Sine of a circle

What does it look like when you take the sine of a circle? Not the angle of points on a circle, but the circle itself as a set of points in the complex plane?

Here's a plot for the sine of circles of radius r centered at the origin, 0 < r < /2.

Here's the same plot but for /2 < r < .

Now let's see what happens when we shift the center of the circle over by 1, first for 0 < r < /2.

And now for /2 < r < .

Here's the Python code that produced the first plot; the code for the other plots is very similar.

 import matplotlib.pyplot as plt from numpy import pi, exp, sin, linspace t = linspace(0, 2*pi, 500) for r in linspace(0.1, pi/2, 17): z = r*exp(1j*t) w = sin(z) plt.plot(w.real, w.imag, 'b') plt.gca().set_aspect("equal") plt.title("$\sin(r\exp(it))$, $0 Related posts

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