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               "]], {}}, {
             Line[{{-3, 0, 0}, {3, 0, 0}}], 
             Line[{{0, -3, 0}, {0, 3, 0}}], 
             Point[{1, 0, 0}], 
             Text[1, {1, 0, 0}, {1, 1}], 
             Point[{0, 1, 0}], 
             Text[I, {0, 1, 0}, {1, 1}], 
             Point[{0, 0, 1}], 
             Text["N", {0, 0, 1}, {0, -1}]}}, {
           DisplayFunction -> Identity, PlotRangePadding -> {{
               Scaled[0.05], 
               Scaled[0.05]}, {
               Scaled[0.05], 
               Scaled[0.05]}, {
               Scaled[0.05], 
               Scaled[0.05]}}, ImagePadding -> Automatic, DisplayFunction -> 
            Identity, Axes -> True, DisplayFunction :> Identity, 
            FaceGridsStyle -> Automatic, 
            Method -> {
             "DefaultGraphicsInteraction" -> {
               "Version" -> 1.2, "TrackMousePosition" -> {True, False}, 
                "Effects" -> {
                 "Highlight" -> {"ratio" -> 2}, 
                  "HighlightPoint" -> {"ratio" -> 2}, 
                  "Droplines" -> {
                   "freeformCursorMode" -> True, 
                    "placement" -> {"x" -> "All", "y" -> "None"}}}}}, 
            PlotRange -> {{-2.9999995714285714`, 
             2.9999995714285714`}, {-2.9999995714285714`, 
             2.9999995714285714`}, {-1., 1.}}, PlotRangePadding -> {
              Scaled[0.02], 
              Scaled[0.02], 
              Scaled[0.02]}, 
            Ticks -> {
             Automatic, Automatic, Automatic}}], $CellContext`circleproj[
           Pattern[$CellContext`c, 
            Blank[]], 
           Pattern[$CellContext`r, 
            Blank[]]] := 
         Module[{$CellContext`xcirc, $CellContext`ycirc, $CellContext`zcirc, \
$CellContext`t}, $CellContext`xcirc = 
            2 ((Part[$CellContext`c, 1] + $CellContext`r 
                Cos[$CellContext`t])/(
              1 + (Part[$CellContext`c, 1] + $CellContext`r 
                 Cos[$CellContext`t])^2 + (
                Part[$CellContext`c, 2] + $CellContext`r 
                 Sin[$CellContext`t])^2)); $CellContext`ycirc = 
            2 ((Part[$CellContext`c, 2] + $CellContext`r 
                Sin[$CellContext`t])/(
              1 + (Part[$CellContext`c, 1] + $CellContext`r 
                 Cos[$CellContext`t])^2 + (
                Part[$CellContext`c, 2] + $CellContext`r 
                 Sin[$CellContext`t])^2)); $CellContext`zcirc = ((
                Part[$CellContext`c, 1] + $CellContext`r 
                 Cos[$CellContext`t])^2 + (
                Part[$CellContext`c, 2] + $CellContext`r 
                 Sin[$CellContext`t])^2 - 1)/(
             1 + (Part[$CellContext`c, 1] + $CellContext`r 
                Cos[$CellContext`t])^2 + (
               Part[$CellContext`c, 2] + $CellContext`r 
                Sin[$CellContext`t])^2); Show[
             
             ParametricPlot3D[{
              Part[$CellContext`c, 1] + $CellContext`r Cos[$CellContext`t], 
               Part[$CellContext`c, 2] + $CellContext`r Sin[$CellContext`t], 
               0}, {$CellContext`t, 0, 2 Pi}], 
             
             ParametricPlot3D[{$CellContext`xcirc, $CellContext`ycirc, \
$CellContext`zcirc}, {$CellContext`t, 0, 2 Pi}, PlotStyle -> {Thick, Red}]]], 
         
         TagSet[$CellContext`r, 
          MessageName[$CellContext`r, "shdw"], 
          "Symbol `1` appears in multiple contexts `2`; definitions in \
context `3` may shadow or be shadowed by other definitions."], \
$CellContext`circlelines[
           Pattern[$CellContext`c, 
            Blank[]], 
           Pattern[$CellContext`r, 
            Blank[]]] := 
         Module[{$CellContext`xcirc, $CellContext`ycirc, $CellContext`zcirc, \
$CellContext`tL, $CellContext`pts, $CellContext`pts2, $CellContext`t, \
$CellContext`lines, $CellContext`lines2}, $CellContext`xcirc = 
            2 ((Part[$CellContext`c, 1] + $CellContext`r 
                Cos[$CellContext`t])/(
              1 + (Part[$CellContext`c, 1] + $CellContext`r 
                 Cos[$CellContext`t])^2 + (
                Part[$CellContext`c, 2] + $CellContext`r 
                 Sin[$CellContext`t])^2)); $CellContext`ycirc = 
            2 ((Part[$CellContext`c, 2] + $CellContext`r 
                Sin[$CellContext`t])/(
              1 + (Part[$CellContext`c, 1] + $CellContext`r 
                 Cos[$CellContext`t])^2 + (
                Part[$CellContext`c, 2] + $CellContext`r 
                 Sin[$CellContext`t])^2)); $CellContext`zcirc = ((
                Part[$CellContext`c, 1] + $CellContext`r 
                 Cos[$CellContext`t])^2 + (
                Part[$CellContext`c, 2] + $CellContext`r 
                 Sin[$CellContext`t])^2 - 1)/(
             1 + (Part[$CellContext`c, 1] + $CellContext`r 
                Cos[$CellContext`t])^2 + (
               Part[$CellContext`c, 2] + $CellContext`r 
                Sin[$CellContext`t])^2); $CellContext`tL = 
            Table[(2 (Pi/10)) $CellContext`k, {$CellContext`k, 1, 
               10}]; $CellContext`pts = 
            Map[ReplaceAll[{
               Part[$CellContext`c, 1] + $CellContext`r Cos[$CellContext`t], 
                Part[$CellContext`c, 2] + $CellContext`r Sin[$CellContext`t], 
                0}, $CellContext`t -> #]& , $CellContext`tL]; \
$CellContext`pts2 = 
            Map[ReplaceAll[{$CellContext`xcirc, $CellContext`ycirc, \
$CellContext`zcirc}, $CellContext`t -> #]& , $CellContext`tL]; \
$CellContext`lines = 
            Map[Line[{#, {0, 0, 
                1}}]& , $CellContext`pts]; $CellContext`lines2 = 
            Map[Line[{#, {0, 0, 1}}]& , $CellContext`pts2]; 
           Graphics3D[{$CellContext`lines, $CellContext`lines2}]], \
$CellContext`lineproj[
           Pattern[$CellContext`p, 
            Blank[]], 
           Pattern[$CellContext`a, 
            Blank[]]] := 
         Module[{$CellContext`line, $CellContext`circ, $CellContext`t, \
$CellContext`t1}, $CellContext`line = InfiniteLine[{
               Part[$CellContext`p, 1], 
               Part[$CellContext`p, 2], 0}, {
               Cos[$CellContext`a], 
               Sin[$CellContext`a], 0}]; $CellContext`circ = {(
               2 (Part[$CellContext`p, 1] + 
                Cos[$CellContext`a] (Cot[$CellContext`t1] + 
                  Csc[$CellContext`t1])))/(
              1 + (Part[$CellContext`p, 1] + 
                Cos[$CellContext`a] (Cot[$CellContext`t1] + 
                  Csc[$CellContext`t1]))^2 + (Part[$CellContext`p, 2] + 
                Cot[$CellContext`t1/2] Sin[$CellContext`a])^2), (
               2 (Part[$CellContext`p, 2] + 
                Cot[$CellContext`t1/2] Sin[$CellContext`a]))/(
              1 + (Part[$CellContext`p, 1] + 
                Cos[$CellContext`a] (Cot[$CellContext`t1] + 
                  Csc[$CellContext`t1]))^2 + (Part[$CellContext`p, 2] + 
                Cot[$CellContext`t1/2] Sin[$CellContext`a])^2), 
              1 - (2 Sin[$CellContext`t1]^2)/((1 + 
                Cos[$CellContext`t1])^2 + ((2 (1 + Cos[$CellContext`t1])) (
                 Part[$CellContext`p, 1] Cos[$CellContext`a] + 
                 Part[$CellContext`p, 2] Sin[$CellContext`a])) 
               Sin[$CellContext`t1] + (1 + Part[$CellContext`p, 1]^2 + 
                Part[$CellContext`p, 2]^2) Sin[$CellContext`t1]^2)}; Show[
             Graphics3D[{Thick, 
               RGBColor[0.368417, 0.506779, 0.709798], $CellContext`line}], 
             
             ParametricPlot3D[$CellContext`circ, {$CellContext`t1, 0, 2 Pi}, 
              PlotStyle -> {Thick, Red}]]], $CellContext`line = 
         Line[{-3, -3}], $CellContext`circ[
           Pattern[$CellContext`c, 
            Blank[]], 
           Pattern[$CellContext`r, 
            Blank[]]] := $CellContext`c + $CellContext`r 
           Cos[(2 Pi) $CellContext`t] + (I $CellContext`r) 
           Sin[(2 Pi) $CellContext`t], $CellContext`lineLink[
           Pattern[$CellContext`p, 
            Blank[]], 
           Pattern[$CellContext`a, 
            Blank[]]] := 
         Module[{$CellContext`xcirc, $CellContext`ycirc, $CellContext`zcirc, \
$CellContext`t1, $CellContext`tL, $CellContext`pts, $CellContext`lines, \
$CellContext`intersects}, $CellContext`xcirc = (
              2 (Part[$CellContext`p, 1] + 
               Cos[$CellContext`a] (Cot[$CellContext`t1] + 
                 Csc[$CellContext`t1])))/(
             1 + (Part[$CellContext`p, 1] + 
               Cos[$CellContext`a] (Cot[$CellContext`t1] + 
                 Csc[$CellContext`t1]))^2 + (Part[$CellContext`p, 2] + 
               Cot[$CellContext`t1/2] 
                Sin[$CellContext`a])^2); $CellContext`ycirc = (
              2 (Part[$CellContext`p, 2] + 
               Cot[$CellContext`t1/2] Sin[$CellContext`a]))/(
             1 + (Part[$CellContext`p, 1] + 
               Cos[$CellContext`a] (Cot[$CellContext`t1] + 
                 Csc[$CellContext`t1]))^2 + (Part[$CellContext`p, 2] + 
               Cot[$CellContext`t1/2] 
                Sin[$CellContext`a])^2); $CellContext`zcirc = 
            1 - (2 Sin[$CellContext`t1]^2)/((1 + 
               Cos[$CellContext`t1])^2 + ((2 (1 + Cos[$CellContext`t1])) (
                Part[$CellContext`p, 1] Cos[$CellContext`a] + 
                Part[$CellContext`p, 2] Sin[$CellContext`a])) 
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            Graphics3D[{
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             Blank[]], 
            Pattern[$CellContext`r, 
             Blank[]]] := 
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                 Cos[$CellContext`t])/(
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                  Cos[$CellContext`t])^2 + (
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                  Cos[$CellContext`t])^2 + (
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                  Sin[$CellContext`t])^2)); $CellContext`zcirc = ((
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                  Sin[$CellContext`t])^2 - 1)/(
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                 Cos[$CellContext`t])^2 + (
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                 Sin[$CellContext`t])^2); Show[
              
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$CellContext`lines, $CellContext`lines2}, $CellContext`xcirc = 
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                  Cos[$CellContext`t])^2 + (
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                  Cos[$CellContext`t])^2 + (
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                  Sin[$CellContext`t])^2 - 1)/(
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   "\"Fig. 6. Lines and circles are mapped into circles on the Riemann sphere \
by the stereographic projection. Try moving and resizing them to see how the \
projection changes. The checkbox \[OpenCurlyDoubleQuote]Show lines\
\[CloseCurlyDoubleQuote] will show how the projection is constructed.\""},
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In Fig. 7, one can see how the stereographic projection preserves angles \
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\>", "Text",
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               "]], {}}, {
             Line[{{-3, 0, 0}, {3, 0, 0}}], 
             Line[{{0, -3, 0}, {0, 3, 0}}], 
             Point[{1, 0, 0}], 
             Text[1, {1, 0, 0}, {1, 1}], 
             Point[{0, 1, 0}], 
             Text[I, {0, 1, 0}, {1, 1}], 
             Point[{0, 0, 1}], 
             Text["N", {0, 0, 1}, {0, -1}]}}, {
           DisplayFunction -> Identity, PlotRangePadding -> {{
               Scaled[0.05], 
               Scaled[0.05]}, {
               Scaled[0.05], 
               Scaled[0.05]}, {
               Scaled[0.05], 
               Scaled[0.05]}}, ImagePadding -> Automatic, DisplayFunction -> 
            Identity, Axes -> True, DisplayFunction :> Identity, 
            FaceGridsStyle -> Automatic, 
            Method -> {
             "DefaultGraphicsInteraction" -> {
               "Version" -> 1.2, "TrackMousePosition" -> {True, False}, 
                "Effects" -> {
                 "Highlight" -> {"ratio" -> 2}, 
                  "HighlightPoint" -> {"ratio" -> 2}, 
                  "Droplines" -> {
                   "freeformCursorMode" -> True, 
                    "placement" -> {"x" -> "All", "y" -> "None"}}}}}, 
            PlotRange -> {{-2.9999995714285714`, 
             2.9999995714285714`}, {-2.9999995714285714`, 
             2.9999995714285714`}, {-1., 1.}}, PlotRangePadding -> {
              Scaled[0.02], 
              Scaled[0.02], 
              Scaled[0.02]}, 
            Ticks -> {
             Automatic, Automatic, Automatic}}], $CellContext`lineproj2[
           Pattern[$CellContext`p, 
            Blank[]], 
           Pattern[$CellContext`a, 
            Blank[]], 
           Pattern[$CellContext`color1, 
            Blank[]], 
           Pattern[$CellContext`color2, 
            Blank[]]] := 
         Module[{$CellContext`line, $CellContext`circ, $CellContext`t, \
$CellContext`t1}, $CellContext`line = InfiniteLine[{
               Part[$CellContext`p, 1], 
               Part[$CellContext`p, 2], 0}, {
               Cos[$CellContext`a], 
               Sin[$CellContext`a], 0}]; $CellContext`circ = {(
               2 (Part[$CellContext`p, 1] + 
                Cos[$CellContext`a] (Cot[$CellContext`t1] + 
                  Csc[$CellContext`t1])))/(
              1 + (Part[$CellContext`p, 1] + 
                Cos[$CellContext`a] (Cot[$CellContext`t1] + 
                  Csc[$CellContext`t1]))^2 + (Part[$CellContext`p, 2] + 
                Cot[$CellContext`t1/2] Sin[$CellContext`a])^2), (
               2 (Part[$CellContext`p, 2] + 
                Cot[$CellContext`t1/2] Sin[$CellContext`a]))/(
              1 + (Part[$CellContext`p, 1] + 
                Cos[$CellContext`a] (Cot[$CellContext`t1] + 
                  Csc[$CellContext`t1]))^2 + (Part[$CellContext`p, 2] + 
                Cot[$CellContext`t1/2] Sin[$CellContext`a])^2), 
              1 - (2 Sin[$CellContext`t1]^2)/((1 + 
                Cos[$CellContext`t1])^2 + ((2 (1 + Cos[$CellContext`t1])) (
                 Part[$CellContext`p, 1] Cos[$CellContext`a] + 
                 Part[$CellContext`p, 2] Sin[$CellContext`a])) 
               Sin[$CellContext`t1] + (1 + Part[$CellContext`p, 1]^2 + 
                Part[$CellContext`p, 2]^2) Sin[$CellContext`t1]^2)}; Show[
             Graphics3D[{Thick, $CellContext`color1, 
               Opacity[0.8], $CellContext`line}], 
             
             ParametricPlot3D[$CellContext`circ, {$CellContext`t1, 0, 2 Pi}, 
              PlotStyle -> {Thick, $CellContext`color2, 
                Opacity[0.8]}]]], $CellContext`line = 
         Line[{-3, -3}], $CellContext`circ[
           Pattern[$CellContext`c, 
            Blank[]], 
           Pattern[$CellContext`r, 
            Blank[]]] := $CellContext`c + $CellContext`r 
           Cos[(2 Pi) $CellContext`t] + (I $CellContext`r) 
           Sin[(2 Pi) $CellContext`t], 
         TagSet[$CellContext`r, 
          MessageName[$CellContext`r, "shdw"], 
          "Symbol `1` appears in multiple contexts `2`; definitions in \
context `3` may shadow or be shadowed by other definitions."], \
$CellContext`pointproj[
           Pattern[$CellContext`p, 
            Blank[]]] := 
         Module[{$CellContext`x, $CellContext`y, $CellContext`z, \
$CellContext`p1}, {$CellContext`x, $CellContext`y} = $CellContext`p; \
$CellContext`p1 = {
             2 ($CellContext`x/(1 + $CellContext`x^2 + $CellContext`y^2)), 
              2 ($CellContext`y/(
               1 + $CellContext`x^2 + $CellContext`y^2)), ($CellContext`x^2 + \
$CellContext`y^2 - 1)/($CellContext`x^2 + $CellContext`y^2 + 1)}; Show[
             Graphics3D[{
               PointSize[0.013], 
               Point[$CellContext`p1]}]]], $CellContext`arcproj[
           Pattern[$CellContext`c, 
            Blank[]], 
           Pattern[$CellContext`th1, 
            Blank[]], 
           Pattern[$CellContext`th2, 
            Blank[]], 
           Pattern[$CellContext`r, 
            Blank[]]] := 
         Module[{$CellContext`xcirc, $CellContext`ycirc, $CellContext`zcirc, \
$CellContext`t}, 
           
           If[$CellContext`th2 > $CellContext`th1, $CellContext`xcirc = 
             2 ((Part[$CellContext`c, 1] + $CellContext`r 
                 Cos[$CellContext`t])/(
               1 + (Part[$CellContext`c, 1] + $CellContext`r 
                  Cos[$CellContext`t])^2 + (
                 Part[$CellContext`c, 2] + $CellContext`r 
                  Sin[$CellContext`t])^2)); $CellContext`ycirc = 
             2 ((Part[$CellContext`c, 2] + $CellContext`r 
                 Sin[$CellContext`t])/(
               1 + (Part[$CellContext`c, 1] + $CellContext`r 
                  Cos[$CellContext`t])^2 + (
                 Part[$CellContext`c, 2] + $CellContext`r 
                  Sin[$CellContext`t])^2)); $CellContext`zcirc = ((
                 Part[$CellContext`c, 1] + $CellContext`r 
                  Cos[$CellContext`t])^2 + (
                 Part[$CellContext`c, 2] + $CellContext`r 
                  Sin[$CellContext`t])^2 - 1)/(
              1 + (Part[$CellContext`c, 1] + $CellContext`r 
                 Cos[$CellContext`t])^2 + (
                Part[$CellContext`c, 2] + $CellContext`r 
                 Sin[$CellContext`t])^2); Show[
              
              ParametricPlot3D[{
               Part[$CellContext`c, 1] + $CellContext`r Cos[$CellContext`t], 
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Cell[CellGroupData[{

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You will see how, often, the shortest way to prove a result about real \
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where the equality is due to the rule for multiplying numbers, which requires \
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Equating to the right-hand side and taking the real part of both:\
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The modulus or absolute value, argument and conjugation have been defined.\
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Complex numbers can be represented as points on the complex plane with \
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