# How can I visualize this complex geometry with mathematica?

Let $$\varepsilon$$ be a complex number of small magnitude and let $$1+\varepsilon$$ have magnitude $$r$$ and argument $$\theta$$. I want to generate the following image in Mathematica:

and then I want to see what happens to the image as I change the magnitude of $$\varepsilon$$ with a slider. In particular, I want to see how the origin-centered circular arc connecting $$1+\varepsilon$$ to the point $$r$$ on the real axis approaches the perpendicular from $$1+\varepsilon$$ to the real axis as the magnitude of $$\varepsilon$$ approaches $$0$$. What would be a good way to do this?

This is what I came up with so far:

e :=
Manipulate[
FromPolarCoordinates[{mag, Pi/6}], {mag, 0, 1}];
eps := {1, 0} + {e[[0]], e[[1]]};
Graphics[Line[{{0, 0}, {1, 0}, {1, 0} + eps}]
]

• I'm new to mathematica so I haven't gotten very far. I tried using the Graphics and Line functions to plot the vectors, but I couldn't get it to work. Commented Aug 11, 2022 at 12:14
• Ok, it's pretty terrible, but this is what I came up with: e := Manipulate[FromPolarCoordinates[{mag, Pi/6}], {mag, 0, 1}]; eps := {1, 0} + {e[[0]], e[[1]]}; Graphics[Line[{{0, 0}, {1, 0}, {1, 0} + eps}]] Commented Aug 11, 2022 at 12:18

What you are trying to achieve seems functionally equivalent to showing graphically that $$\lim_{\theta\to 0}\frac{\sin \theta}{\theta}=1$$. Here is an example to achieve that:

Manipulate[
Graphics[{
{Thick, Blue, Arrow[{{0, 0}, {1, 0}}]},
{Red, Arrow[{{0, 0}, {Cos[theta], Sin[theta]}}]},
{Black, Dashed, Line[{{Cos[theta], Sin[theta]}, {Cos[theta], 0}}]},
{Black, Circle[{0, 0}, 1, {0, theta}]}
},
Axes -> True, PlotRange -> {0, 1.1}
],
{theta, Pi/2, 0}
]


I do not know if I understood you right. Can modify as needed.

Manipulate[
Module[{x = 1 + realEps, y = imEps, r},
r = Sqrt[x^2 + y^2];
Grid[{{Row[{"r =", r}]},
{Graphics[{
{Red, Arrow[{{0, 0}, {x, y}}]},
{Dashed, Circle[{0, 0}, r, {0, ArcTan[x, y]}]},
{Text[
Style[Row[{ArcTan[x, y]*180/Pi , " deg"}], Small], {.7, 0.2}]}
}, Axes -> True, PlotRange -> {{0, 4}, {0, 3}},
GridLines -> Automatic, GridLinesStyle -> LightGray,
ImageSize -> 300
]}
}]
]
,
{{realEps, .5, "real part \[Epsilon]"}, 0, 2, .01,
Appearance -> "Labeled"},
{{imEps, 1, "imaginary part \[Epsilon]"}, 0, 3, .01,
Appearance -> "Labeled"},
TrackedSymbols :> {realEps, imEps}
]