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count="2"/>Линейная функция. </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">Отображение, осуществляемое линейной функцией W(z)=az+b </inlineAttr> </f> </text> </region> <region region-id="37" left="48" top="66.75" width="461.25" height="88.5" align-x="48" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Линейная функция однолистна, аналитична и нигде не обращается в ноль во</f> <f family="Arial Cyr" charset="204" size="12">всей расширенной комплексной плоскости. Таким образом она осуществляет конформное отображение.<sp count="2"/>Причём имеет две неподвижные точки, определяемые из уравнения a*z + b = z : </f> <region region-id="38" left="48" top="121.5" width="54" height="33.75" align-x="66" align-y="141" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">z1</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">b</ml:id> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:id xml:space="preserve">a</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="2"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>(при </inlineAttr> </f> <region region-id="39" left="134.25" top="129.75" width="27.75" height="15.75" align-x="147" align-y="141" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:notEqual/> <ml:id xml:space="preserve">a</ml:id> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="3"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">)<sp count="3"/>и </inlineAttr> </f> <region region-id="40" left="184.5" top="129" width="36" height="16.5" align-x="202.5" align-y="141" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">z2</ml:id> <ml:id xml:space="preserve">∞</ml:id> </ml:apply> </math> <rendering item-idref="4"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">.</inlineAttr> </f> </text> </region> <region region-id="24" left="54" top="168.75" width="419.25" height="27" align-x="54" align-y="180" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Отобразим область первого треугольника на область второго рис.1. Укажем</f> <f family="Arial Cyr" charset="204" size="12">на рис.1 положение: </f> </text> </region> <region region-id="31" left="54" top="222.75" width="192" height="13.5" align-x="54" align-y="234" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">- первого (черный) треугольника:</f> </text> </region> <region region-id="127" left="270" top="222.75" width="81.75" height="15.75" align-x="283.5" align-y="234" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">x</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.01</ml:real> </ml:sequence> <ml:real>0.5</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="128" left="378" top="214.5" width="121.5" height="33.75" align-x="412.5" align-y="234" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">y1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:neg/> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="20" left="54" top="264.75" width="215.25" height="13.5" align-x="54" align-y="276" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">- второго (коричневый) треугольника:</f> </text> </region> <region region-id="132" left="288" top="264.75" width="116.25" height="15.75" align-x="307.5" align-y="276" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">x1</ml:id> <ml:range> <ml:sequence> <ml:real>0.5</ml:real> <ml:apply> <ml:plus/> <ml:real>0.5</ml:real> <ml:real>0.01</ml:real> </ml:apply> </ml:sequence> <ml:real>2</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="131" left="426" top="262.5" width="69" height="18" align-x="460.5" align-y="276" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">y2</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="34" left="60" top="302.25" width="129" height="36" align-x="100.5" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">y3</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x1</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:neg/> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:real>3</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x1</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="14" left="30" top="354" width="368.25" height="295.5" align-x="30" align-y="354" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="10"/> <rendering item-idref="11"/> </region> <region region-id="120" left="126" top="674.25" width="185.25" height="12" align-x="126" align-y="684" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Ðèñ.1. Îòîáðàæåíèÿ ëèíåéíîé ôóíêöèåé. </text> </region> <region region-id="133" left="36" top="708" width="468.75" height="168" align-x="36" align-y="720" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Гиппотенузу </f> <f family="Arial Cyr" charset="204" size="12"> y </f> <f family="Arial Cyr" charset="204" size="12">1(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">) черного треугольника рис.1 запишем в полярной системе координат</f> <region region-id="134" left="36" top="723" width="231" height="38.25" align-x="203.25" align-y="744.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:sequence> </math> <rendering item-idref="12"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Здесь </inlineAttr> </f> <region region-id="135" left="310.5" top="732.75" width="24.75" height="16.5" align-x="314.25" align-y="744.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </math> <rendering item-idref="13"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> на комплексной плоскости </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">фактически определяет модуль радиус-вектора, конец которого находится на пря-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">мой </inlineAttr> </f> <region region-id="136" left="60" top="775.5" width="87.75" height="33.75" align-x="72.75" align-y="795" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:neg/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </math> <rendering item-idref="14"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> в декартовой системе координат. Таким образом, чтобы за-</inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">писать </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12">радиус-вектор как комплексную переменную на комплексной плоскости, надо</f> <f family="Arial Cyr" charset="204" size="12">найти проекции </f> <region region-id="137" left="122.25" top="823.5" width="24.75" height="16.5" align-x="126" align-y="835.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </math> <rendering item-idref="15"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> на действительную и мнимую оси, то есть представить</f> </inlineAttr> <region region-id="138" left="36" top="840.75" width="189.75" height="21.75" align-x="169.5" align-y="858" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="16"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">. Именно это представление и будет использо-</f> </inlineAttr> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">вано ниже.</f> </inlineAttr> </text> </region> <region region-id="35" left="30" top="888.75" width="474.75" height="27" align-x="30" align-y="900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Найдем в классе линейных функций ту, которая выполнит указанное отображение, </f> <f family="Arial Cyr" charset="204" size="12">используя композицию отображений поворота, </f> <f family="Arial Cyr" charset="204" size="12">растяжения и </f> <f family="Arial Cyr" charset="204" size="12">параллельного переноса.</f> </text> </region> <region region-id="41" left="42" top="942.75" width="423" height="150.75" align-x="42" align-y="954" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Отображение поворота </f> <f family="Arial Cyr" charset="204" size="12">. </f> <f family="Arial Cyr" charset="204" size="12">Из рис.1 следует, что первый треугольник надо повернуть на положительный</f> <f family="Arial Cyr" charset="204" size="12"> угол </f> <region region-id="42" left="70.5" top="970.5" width="108" height="34.5" align-x="99" align-y="990.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">π</ml:id> <ml:apply> <ml:id xml:space="preserve">arctg</ml:id> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="17"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">.<sp count="2"/>Запишем соответствующее этому отображение:</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <region region-id="81" left="45" top="1005.75" width="84" height="37.5" align-x="81.75" align-y="1038.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">w1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">z</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="18"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">. </f> </inlineAttr> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">Или </f> </inlineAttr> <region region-id="139" left="66" top="1044" width="129.75" height="36" align-x="102" align-y="1065.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">w1</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>-1</ml:real> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve">z</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="19"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Чтобы представить на рис.2 каждый элемент повернутого треугольника, запишем соответствующие для них выражения:</inlineAttr> </f> </text> </region> <region region-id="153" left="42" top="1128.75" width="120.75" height="13.5" align-x="42" align-y="1140" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Первый катет<sp count="2"/>рис.1:</f> </text> </region> <region region-id="154" left="174" top="1118.25" width="90" height="36" align-x="192.75" align-y="1140" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">z1</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.01</ml:real> </ml:sequence> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="155" left="294" top="1128.75" width="60" height="15.75" align-x="327" align-y="1140" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">z</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z1</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:id xml:space="preserve">z1</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="156" left="378" top="1106.25" width="122.25" height="38.25" align-x="378" align-y="1140" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Здесь </f> <region region-id="141" left="414.75" top="1107" width="54.75" height="37.5" align-x="432.75" align-y="1140" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">z</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="22"/> </region> </text> </region> <region region-id="157" left="36" top="1182.75" width="120.75" height="13.5" align-x="36" align-y="1194" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Второй катет </f> <f family="Arial Cyr" charset="204" size="12">рис.1</f> <f family="Arial Cyr" charset="204" size="12">:</f> </text> </region> <region region-id="158" left="204" top="1174.5" width="81" height="33.75" align-x="222.75" align-y="1194" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">z2</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.01</ml:real> </ml:sequence> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="94" left="36" top="1230.75" width="69.75" height="13.5" align-x="36" align-y="1242" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Гиппотенуза:</f> </text> </region> <region region-id="95" left="114" top="1221.75" width="81.75" height="34.5" align-x="131.25" align-y="1242" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">α</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.01</ml:real> </ml:sequence> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="96" left="222" top="1220.25" width="275.25" height="38.25" align-x="261" align-y="1242" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">z3</ml:id> <ml:boundVars> <ml:id xml:space="preserve">α</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="152" left="42" top="1284.75" width="441" height="40.5" align-x="42" align-y="1296" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">А далее осуществим отображение порота, записывая всякий раз каждый из эле-</f> <f family="Arial Cyr" charset="204" size="12">ментов треугольника как аргумент в функцию w1(z) и помещая полученную таким</f> <f family="Arial Cyr" charset="204" size="12">образом функцию на соответствующую ось рис.2.</f> </text> </region> <region region-id="56" left="30" top="1416" width="388.5" height="278.25" align-x="30" align-y="1416" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="26"/> <rendering item-idref="27"/> </region> <region region-id="125" left="180" top="1722.75" width="216" height="13.5" align-x="180" align-y="1734" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f size="12">Ðèñ.2. Îòîáðàæåíèå ïîâîðîòà</f> </text> </region> <region region-id="97" left="42" top="1752.75" width="438.75" height="50.25" align-x="42" align-y="1764" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Теперь необходимо осуществить отображение </f> <f family="Arial Cyr" charset="204" size="12"> растяжения "а" и </f> <f family="Arial Cyr" charset="204" size="12">параллельного переноса на вектор </f> <region region-id="98" left="153" top="1767" width="71.25" height="36" align-x="167.25" align-y="1788.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">b</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="28"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">:</inlineAttr> </f> </text> </region> <region region-id="104" left="48" top="1824.75" width="30" height="15.75" align-x="61.5" align-y="1836" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="103" left="174" top="1824.75" width="104.25" height="15.75" align-x="210.75" align-y="1836" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">w2</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">a</ml:id> <ml:apply> <ml:id xml:space="preserve">w1</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">b</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="102" left="36" top="1866" width="400.5" height="164.25" align-x="36" align-y="1866" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="31"/> <rendering item-idref="32"/> </region> <region region-id="126" left="162" top="2070.75" width="234" height="14.25" align-x="162" align-y="2082" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <sp count="2"/> <f family="Times New Roman Cyr" charset="204" size="12">Рис.3. Отображение растяжения и переноса.</f> </text> </region> <region region-id="106" left="48" top="2112.75" width="411.75" height="13.5" align-x="48" align-y="2124" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Итак, </f> <f family="Arial Cyr" charset="204" size="12">линейная функция, осуществляющая заданное отображение найдена:</f> </text> </region> <region region-id="109" left="72" top="2145" width="147.75" height="47.25" align-x="102.75" align-y="2178" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">w</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="110" left="252" top="2166.75" width="18" height="13.5" align-x="252" align-y="2178" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">или</f> </text> </region> <region region-id="112" left="300" top="2158.5" width="135.75" height="33.75" align-x="330.75" align-y="2178" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">w</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:parens> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">z</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="113" left="48" top="2202.75" width="429" height="13.5" align-x="48" align-y="2214" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">В качестве проверки найдем отображение трёх вершин первого треугольника:</f> </text> </region> <region region-id="115" left="48" top="2223.75" width="98.25" height="46.5" align-x="80.25" align-y="2256" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">w</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:symResult> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:pow/> <ml:real>3</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="35"/> </region> <region region-id="116" left="180" top="2234.25" width="75.75" height="36" align-x="234" align-y="2256" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">w</ml:id> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.9999999999999998</ml:real> </result> </ml:eval> </math> <rendering item-idref="36"/> </region> <region region-id="117" left="282" top="2236.5" width="60.75" height="33.75" align-x="321" align-y="2256" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">w</ml:id> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0</ml:real> </result> </ml:eval> </math> <rendering item-idref="37"/> </region> <region region-id="118" left="48" top="2286.75" width="441" height="67.5" align-x="48" align-y="2298" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Как легко заметить, они отобразились в соответствующие вершины второго треугольника. </f> <f family="Arial Cyr" charset="204" size="12">Конечно, можно было бы и сразу записать это отображение, если известен угол </f> <f family="Arial Cyr" charset="204" size="12">поворота и вектор параллельного переноса, то есть известна</f> <f family="Arial Cyr" charset="204" size="12">функция w(z). Тогда по этой функции можно провести отображение каждого</f> <f family="Arial Cyr" charset="204" size="12">фрагмента фигуры рис.1. Запишем эти фрагменты ещё раз:</f> </text> </region> <region region-id="159" left="48" top="2370.75" width="151.5" height="13.5" align-x="48" align-y="2382" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">горизонтальный катет рис.1</f> </text> </region> <region region-id="160" left="228" top="2370.75" width="89.25" height="15.75" align-x="249" align-y="2382" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Z1</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.01</ml:real> </ml:sequence> <ml:real>0.5</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="38"/> </region> <region region-id="169" left="42" top="2412.75" width="141" height="13.5" align-x="42" align-y="2424" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">вертикальный катет рис.1</f> </text> </region> <region region-id="168" left="216" top="2402.25" width="92.25" height="36" align-x="237" align-y="2424" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Z2</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.01</ml:real> </ml:sequence> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="167" left="342" top="2412.75" width="72.75" height="15.75" align-x="385.5" align-y="2424" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">Z3</ml:id> <ml:boundVars> <ml:id xml:space="preserve">Z2</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:id xml:space="preserve">Z2</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="40"/> </region> <region region-id="172" left="42" top="2454.75" width="96" height="13.5" align-x="42" align-y="2466" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">гиппотенуза рис.1</f> </text> </region> <region region-id="173" left="168" top="2445.75" width="81.75" height="34.5" align-x="185.25" align-y="2466" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">α</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.01</ml:real> </ml:sequence> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="174" left="276" top="2444.25" width="198.75" height="38.25" align-x="317.25" align-y="2466" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">Z4</ml:id> <ml:boundVars> <ml:id xml:space="preserve">α</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="42"/> </region> <region region-id="162" left="48" top="2514" width="445.5" height="188.25" align-x="48" align-y="2514" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="43"/> <rendering item-idref="44"/> </region> <region region-id="175" left="84" top="2736.75" width="321.75" height="13.5" align-x="84" align-y="2748" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Рис.4. 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customized="false"/> </units> </calculation> <editor protection-level="none" view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="cm"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="false" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="175" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="23" left="60" top="12.75" width="342" height="39.75" align-x="60" align-y="24" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12"> 2.<sp count="2"/>Линейная функция. </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">Отображение, осуществляемое линейной функцией W(z)=az+b </inlineAttr> </f> </text> </region> <region region-id="37" left="48" top="66.75" width="461.25" height="88.5" align-x="48" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Линейная функция однолистна, аналитична и нигде не обращается в ноль во</f> <f family="Arial Cyr" charset="204" size="12">всей расширенной комплексной плоскости. Таким образом она осуществляет конформное отображение.<sp count="2"/>Причём имеет две неподвижные точки, определяемые из уравнения a*z + b = z : </f> <region region-id="38" left="48" top="121.5" width="54" height="33.75" align-x="66" align-y="141" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">z1</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">b</ml:id> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:id xml:space="preserve">a</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="2"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>(при </inlineAttr> </f> <region region-id="39" left="134.25" top="129.75" width="27.75" height="15.75" align-x="147" align-y="141" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:notEqual/> <ml:id xml:space="preserve">a</ml:id> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="3"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">)<sp count="3"/>и </inlineAttr> </f> <region region-id="40" left="184.5" top="129" width="36" height="16.5" align-x="202.5" align-y="141" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">z2</ml:id> <ml:id xml:space="preserve">∞</ml:id> </ml:apply> </math> <rendering item-idref="4"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">.</inlineAttr> </f> </text> </region> <region region-id="24" left="54" top="168.75" width="419.25" height="27" align-x="54" align-y="180" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Отобразим область первого треугольника на область второго рис.1. Укажем</f> <f family="Arial Cyr" charset="204" size="12">на рис.1 положение: </f> </text> </region> <region region-id="31" left="54" top="222.75" width="192" height="13.5" align-x="54" align-y="234" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">- первого (черный) треугольника:</f> </text> </region> <region region-id="127" left="270" top="222.75" width="81.75" height="15.75" align-x="283.5" align-y="234" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">x</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.01</ml:real> </ml:sequence> <ml:real>0.5</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="128" left="378" top="214.5" width="121.5" height="33.75" align-x="412.5" align-y="234" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">y1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:neg/> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="20" left="54" top="264.75" width="215.25" height="13.5" align-x="54" align-y="276" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">- второго (коричневый) треугольника:</f> </text> </region> <region region-id="132" left="288" top="264.75" width="116.25" height="15.75" align-x="307.5" align-y="276" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">x1</ml:id> <ml:range> <ml:sequence> <ml:real>0.5</ml:real> <ml:apply> <ml:plus/> <ml:real>0.5</ml:real> <ml:real>0.01</ml:real> </ml:apply> </ml:sequence> <ml:real>2</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="131" left="426" top="262.5" width="69" height="18" align-x="460.5" align-y="276" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">y2</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="34" left="60" top="302.25" width="129" height="36" align-x="100.5" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">y3</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x1</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:neg/> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:real>3</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x1</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="14" left="30" top="354" width="368.25" height="295.5" align-x="30" align-y="354" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="10"/> <rendering item-idref="11"/> </region> <region region-id="120" left="126" top="674.25" width="185.25" height="12" align-x="126" align-y="684" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Ðèñ.1. Îòîáðàæåíèÿ ëèíåéíîé ôóíêöèåé. </text> </region> <region region-id="133" left="36" top="708" width="468.75" height="168" align-x="36" align-y="720" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Гиппотенузу </f> <f family="Arial Cyr" charset="204" size="12"> y </f> <f family="Arial Cyr" charset="204" size="12">1(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">) черного треугольника рис.1 запишем в полярной системе координат</f> <region region-id="134" left="36" top="723" width="231" height="38.25" align-x="203.25" align-y="744.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:sequence> </math> <rendering item-idref="12"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Здесь </inlineAttr> </f> <region region-id="135" left="310.5" top="732.75" width="24.75" height="16.5" align-x="314.25" align-y="744.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </math> <rendering item-idref="13"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> на комплексной плоскости </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">фактически определяет модуль радиус-вектора, конец которого находится на пря-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">мой </inlineAttr> </f> <region region-id="136" left="60" top="775.5" width="87.75" height="33.75" align-x="72.75" align-y="795" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:neg/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </math> <rendering item-idref="14"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> в декартовой системе координат. Таким образом, чтобы за-</inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">писать </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12">радиус-вектор как комплексную переменную на комплексной плоскости, надо</f> <f family="Arial Cyr" charset="204" size="12">найти проекции </f> <region region-id="137" left="122.25" top="823.5" width="24.75" height="16.5" align-x="126" align-y="835.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </math> <rendering item-idref="15"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> на действительную и мнимую оси, то есть представить</f> </inlineAttr> <region region-id="138" left="36" top="840.75" width="189.75" height="21.75" align-x="169.5" align-y="858" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="16"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">. Именно это представление и будет использо-</f> </inlineAttr> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">вано ниже.</f> </inlineAttr> </text> </region> <region region-id="35" left="30" top="888.75" width="474.75" height="27" align-x="30" align-y="900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Найдем в классе линейных функций ту, которая выполнит указанное отображение, </f> <f family="Arial Cyr" charset="204" size="12">используя композицию отображений поворота, </f> <f family="Arial Cyr" charset="204" size="12">растяжения и </f> <f family="Arial Cyr" charset="204" size="12">параллельного переноса.</f> </text> </region> <region region-id="41" left="42" top="942.75" width="423" height="150.75" align-x="42" align-y="954" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Отображение поворота </f> <f family="Arial Cyr" charset="204" size="12">. </f> <f family="Arial Cyr" charset="204" size="12">Из рис.1 следует, что первый треугольник надо повернуть на положительный</f> <f family="Arial Cyr" charset="204" size="12"> угол </f> <region region-id="42" left="70.5" top="970.5" width="108" height="34.5" align-x="99" align-y="990.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">π</ml:id> <ml:apply> <ml:id xml:space="preserve">arctg</ml:id> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="17"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">.<sp count="2"/>Запишем соответствующее этому отображение:</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <region region-id="81" left="45" top="1005.75" width="84" height="37.5" align-x="81.75" align-y="1038.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">w1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">z</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="18"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">. </f> </inlineAttr> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">Или </f> </inlineAttr> <region region-id="139" left="66" top="1044" width="129.75" height="36" align-x="102" align-y="1065.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">w1</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>-1</ml:real> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve">z</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="19"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Чтобы представить на рис.2 каждый элемент повернутого треугольника, запишем соответствующие для них выражения:</inlineAttr> </f> </text> </region> <region region-id="153" left="42" top="1128.75" width="120.75" height="13.5" align-x="42" align-y="1140" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Первый катет<sp count="2"/>рис.1:</f> </text> </region> <region region-id="154" left="174" top="1118.25" width="90" height="36" align-x="192.75" align-y="1140" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">z1</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.01</ml:real> </ml:sequence> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="155" left="294" top="1128.75" width="60" height="15.75" align-x="327" align-y="1140" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">z</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z1</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:id xml:space="preserve">z1</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="156" left="378" top="1106.25" width="122.25" height="38.25" align-x="378" align-y="1140" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Здесь </f> <region region-id="141" left="414.75" top="1107" width="54.75" height="37.5" align-x="432.75" align-y="1140" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">z</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="22"/> </region> </text> </region> <region region-id="157" left="36" top="1182.75" width="120.75" height="13.5" align-x="36" align-y="1194" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Второй катет </f> <f family="Arial Cyr" charset="204" size="12">рис.1</f> <f family="Arial Cyr" charset="204" size="12">:</f> </text> </region> <region region-id="158" left="204" top="1174.5" width="81" height="33.75" align-x="222.75" align-y="1194" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">z2</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.01</ml:real> </ml:sequence> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="94" left="36" top="1230.75" width="69.75" height="13.5" align-x="36" align-y="1242" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Гиппотенуза:</f> </text> </region> <region region-id="95" left="114" top="1221.75" width="81.75" height="34.5" align-x="131.25" align-y="1242" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">α</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.01</ml:real> </ml:sequence> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="96" left="222" top="1220.25" width="275.25" height="38.25" align-x="261" align-y="1242" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">z3</ml:id> <ml:boundVars> <ml:id xml:space="preserve">α</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="152" left="42" top="1284.75" width="441" height="40.5" align-x="42" align-y="1296" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">А далее осуществим отображение порота, записывая всякий раз каждый из эле-</f> <f family="Arial Cyr" charset="204" size="12">ментов треугольника как аргумент в функцию w1(z) и помещая полученную таким</f> <f family="Arial Cyr" charset="204" size="12">образом функцию на соответствующую ось рис.2.</f> </text> </region> <region region-id="56" left="30" top="1416" width="388.5" height="278.25" align-x="30" align-y="1416" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="26"/> <rendering item-idref="27"/> </region> <region region-id="125" left="180" top="1722.75" width="216" height="13.5" align-x="180" align-y="1734" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f size="12">Ðèñ.2. Îòîáðàæåíèå ïîâîðîòà</f> </text> </region> <region region-id="97" left="42" top="1752.75" width="438.75" height="50.25" align-x="42" align-y="1764" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Теперь необходимо осуществить отображение </f> <f family="Arial Cyr" charset="204" size="12"> растяжения "а" и </f> <f family="Arial Cyr" charset="204" size="12">параллельного переноса на вектор </f> <region region-id="98" left="153" top="1767" width="71.25" height="36" align-x="167.25" align-y="1788.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">b</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="28"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">:</inlineAttr> </f> </text> </region> <region region-id="104" left="48" top="1824.75" width="30" height="15.75" align-x="61.5" align-y="1836" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="103" left="174" top="1824.75" width="104.25" height="15.75" align-x="210.75" align-y="1836" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">w2</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">a</ml:id> <ml:apply> <ml:id xml:space="preserve">w1</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">b</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="102" left="36" top="1866" width="400.5" height="164.25" align-x="36" align-y="1866" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="31"/> <rendering item-idref="32"/> </region> <region region-id="126" left="162" top="2070.75" width="234" height="14.25" align-x="162" align-y="2082" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <sp count="2"/> <f family="Times New Roman Cyr" charset="204" size="12">Рис.3. Отображение растяжения и переноса.</f> </text> </region> <region region-id="106" left="48" top="2112.75" width="411.75" height="13.5" align-x="48" align-y="2124" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Итак, </f> <f family="Arial Cyr" charset="204" size="12">линейная функция, осуществляющая заданное отображение найдена:</f> </text> </region> <region region-id="109" left="72" top="2145" width="147.75" height="47.25" align-x="102.75" align-y="2178" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">w</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="110" left="252" top="2166.75" width="18" height="13.5" align-x="252" align-y="2178" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">или</f> </text> </region> <region region-id="112" left="300" top="2158.5" width="135.75" height="33.75" align-x="330.75" align-y="2178" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">w</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:parens> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">z</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="113" left="48" top="2202.75" width="429" height="13.5" align-x="48" align-y="2214" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">В качестве проверки найдем отображение трёх вершин первого треугольника:</f> </text> </region> <region region-id="115" left="48" top="2223.75" width="98.25" height="46.5" align-x="80.25" align-y="2256" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">w</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:symResult> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:pow/> <ml:real>3</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="35"/> </region> <region region-id="116" left="180" top="2234.25" width="75.75" height="36" align-x="234" align-y="2256" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">w</ml:id> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.9999999999999998</ml:real> </result> </ml:eval> </math> <rendering item-idref="36"/> </region> <region region-id="117" left="282" top="2236.5" width="60.75" height="33.75" align-x="321" align-y="2256" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">w</ml:id> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0</ml:real> </result> </ml:eval> </math> <rendering item-idref="37"/> </region> <region region-id="118" left="48" top="2286.75" width="441" height="67.5" align-x="48" align-y="2298" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Как легко заметить, они отобразились в соответствующие вершины второго треугольника. </f> <f family="Arial Cyr" charset="204" size="12">Конечно, можно было бы и сразу записать это отображение, если известен угол </f> <f family="Arial Cyr" charset="204" size="12">поворота и вектор параллельного переноса, то есть известна</f> <f family="Arial Cyr" charset="204" size="12">функция w(z). Тогда по этой функции можно провести отображение каждого</f> <f family="Arial Cyr" charset="204" size="12">фрагмента фигуры рис.1. Запишем эти фрагменты ещё раз:</f> </text> </region> <region region-id="159" left="48" top="2370.75" width="151.5" height="13.5" align-x="48" align-y="2382" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">горизонтальный катет рис.1</f> </text> </region> <region region-id="160" left="228" top="2370.75" width="89.25" height="15.75" align-x="249" align-y="2382" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Z1</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.01</ml:real> </ml:sequence> <ml:real>0.5</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="38"/> </region> <region region-id="169" left="42" top="2412.75" width="141" height="13.5" align-x="42" align-y="2424" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">вертикальный катет рис.1</f> </text> </region> <region region-id="168" left="216" top="2402.25" width="92.25" height="36" align-x="237" align-y="2424" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Z2</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.01</ml:real> </ml:sequence> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="167" left="342" top="2412.75" width="72.75" height="15.75" align-x="385.5" align-y="2424" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">Z3</ml:id> <ml:boundVars> <ml:id xml:space="preserve">Z2</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:id xml:space="preserve">Z2</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="40"/> </region> <region region-id="172" left="42" top="2454.75" width="96" height="13.5" align-x="42" align-y="2466" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">гиппотенуза рис.1</f> </text> </region> <region region-id="173" left="168" top="2445.75" width="81.75" height="34.5" align-x="185.25" align-y="2466" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">α</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.01</ml:real> </ml:sequence> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="174" left="276" top="2444.25" width="198.75" height="38.25" align-x="317.25" align-y="2466" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">Z4</ml:id> <ml:boundVars> <ml:id xml:space="preserve">α</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="i">1</ml:imag> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="42"/> </region> <region region-id="162" left="48" top="2514" width="445.5" height="188.25" align-x="48" align-y="2514" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="43"/> <rendering item-idref="44"/> </region> <region region-id="175" left="84" top="2736.75" width="321.75" height="13.5" align-x="84" align-y="2748" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Рис.4. Построение отображения по известной функции w(z)</f> </text> </region> </regions> <binaryContent> <item item-id="1" content-encoding="gzip">H4sIAAAAAAAA/4yNvQ6CQBCEdzlQwd/GGhMbj9InsCBWJj6CnvifEA1SWPrm5+yCvUf2GGa/ GQZExBiHSVQb3OPjcnep3PO6Oz+q0tURyZljYrVzVzu1aIHplcX2cD8VdWPtmxcFM1ym7Z9K q2r5Stz79lprtVE2w2y4ZZvj8RBn/7u/8m5dueK0zANd2RZMeUJs2Ytitlj6lAO2BtGUDdvQ ixOCIQ8RgfmssOoII6muMhA9YQSOpUfiCRiN98FofABG40P9F8RIGYixMAJ/AQAA//8DAMuS Xhh8AQAA</item> <item item-id="2">iVBORw0KGgoAAAANSUhEUgAAAEgAAAAtCAYAAAAN1C2uAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AWtJREFUaEPtmlESwyAIRNv7H7pNOs2MmpA1gBri5tcK+lhAnb4/y/fiJxNYAfGTCbwI55wA AQGFENBsgJZq61o1fK25Lu26sRUOAaGUoIJA1yGgOkBe6faoGvS/FWR1yFqTHgmo1JgFEgGh on+9md57xpFaqKAkZiUMC5xfTbu3HnSr2zqYFY4KkIdT3bbHzKpWkGdUxmxV57Ua0GaeCkJt z/kor4trv1mZgtI0ko7qtQo6shUxTXeAUCpNDQjB2e46PQSOFNhj/LDNI4Wg8R7wevrYdTEE AI2nKjyLcs9NWnyJNUhKp2kBSdFO6UfsQhb1qK4aVofR5l8+SUfboHW9oQDV1j8rlKyseBpr 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