Manipulate Exp[I θ]+b Exp[-I θ]

Manipulate[ Module[{ p1 = {Cos[theta ], Sin[theta]}, p2 = {b Cos[-theta], b Sin[-theta ]}, p3 = {(1 + b) Cos[theta ], (1 - b) Sin[theta ]}}, Graphics[{ Circle[{0, 0}], {Red, Circle[{0, 0}, b]}, Arrow[{{0, 0}, p1}], {Red, Arrow[{{0, 0}, p2}]}, Line[{{0, 0}, p1, p3, p2, {0, 0}}], {Dashed, Line[{{0, 0}, p3}]}}, PlotRange -> {{-1.5, 1.5}, {-1.5, 1.5}}, Axes -> True] ], Grid[{ {"b ", Control[{{b, .5, ""}, 0, 1, .1}], Dynamic[b]}, {"theta ", Control[{{theta, Pi/4, ""}, 0, 2 Pi, 0.1}], Row[{Dynamic[theta*180./Pi], Degree}] }}], ContinuousAction -> True ] 

Blatantly copying Nasser's code, here is a slightly different style.

I used both Text and Tooltip merely to demonstrate what is possible.

Manipulate[ With[{ p1 = {Cos[θ], Sin[θ]}, p2 = {b Cos[-θ], b Sin[-θ]}, p3 = {(1 + b) Cos[θ], (1 - b) Sin[θ]} }, Graphics[{ Circle[{0, 0}], {Red, Circle[{0, 0}, b], Arrow[{{0, 0}, p1}], Arrow[{{0, 0}, p2}]}, Arrow[{{0, 0}, p3}], {Dashed, Line[{p1, p3, p2}], Circle[{0, 0}, Abs @ {b + 1, b - 1}]}, {Red, Text[E^(I "θ"), 1.1 p1], Text["b"/E^(I "θ"), 1.1 p2]}, Tooltip[Point[p3], E^(I "θ") + "b"/E^(I "θ")] }, PlotRange -> {{-2, 2}, {-1.3, 1.3}}, BaseStyle -> 20, ImageSize -> 500 ] ], Grid[{ {"b ", Control[{{b, 0.5, ""}, 0, 1, 0.1}], Dynamic[b]}, {"θ", Control[{{θ, Pi / 4, ""}, 0, 2 Pi, 0.1}], Row[{Dynamic[θ 180` / Pi], Degree}]} }, BaseStyle -> 20] ] 

I used this once:

ComplexParametricPlot[f_, stuff__] := ParametricPlot[{Re[#], Im[#]} &@f, stuff] 

Then putting it in the previous Manipulate (from "part I"), and giving b a complex control, we get:

Manipulate[ Show[Graphics[ GraphicsComplex[ complexparts /@ {E^(I π θ), b E^(-I π θ), E^(I π θ) + b E^(-I π θ), 0}, {Circle[4], {Red, Circle[4, Norm[b]]}, Arrow[{4, 1}], {Red, Arrow[{4, 2}]}, Line[{1, 3, 2}], {Dashed, Line[{4, 3}]}}], PlotRange -> 2, Axes -> True], ComplexParametricPlot[ E^(I π t) + b E^(-I π t), {t, 0, θ + 10^-6}], ImageSize -> 250 ], {{b, 0.5}, LocatorPane[ Dynamic[{Re[b], Im[b]}, (b = If[Norm[#] > 1, Complex @@ Normalize[#], Complex @@ #]) &], Labeled[ Graphics[{GrayLevel[0.97], Rectangle[{-1, -1}, {1, 1}], White, Disk[]}, Axes -> True, ImageSize -> 130], Row[{"b = ", Dynamic[b]}]]] &, ControlPlacement -> Left}, {{θ, 1/4.}, 0., 2, Row[{Manipulator[##], " ", InputField[#1, Number, FieldSize -> {{0, 6}, 1}, Appearance -> "Frameless"], " π"}] &}, {{complexparts, {Re[#], Im[#]} &}, None}] 

Addendum

Here's another version, inspired by Mr.Wizard, kguler, and the OP. Looks nearly the same, but it uses Circle and Rotate to trace the ellipse.

Manipulate[ Graphics[GraphicsComplex[ complexparts /@ {E^(I π θ), b E^(-I π θ), E^(I π θ) + b E^(-I π θ), 0}, {Circle[4], {Red, Circle[4, Norm[b]]}, Arrow[{4, 1}], {Red, Arrow[{4, 2}]}, Line[{1, 3, 2}], {Dashed, Blue, Line[{4, 3}], Rotate[ Circle[4, Abs@{E^(I #) + b E^(-I #), E^(I #) - b E^(-I #)}, {-#, π θ - #}], #, {0, 0}] &[Arg[b]/2]}}], PlotRange -> 2, Axes -> True, ImageSize -> 250], {{b, 0.5 + 0.2 I}, LocatorPane[ Dynamic[{Re[b], Im[b]}, (b = If[Norm[#] > 1, Complex @@ Normalize[#], Complex @@ #]) &], Labeled[ Graphics[{GrayLevel[0.97], Rectangle[{-1, -1}, {1, 1}], White, Disk[]}, Axes -> True, ImageSize -> 130], Row[{"b = ", Dynamic[b]}]]] &, ControlPlacement -> Left}, {{θ, 0.25}, 0., 2., Row[{Manipulator[##], " ", InputField[#1, Number, FieldSize -> {{0, 6}, 1}, Appearance -> "Frameless"], " π"}] &}, {{complexparts, {Re[#], Im[#]} &}, None}] 

My own little modification. Deals with complex numbers E^(I θ) as such, and maps the Complex graphics to Real coordinates in a, ahem, GraphicsComplex-- sorry, can't resist a bad pun. Couldn't figure out how to get the InputField that serves as a label to format properly. (Doesn't round numbers the way a regular label will. Feel free to fix it, if you can.)

Manipulate[ Graphics[GraphicsComplex[ complexparts /@ {E^(I π θ), b E^(-I π θ), E^(I π θ) + b E^(-I π θ), 0}, {Circle[4], {Red, Circle[4, b]}, Arrow[{4, 1}], {Red, Arrow[{4, 2}]}, Line[{1, 3, 2}], {Dashed, Line[{4, 3}]}}], PlotRange -> {{-1.5, 1.5}, {-1.5, 1.5}}, Axes -> True], {{b, .5}, 0, 1, Appearance -> "Labeled"}, {{θ, 1/4.}, 0., 2, Row[{Manipulator[##], " ", InputField[#1, Number, FieldSize -> {{0, 6}, 1}, Appearance -> "Frameless"], " π"}] &}, {{complexparts, {Re[#], Im[#]} &}, None}] 

Yep, the code ParametricPlot[{(1 + b) Cos[theta], (1 - b) Sin[theta]}, {theta, 0, 2 Pi}] was definitely my thought as well, but what was tough was where to put it. I finally got it to work as follows:

Manipulate[ Module[{ p1 = {Cos[theta], Sin[theta]}, p2 = {b Cos[-theta], b Sin[-theta]}, p3 = {(1 + b) Cos[theta], (1 - b) Sin[theta]}}, Show[{ ParametricPlot[{(1 + b) Cos[theta], (1 - b) Sin[theta]}, {theta, 0, 2 Pi}, PlotRange -> 1 + b], Graphics[{ Circle[{0, 0}], {Red, Circle[{0, 0}, b]}, Arrow[{{0, 0}, p1}], {Red, Arrow[{{0, 0}, p2}]}, {Black, Dashed, Line[{{p1, p3}, {p2, p3}}]}, {Blue, Arrow[{{0, 0}, p3}]} }] }] ], {{b, 0.5, "b"}, 0, 2}, {{theta, 0, "theta"}, 0, 2 Pi} ] 

I now have a working demonstration of the graph of $e^{i\theta}+be^{-i\theta}$ as $0\le\theta\le 2\pi$. There are three cases to consider: (1) $0<b<1$, (2) $b=1$, and (3) $b>1$, where $b$ is a real number. In either case, we can perform the following analysis:

$$ \begin{align*} e^{i\theta}+be^{-i\theta} &=(\cos\theta+i\sin\theta)+b(\cos(-\theta)+i\sin(-\theta))\\ &=(1+b)\cos\theta+i(1-b)\sin\theta \end{align*} $$

When we compare $z=x+iy$ with the last result, we get: $$ \begin{align*} x&=(1+b)\cos\theta\\ y&=(1-b)\sin\theta \end{align*} $$ Thus, $$ \begin{align*} \frac{x^2}{(1+b)^2}&=\cos^2\theta\\ \frac{y^2}{(1-b)^2}&=\sin^2\theta,\\ \end{align*} $$ so we can see that $$ \frac{x^2}{(1+b)^2}+\frac{x^2}{(1-b)^2}=1. $$ Thus, we have an ellipse with $x$-intercepts at $(-1-b,0)$ and $(1+b,0)$ and $y$-intercepts at $(0,-1+b)$ and $(0,1-b)$. Thus, PlotRange->1+b is a good move, as seen in the image below.

By visualizing the geometry, one can determine that the ellipse will be traced out in a counter-clockwise direction when $b<1$ and it will be traced out in a clockwise direction when $b>1$, and finally, it will trace out the line segment on the real axis between $-2$ and $2$ when $b=1$. But these cases can also be determined via the equations

$$ \begin{align*} x&=(1+b)\cos\theta\\ y&=(1-b)\sin\theta \end{align*} $$

For example, when $b=1$, these become:

$$ \begin{align*} x&=2\cos\theta\\ y&=0 \end{align*} $$

This defines the segment on the real axis between $-2$ and $2$.

There is one more step to be accomplished before I am done examining this question. I now need to select a $b$-value that is complex, that is, of the form $b=re^{i\phi}$. In this case, one can show that

$$ \begin{align*} e^{i\theta}+be^{-i\theta} &=e^{i\theta}+re^{i\phi}e^{-i\theta}\\ &=e^{i\phi}(e^{i(\theta-\phi)}+re^{-i(\theta-\phi)}), \end{align*} $$

which means that the entire image will simply be rotated through $\phi$ radians. So my final need will be to replace the $b$ slider with $r$ and $\phi$ sliders, where $r$ is a real number between 0 and 2, while $\phi$ is an angle between $-\pi/2$ and $\pi/2$. The $r$-value should affect the image in the same way as the $b$-value of the current image, while the $\phi$-value should rotate the image through $\phi$ radians.