFE7lFQNfLwrpGe75mPE9AVocH0oAC6nEeY06BCr5+TODWZf9etPXerztqcxrhAnjczcZJIUUDUMcFYnTgWs8LA6PXQ0gKiMmRnGAtyPPFZHuT0USyPuU84s9QNdfsonX1BQYYcT2RU8NtUoQNggPAHD9RcPOcBqxY8rpYu5EaGMmIGMQlxGpShuuO5cXHZW1GN2/PK3K2zOA6n62Nm4uTS1kWh1XD5jCg8lNE/ilFlc2/Pf6VtWSFOykww/rjaTHREFw6lsy+osKiGx7AiQhlH9xFZQsIr0Cg9kxVWZVeBGx6M1ztXWrsPUDGizpekLHg2iy/PLUY80XnP6CyOQSM7kLMjLntLhcvlFxoCVUD+vbGoxCf3PLEPG/r0sXztavDw9syJ8gzAtMKdBtkpbonspHS2Fgh8Il4I5g5AJspIKjHlLhCVFfZEaPDtK+IwGsO/F10ra/mcwK6DjRr5kuR5D9nXiTL6XMGTOmtzRtry5XmrUhanCx+1xmlbYoBXL8NcXjOnckrdndCIqJQ7t4qmTLKb2o7OAoNg01oeMLCGDma5w2lgo3S2DkYYCg7AAhLvXCVy2yXSschFzM4/mNh0t34KGKOjhmG1Y2KXt7x2JxDW+Nzdu7M2Fus8XZbBwzqLU4bq7rEur05bDMPcFq/vPU/GomWARSZmu9flZTchokkTlLu229rQhK/UCEdFYmvgl88SvHJbA6WzLwHUATcNxo9uIJfRN7FbDbwB/ZD7t936Kex8RVgINvkS0FsSG1eK6J5y76xIXSSWF9qJq7fiOqToLE4ZgmzPXHcsb19piwxbMFaHDmJFhmLVqd27GmbUsv16oYmPSDlLbuLUYWwFLCiAblK5K+2e0tm6sPPl4IzTgtpWDq5eI5k9Q74+1n1e2mIUHjMcW0kYNbzZky8wg5YpGWvTlyawr5ehpJWOJlFnbc5oBDXMX5Y8P5l3vwX+6Ps3JEAsQoPdKBYAfdozVo9urtao9QUqrKqBfbEwxAO73ZgLSulsXcpM8LyZWC2JaZPYbal8l+rEUW5YiPuWL8QseRiNAaBnNy+Q06yVn8jfvyl9VZW+iT1W3Q25OotTgqjzxM9i05ZKlEWNbGX2KmAqBgYtmPyuW8lz3yqu5vxZyfw5bWkbhXPHRGDQBvnTIaEbXS6UztZDvrNSPSD1cdsxaS0SPqtbVzeFPYKbfLizqvfoYaxig6vqACy4dMGjmYmTiNqxtZV4g87ilKPo8bz95wuPmbSuPrSAtt68hOeGMUyw2/qGqMjabs4dlJlU/aMxY3bfToFbE5cpna3v7BvhRXOxtPFxo9htZxNcvcaQn+2majK5KVJ80sovckkX7Dq1VEmf+3QasN1aUJDbTXiPzuJUIPoZiZPSBQlW17aHAdPePuFYtMyiORx3TIGtKrmRUdCW9qrZvQUrPB0UQNdI3Lt+SOls/bDzFcCeBabB5bOe3nbFvbjBswYMgUH9MaNg6iSOK0aBWaPYkbX+dP5BT8ZsuYK36SyOQska+2i4EmK7YvU/voN5aUOC6GIW8WU6tHducIIDbQQVCycdBwpFOh9LJ+OF7k6lo3S2foBebFmPBR5ERTAI3MadXMwcOqtnD4uS4JC1+DispHcojaFoKtHWqoWe8e/OSpziyu6znsc7ddbmdLA84lxbm7bcom3iDIOxOtxZXGLCaPeEgqIk7IHkllOKwLOcpc6C/OkGyO3LepTONgg4++AagCuxaqmnMsfdPbbMKLR/j43QbcZL9NVz1YN7hI0vv8gh5qiHQxUQi/Tz0BBeq7M4DkS98/nGBM6NxsNsWbny0GA6sGpPHW6LFZEIIuupLIxGDw9lZCR5Yg2G0tnGeHBTjLsdM595IqPR5wDCOtKZmBDkz2gknKgM0S14NjOJd8eq8+p4IC/XWRy9RjojaRJPltuQGwHYBAd2YcWUwfMPTI2J+lzk4V1O7142Y1uINKi0qgY4Y7mmTmB7pm4vpbONnh1D9Va4fSMZpQav1ggXMYt5gpHDrWpivP5Xz2ML3ICGNrUtRXUn8g/szNrgjnKxhOMTOouhhfNEyTMTJzUU+GXVwrQgbCo2diRBJYCBui9eJFu5rA0EGwDjYNkiLMs20L+oSU8XUVA62wSwSOnfmx4Rxpg7k9MGih7YzSg3lKa7c7P1hzLCELiT8Yp8r/6vQ6dmSDIWPpulUDKbFQdKIj6js07MGsXFohM7stZX1leJJi9dFhJMB2qybQO/9Z/lsOg5tMC2kQb2+K4UnBYwbu9ea22pI9ehdLYJwJW4f10SRmOE0xinDgtJHyWtBVXLViyVzJvTyu0VykzwYGeMQUgg3aKpx9Jf/GxuKv++mwpyuwnf0lmb02MLwZxNGauTuLfrVKJxoPCebQI8cyE/o7WTCc3l82y/3lafupr1YlJB+HLCvBkcT9bko3S2aYr1cOxyPma7hTIKMmW+Hj+IpU52/6k1UQfgHl7gdKeEBDHqxA9VGUzbs9afKzjS5Mq4F+JzOotTiuhyxU/XpC1F1eLaDzaLBo4ZyMRjZorRVkikQSuaON6ZtO0tkc4to8ICY1UgwhgBvemIwqNuQEpnXcKshgb2w4ZsdCTD4uM6a7fq+YMHwkfjW1z2GxhKuFv21mVxTd2mMlSXwL2xLWu9XMlykFEFpvX4qM7iFCOaS/STu59vqqiVvqwSY14vMG6HxzCByrT44FYtZJEKSP+OrQHMwGZMYWNZCf70hpYT3Aels64CCZV4adqYgYwKT+55RziISnv9Mpr2rGU6e+eaBD8Pm2Kr8+jtiEqiKAJmbJrgUcvKTXkJPq2zNmeYrU4t2pa19gr9dKVTbcEFyk2VBfkXAX2ZN5PTsjUxE4fu68tfYAa2eS2W+hVOA8aBxPNV/CmddRW7Di56Lgv2x9YWJo1j+3bVRFxhm3/zFGTKQ4IwkZ0ynoXHFAPT9Whe3Mn8Az6tsDi+rrM4pYiGIUlfk7aUKc0CRi64TNfOY8XXwbjdvqHZWfx2i54TFgIezKR/rxYDVPWXE1j9fnAGNq9zY5GdRqB0thmAIfvkroQWiEntlPFsnw4/QBLuy9fHNms1TMxSBAdgk9AhA5jgVJShyBPOrdWpi0n/LkTRNnQWp1yPnimI3/t8q1mrBBOOXVv54U5Xz95tzZRaLWSRiywKX80+B6qaeE+CBxjMncElqxuUzjaPYhS+dEYU6oyYmTaJXeKeXS48gD7tmXLjervRVZ3lYmV5scHarw+jzKDhyLJXpi4SyvO9pwpM62lLOotTgmhWpCxM4t4p1WuX/8zFveoH97hcaBVVWxQST25KTyxAZIFhFEZzhhKPYJG46R+ls80GSO254yJ8IjZ9skejQ4gHK/vd9MIrM0eOpyD3j2KqYf6BnO03WOdbs6WVd9L2dNbmdNpqYP78p9N5iuzF8/i41MbvdalyirEwh/n9t2YBh/Rv0ZIvjsD3rovxtYQRMUwCU+NaAKWzLQGM0VNHRPglHDmUZYR9MlXMCknFc2dZm6qUmJMqA/Y7+KYDopnPuHdWpS4uJ2/LA7fSJnUWpxjRPOHe2vl8/dTpBZHOSfSGNU3lL6Bq8dTJkvlzfXERzK6Dz50Q4+4Cp8iS3B9KZ1t6IRH4xoXqCxkeSkdasb0raV/BoudGRUJ7dzc0MbRju6GIw8DjJJQZPjB1fGKML0bFuk4b1lkcoLZbMlYPXXwhIgx7cA4bwmqkKoVVIWH79TYWZJPe7RawdiUPT1YEZpA3zDgpnW0V2alS/HI2kuPvzWiuXuKGh9rq24YEjM71q3jA9gkPz/9syUS6NJX03rqbNq+zOFatsuvBfiHR6WDchtEYWml99dsQlXjWDNGE8a578L0E8OQYM4KJ35UzpniLW4/S2dYC/xpXC9i73cdiue1mlBMcCO3fW+f3egiKjmQCM/aH8buClm5xeMdgdTftRGdxbj/K6bhkGDBsgYnw7GE95eg4gQGGnEzS+9kswM0Inhz4zbhzsxfdjJTOEgCqVPaLqn6EDurHdGVlyXtQXzrPDaXV+OCApD59KOkTyggY8LDjgpEH9vK9ZPMuD9CudBYg48m+Hbfj20mbI0NZs2dw6lxoK+wDJdZqcKBw/D4Bfg8CuyfLywqZUjpLDBYNHLuci7tr+4RjCamku95dBVHpHtytNOu1UmGxQTd9IjssrLDj4uFB0U9FDMhLpl2eob3prA0rZg8PHZ7333nTew+60b8Pi5mrqLIZsM065b4UMAumX0MGsPC7LyqcoZV63SoCpbOEASzBggwZHv8EnqgjYpgmDfm9ahI7ojpyYH/88o6cK5//8NknX89e02XEsUnjmHqlL1nlhNAOddbmdMRfOy/1i0zsuHjk2+8H393zH+alL775x+8dvlA1ptQI79lebcaGhTDWr+Gb1eT36lUonSUYRA5NHMvGPbZYnl8szy3bNBFHlcPQz++PVQU/VuX+qE785sOOI57ckfmMMU4o7VNnbU6plfEU/aOEx9d0rMrDRoLgxlfpiQ9I71gjlJng9ETZsMFM/F4LDaYzchSe2RyhBVA6SzzgGZt0X4LXAYgIYQyIZpw4JCozel1SDRiUsFC5YJZ0bOTvwa0FsKR+O3Lo0HblK6hNu9VZnCqH+cnB/+IjoTz7+/Dgiw9uSsrN5HesDkBhCzPlk8axQ5x7RtAC6Qd2CQyQ1/kKakPprLsAahUfJ8QLAoTRGDGDmaePioAEe4OKgZuHnadY/jM3JJj++Zyf//PNgMubP9YndupP+6dG1n4372vnOutA1f4/fJVx6lPVg6/+98FfPp0xx69P4riR3Hs3JJVWr7ASKixwaoJ0xhQ2LRAzYmhBWO47n67w/ukXpbNuxIFg1RR3buLjahseQo+KYOzaKpByFaSYCXYUU9iE29I5M9ghvfg/jt/z+dTl0dFFty5Dd65e2b5pc2bPHprLF0g/b2TRznXWZtDJ+/Y5tXrlgd27CjJly+dKvhl24IuZS2m0/KGDWMcPCQ0QREpBD8wbgMCXz4rGjWThNiz4OSKGmfpYWm4i+6S5BqWz7h8lKDY9X7uSFxxIx4uzBfjRZ0zhPL4tMcAQMdvkNQp42ldaVPwixfF4YXQkIzyQ4zf49n+nLuo3NuH6OVmpUVVtYqNq9ZkT3KhIW1OZuG2Vdq2ziEo8eyavb5+aqMQyo4qZK18wh/OfSYu7jI6PpHH8e9NXLuGmJ2Lu+xL3e0LBjQNktCBDtnuLICSIgQfGgpto6kT204fSSivsQzubUDrrIcAzGYzOfTuEA/oy8XIBYTS6X2/6wjmc+9clEpYCCC6BXnwgncD0sGggMEyPHxQO6sfCAiFCmaEROf+bsCJm5fFnd6Eq26sjVcXvFy1ZOK99Sm171llDQS4n0N/EKqzz+1KDCuLDqzen/3fW7N4Db/cJ4YDRG+hHj13JS34khUXKMhNM4LQdjNsyk0qvhDKfynZvFYTTMPerc52DDj535WJuUZbcF0s/UzrrWbQwULeE25LZ0zjhIXS8YhuwcIMD6BFh9HUrec8eSjn5cp1MicsuGHau+HPBe8CbgbACE0DBU4KxeOuSeN5MLjBAcE13VpJndRm7P+bIYgVL00jlXGNRHrt3T0NOBvnnyuO0Z5216TXK3TsbrHSBwJUGZOPFsz/9vIIWmRERwsSDF4HgDhnA2rONn/VUKqQrjM75GW5SuPKh+LgFfwIMAilbkZsmO3dMOH4Uy693Uc2tERpEHz2ctX+XwKKGvN8P2xCUzpIDGF5VNtXVX8TTJ3H6RzPwgrZ4GGBIEGYvRIYyli7iHtknfHJXUpgl5+bLwcRfwlbWIGYqwG9YufKCTPn1C6Ltm/jTJnECetOB3YprK34nhAXTRw/lDpuWMP3hNBMidTRZS1Svgfbs0N68Tvop8jztVmdfNWPrxaqDqoyGnXlrpu44NXYMlhEA1LC6lFIIZniC4de/L2v9at65Y6KMJCl43vMKFXXGrZCB/YaeLc9JlV05J966gT9qOBP8ITA1cM8AOCA4LLBCRo9gbYzlq6VQmY84YRuB0lmSwSwFC8zOk8cu540YygSaC4Ys7satUd5qxaRhclwbbLfzkBfvwf8E/KaPc8uDCaPZvxwTa2HRmqz5WcInxc3dG9F3nF9E0T511qoQYx4DHqsZJ0oj///2zjy4iSvP4+SvHJX9Y2qnslUzmdr/piqV2d2qSVLZZUJwIJyBQA5COAOEYG7iGAewHU5zX8Hhxlw5gNjAxA5gG982si2f2NZ9391SH7rlS7b2yWbAxWGQ3FIf+n3rU6mOLEtP7eajX79+/d66yq+M9mZRiSl1reLzWdIZHzw4bocekOgQfei4RQ4dnLt56NMGf3falI7Zn0i/XiW/dV2HTs74O4n+o4BnOQQ6LULnUJ2krbrYcGiPeuUy+ayPJR9Nl8ycJpk+tWP6lPAdvUNBj3z4QcfM6ZJPZko++1iSnqY8f0orbzUH3Vg3jXkJy7WOnw6Ks7opItKW+GwmxfSprqZ61vdJPElMzyJcTRHPF9NJ4BJD3frK1QSmRQdbr8vmtNmKC/Q7Nqvmfy759CPJRx9KZqDjdkrYnkMPWvS/A8etZPC4RY7evEmJCluLyhL0YF287RkYHvAs10HyRXTR93pghzL4IPrpQ3246Pzurv5OculinyPae2dJjLieO7DaAvt7IG4koGetx37w26Mf4d9F2s82Hz0i3vPQHDTogHzicUs/8bgVMOBZoREgsIXFn5usEZwGDoPXauDv8lCRkmiedeRdUUx832sa6Z0pHodlXXlyleom65+Is4BnhUM36dhVu6VQnudjaGJGr1ErT3rXuCWT9Y8WHxLKsz6rQTFpAlnMmBzNVtncok/NlkTZgREBnhUCXSReIL28p24riTM8nZ1TLFJMGE9VlrH+GeNAAnmWxBRTJxtSU5h9WZ/DWqH4fUXpkj6KYv8zcgnwLL8JoLM/TPN12XKjpSNAxOQE33b6uGLyJD+vZn2OjsTxrHbpEvnECf6ou++HhcJ1h8W7TjYd7o38AqxQAc/ymCBJplWuua243hnpmK0I0Xy1RPnhNMHfJJY4nvXj5pF3yw5DJ4GrTc3fVK5q04kiHlAoRMCzvKSHIs63HjvddITG9f7YlLFD8eFmsmRgNlJBXxNLBM86rl6hK0rj817d4cXMf/u6fLnLwb8lShneFeBZftFF2lt0NQfrszTm1rhXCph53x7W90DsSATPWrMPO3IvxfMdfQ7rqcbD34t399M06x+fLcCzvCFAYA5cs+NORqH0Kis9X27JXfn4JFTbsr4rYvUBBe1Zr1EdHhBN4fE/KekkMK25bfOdDWWK/J6E7LQFz/IDVMb+1HrqYF1WH83qldyBLtqYdu2xiIA96zPp5eOSHAVsTluBigNUImwVbTRbpTG6ZstZwLNcp5t0lCsK0qrWGjkyMtFhlY99l7jN6cWjokOonvVhJsXUybqVywMe9odbhWjXvvptOc3ZTI3y5gXgWe7SSeAqU3NmTVqV8iayLevtuY/t1An5e0mk4AbVCtKzPptR+cFUzRcLAj4n640ZBBWzHod5bXlyhfL3LiIh7u0Gz3ITzGk3nGo6nC3eG6K58s9jKLZj2fL3xtKiatZbwiAC9CyJqT79RDN/bsDHuWtQnSTeoq/6tmINKiY6hd6NAJ7lHD6H9Y66eHnJ4oEalrvnVpYD+8Lz6anlrLeEKYTm2YHbEHyYye/i7qWnIEXmNGcfbdjvtAv5RhjwLLcwWiRf3l6gMDbx4kIBebMgvBGb24rij5A868PNiimTfHZ+/GlCtGtN+bIq5U2hdtqCZ7kCKmMzqlN/6/ili+RZj5VhYxpRWMB6M0aOkDzrp+zGjE0BDleyj0LhuqUlX5g4cr2XUcCz7INK19z2C+nV3/D0DkUi/5qrtYH1ZowcYXjW1dZkyznFejOiplZ9O6V8JevNYBbwLMtoTC2LiufZbSrWWzJCAh7KvHMbr+dA4L1nSYwqK5aPf8+0JYPXfwh0bpct3nuq6fsekk/1+DCAZ1k6kgb6oZaXLkHf3ly+2PXseM06+YTx6gXzol/EgW147dmA02E7dVw+Lgk7nyOAaSjQSR6BaTdWfd2krRTAaATwLAuEaOeh+p0XW064hHWN1WsxKD+cppw+zaOSsd6YKOCvZ5FkjZs2yMaOIcuKBSDZB5+LwDr0tesqlvN9VVDwbHx3N+WoVRV/U7FKb2nrJHh8ZvckfLhZtyIZnbq6WsSsNyZSeOlZJCBX+B4WZ12Ns7FOSJK9j9dhud7xy966rX007WO7MdEBno0T6JuZwnVry5PvqIt4N6IgMtyk/dqv4Q0K9zs5dBvbU+GjZ6mKUvXc2eHJfYRo2PugogSzKQ+Ld9+U5vZSJOvtiRTwbDwIOd07RZnnW48HYv9enAD9mycxa/Zh9fw5XB4k/xA886ybQDvZZ7eg/SyYIczD0006WnTVO0QZRksHv04HwbOxJUiRBdIr6JTHYpXx4tYDBvGo5VT5beQCr0nLi4GcvPEsiXnUMs28OfdWUeTncMDo8A1Mcn+l7dwh8U4ezY0Ano0VnSSuMbWgMlakKuLULDBxJuAm1fPmGL5d7+X8CmO88Czan/arV2Rjx2i/+tKjVbLeHnZ2AoF5Ccvu2i35kstBPnQjgGdjgsduPtvyw5mmI3zsS2IcV2uD6rNPFVMnU2VFXO6x5b5nPTqFbvUKJFnbmZNIuKy3h126SLvK2LT1ziaZsZ7jpQx4lmF6SEel8sb6yjWd/DmpiT3WgJc2Z22TJb1r2JgWntWf/SY9Bk57duC+A49KpvlifnjYnKCvekVEL0nmtl843njQgWk42zUHnmWMThJXmBrXV601xGyJb35DYi5pm3nXDuRZn9XIwQE6nPWs125VzpzuVkrD17ucHP2WYpcektgmSi+V58d67efoAM8yA4FpDtRn/dZxqVvYY7YYgcKtRw5qv1zCtT4EznmWxLxGjc9uCbhJY8YGr1nPfpO4DW3XL7o9V29u49otZODZkeIlLIWyvG01mxJzgbnoQO5wNtSiDefdJq9Zx5GzYE551oeZsAs58nFJ9wYVAM8GOpVs0lWkV6dSOIe+lsCzI0Jrbl1cPJfjffCcJeAi1PPnyN8fZ0zf4LdbWG8PRzyLvoSIW/nyCeMVkyfaf76YUMO2mKKTwI+I9+R3XPI62D+u/ODZqHHbTavKv0JnKKy3hO+4musVUycrJr6vX7cmwOolMo541rTtO7Q3bCeOJcjdBzHly9sLUDHEejPAs5HvMsJ+rOFAbvsFjnxVCgGH1dVYp5wx/d5QBIqdbgR2PWvPu6RbkewfqGfDi82w/kcRCnpz+8Kizz0OM4ttAM9GQCeJ16lLMmvSnLg+Ue6gjScD5ZtXr5KPS/KopOFH4lvesuBZEgu4SMf1vPC7S9psZ0+z/1cQIqgkuiXL2yXKZKuLDzz7TCCrkrhuedkSjamVa5cyhQaqbWXt4Tv3cbPyk4+wi+cCXjo+F8ri6dmAm3SKRZoFcxUT31eMH3fvcW5cDxQkAQKzY+rN1d+WKfLjb1vw7NPpIYnNNd8Wya7CqNh4gk6fLdmH5WPeUUyZpF+9gqoqi/U7xsOzAxW6Ry1TzfpE9s5o3eqVtFjE67UP+EUXaZca6r+uWEngcb2pATw7HP00ldd+8UTTIRLXgmRZYEBAVEWJbmWy9P/eHpz9z9nS4I/N3yJ2nh28vudsqjekpqBtn82InT/tlndwbQRxgoDqWVQ2ba7Z0BOvwhYjQ8KyAAAI+UlEQVQ8+3jQ957MKN4uSkfffjBsi3UCLsKjkfvDN/grJW+94TVp/SROlhQGPBSDM4Ex7FnaHvA53fJ224mjpq3foe8Mj0pq2bf73pgKqGHZhrYbTjQeutx2NkhRsX4v8OyjYC7ceKA+q0ByJUiRHLw9NLHBwrefEuHZxGVj3pGiU++VybZjPzgba0feucmAZ/81tbmzsc6YvlE5fZr07bdUn35szT4cGCxdQa9cApVTEkPdTtF3bQZRTMsp8OwDfIS1lyR/bTv7fcNuvq9HJHwcVi9mJAoLzNu3qGd/Jnnz7662Zj+N43mXXe3N4Sc4HeFSNxKvRexZZHZUtHopn83oM+vR/+KXfrQc2o/elK6p0K1Itp0+gepZVNWCXrlMP01fa//xkHin026IUfcgePYePRTRqq3OrFlvtEi4ORUF8HhILOClPQYNMm/AQ+lWLidv30IPWvbt0q1eYT12xPHPPPpOpXtwaUgKf9DPgNw3sO7D/Zd6jGfRT9HT0Jn+YDXqsIa30X8HVvDGzp0xbclE5SoqWu2XfkJPIAr+iV08F34O+i0+TG0ODBKusShyb9223PYLXTFYqQE8Gx7wgdtU++t2FMuv8WLOYGA4BmtYCiduFRgzNulWJqvmzJZPnGDM2BiWY3mxdtnSgJv0GrXW7w/hv1x0XP+VLCmkym/7MBPyrOZWgau9xT8wjJfIv+a4ehnLOWk5sNeQlhqez8VHS0f/r+tuIyqc9evWaBbON6SmWI//QFWgXx8cBg8XS3kMUgFmU2yu2dCorWT2Elmie9ZjN19tv5BVm9kLXbGCBJ3Xu8mA3+WnBq74t7fYzpxED4bXfVk4X7NgrmbeHPXsWaqPZnhtRjdmbpg9KzxDbnj52Gr1nNma+XO1ixZqly7RLvvSZzMifSNT+6wG/8AAWF4sxgNECjKsSF28q/Y7zKYMMDSiOXE920ngcqM4uWQx3D6boFB4AFnYaUe6HFybANWzZp0yMLBMd7g7wulAP73XvcB6a4H40kdTB+uzLt3NcTNxw24iejaAqhW7Ibl0sdzYAKNigftwZB4ZgDsgP6wpW9aqqxnh6yScZ1H1er7l2K/tF/i1LjEQB8CzwGMxW6WLi+e57dFP7pNYnm0ziDZVp0BHAfBYwLPAk0CF7Q3Zr1mizOgGIyWKZ9Fumlf0mdUqY/0PBnAW8CwwPKik3V+3PV96KVLbCt+zvRSRWrVapC72ETBrMjAc4FngqQQIm9ki/apkod5y99mH8QnBswEa95M2r/3hy4LdpOO65OfjjQdpO4dWCgI4C3gWeEbQ+XGtumR1+bIQTT/L83ns2S6XPRTq8tstEmnIjjZDwaF7QWe+u6nqG5WpGW7uAp4R8CwQEU674VzzsXPNR3ufdn/TYz27alVo8+bQnj3MiDEmnu124bl5/fn5ocryXped9lJUt+ve3Ps9FJFe9U2F8kbcJj0DhAF4FoiUTgJXmpp2iDLadHe6yCcu/zHUszduhH76KWSxhMzmkM0WwhmqcWPi2VAX9fbbobS00IwZId+/lkpF3yqnm7PPNB+BEQVAFIBngejophylioItNRtcduPjnzDEs/PmhTZtCqEakdnExLN9fmLL1lAwGCq82eN3WLtJe72mZH/ddoOlIwAdBUBUgGeBkdBNOk40HTrZdOjRCW2HeralJRQIhPR6Zr0Yo/5Zh9WsD+zICvX6vDim2l+/o1B29am9JAAwDOBZYIQECMxqle+4k16lvDl0Qtuhnq2oCO3bx6wUw2HYs6LtX6h+O3Nvfli79Ze7Zw6Jd4WcLtZ3McB3wLMAI/SQRLXy5q7azTpz2+Cyrb1+T8m6SZ20nVkZDg2Tni0qKvrrn5MmvztDJ2mrV5dm1KQ+OpwLAKIDPAswSIh2HqzfebHlpJ+27c3a/ud/H5uasr6rq5tBHw4NY55taWn94x93vvlm6K23Qv/2l9fFit89eqVTJwMARiBU7drmWtabAQgJt1752sd//e//sgyKa82aVKZ8+FAY82x29h7U1jfeCPPqf+ZVrp3we3ISADBFwbKx1xb/g/VmAELiRnLSf7w86r64XnrxH0z58KEw49m+/r5LRZf+8uovg8197bVljLwsBHI/vb29tNvLdisgQsvSpSv/9jdsUFyjZ04JhoKxeBcGPOvp9qwVrXX3uHNyLvzhD6++8sqfaZoa+ctCIEODPEuSJNutgAgwkyZ9+Kc//c/rr/891B9Kq0+rwWoYf4uRenaDeIPYLmakKRAIBMJ6XF2upVVLsQDG4GtG79nLmsvZHdl9/X0MtgYCgUBYT39/fzvVjmpbpl4wGs9q3drUulS6i2aqERAIBMK19Pb1XlRdzJHnjPylIvZsiigFmX7kbwyBQCDcj6fHs7Vpq4yWjeRFIvDsnrt7crW5PX09I3k/CAQC4V2QZ9Mb0p1dzuh+/Zk8W24t3968Per3gECiTl9f3/Hjx48ePVpSUsJ2WyAJnWBfsNhUvO9uNNMfPMWzJq8poyGjhWiJqmEQyEgTDAZzc3NHjRp1+fJlttsCgYS7EU5ITxSaCiP6rSd6tr+/f3fr7p/VP8OIAgi7EYvFzz33HNutgEAeROlSHmg7oHapn/H5j/dsmbVsW9O2rmAXcw2DQKLMrFmzXnrpJbQRCATYbgsE8iD5+vxzinPdwafPPvOwZ3Ue3c6WnVJaGpuGQSARZ9SoUc8///zMmTMXL148ZswYtpsDgTxId183KmxLLaXDP+2BZz09npPSkz8qf4xxwyCQCELTNPLs2bNnQwN9tWh7XyzmYYZARhCqk0pvSNe6tU96wgPPZjZm9of649IqCORZ09bWhtw6uN3d3Y22Fy1axG6TIJDHphqrPio5igrWR3/E8HoKEAizSUlJefnllwe3dTod8mxdXR27TYJAnhRUqu5u3V2D1Tw0fAA8C+F05syZ88orrwxujx49+sUXX2S3PRDIUxPsDy6qWGTz2+4/Ap6FcDo2mw3Vs8FgsLCw8IUXXujvh64tCD9i9prvb4NnIRAIJLYBz0IgEEhsA56FQCCQ2AY8C4FAILENeBYCgUBim/8HKdA8wDr/Ta8AAAAASUVORK5CYII=[/img][/center][/td][/tr][/table][/size]
[size=85]Eine [i][b]konforme Abbildung[/b][/i] ist[br][br] [math]\mathbf{Torus}\left(u,v\right):=\frac{\sqrt{R^2-r^2}}{R-r\cdot \cos\left(\frac{v}{a}\cdot\pi\right)}\cdot\left(\begin{matrix}\sqrt{R^2-r^2}\cdot \cos\left(\frac{u}{b}\cdot\pi\right)\\\sqrt{R^2-r^2}\cdot \sin\left(\frac{u}{b}\cdot\pi\right)\\r\cdot \sin\left(\frac{v}{a}\cdot\pi\right)\end{matrix}\right)[/math] mit [math]-\frac{a}{2}\le u\le\frac{a}{2}[/math] und [math]-\frac{b}{2}\le v\le\frac{b}{2}[/math][br][i][b][br]Konform[/b][/i] bedeutet [color=#ff0000][i][b]winkeltreu[/b][/i][/color], dh. die [color=#ff0000][i][b]Winkel[/b][/i][/color] zwischen Geradenstücken im Rechteck stimmen mit den Winkeln zwischen den Bildkurven der Abbildung überein.[br]Die [color=#ff0000][i][b]senkrechten[/b][/i][/color] bzw. die [color=#3c78d8][i][b]waagrechten[/b][/i][/color] Geradenstücke werden abgebildet auf die [color=#ff0000][i][b]Quer[/b][/i][/color]- bzw. [color=#3c78d8][i][b]Längskreise[/b][/i][/color] auf dem [color=#1155Cc][i][b]Torus[/b][/i][/color].[br]Die Bilder der [i][b]schrägen[/b][/i] Geradenstücke schneiden diese Kreise unter konstantem Winkel, es sind die [color=#BF9000][i][b]Torus-Loxodrome[/b][/i][/color].[br][br]Ist die [color=#9900ff][i][b]Steigung[/b][/i][/color] [math]s[/math] dieser Geradenstücke ein [i][b]rationales[/b][/i] Vielfaches von [math]\frac{a}{b}[/math], dh. gilt [math]s=\frac{m}{n}\cdot\frac{a}{b}[/math] für [math]m,n\in\mathbb{Z}[/math], so sind die Kurven [color=#00ff00][i][b]geschlossen[/b][/i][/color] - ein hinreichend großes Parameterintervall vorausgesetzt. Diese [color=#0000ff][i][b]Loxodrome[/b][/i][/color] winden sich also [i]endlich[/i] oft um den [color=#1155Cc][i][b]Torus[/b][/i][/color].[br]Für [math]n=m[/math] erhält man einen [i][b]Villarceau[/b][/i]schen [color=#ff0000][i][b]Kreis[/b][/i][/color].[br]Ist die [color=#9900ff][i][b]Steigung[/b][/i][/color] [math]s[/math] ein [i][b]irrationales[/b][/i] Vielfaches von [math]\frac{a}{b}[/math], so sind die [color=#0000ff][i][b]Loxodrome[/b][/i][/color] unendlich lang (theoretisch): sie winden sich unendlich oft und [i][b]dicht[/b][/i] um den [color=#0B5394][i][b]Torus[/b][/i][/color].[br]Im Applet oben ist [math]s=\sqrt{2}\cdot\frac{a}{b}[/math], der [color=#1155Cc][i][b]Torus[/b][/i][/color] wird mit Hilfe der irrationalen Zahl [math]\sqrt{2}[/math] "eingewickelt"! Der Parameterbereich für die [color=#ffff00][i][b]gelben Loxodrome[/b][/i][/color] ist [math]10\cdot\pi[/math] mal so groß wie der für die [i][b]schwarzen[/b][/i] mit der [color=#9900ff][i][b]Steigung[/b][/i][/color] [math]-s[/math].[br][br]Natürlich wirkt es sich für große Parameterintervalle aus, dass auch die [color=#980000][i][b]ge[icon]/images/ggb/toolbar/mode_zoom.png[/icon]gebra[/b][/i][/color]-[math]\sqrt{2}[/math] nur ein Näherungswert ist![br]Würde sich ein erkennbar anderes [color=#ffff00][i][b]Wickelbild[/b][/i][/color] ergeben, wenn als Faktor [br]sogar eine [i][b]transzendente[/b][/i] Zahl, etwa [math]e[/math] oder [math]\pi[/math] gewählt würde?[/size]

Information: Einen Torus mit √2 wickeln