Mapping Lines and Circles onto the Riemann Sphere

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One of the great miracles of mathematics is the fact that an infinitely extended plane, which is densely packed with complex numbers, can be mapped onto a sphere with radius and hence an area of . Here you can see a decent fraction of the complex plane and the south pole of the Riemann sphere placed at the origin.

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You can observe how straight lines or circles in the complex plane are transformed into circles on the sphere. The "radius" control changes the radius of the circle while "angle" alters the angle between the straight line and the real axis. "Point" is the center of the circle or one point of the line that you can move in the part of the complex plane shown.

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Contributed by: Hans-Joachim Domke (July 2009)
Open content licensed under CC BY-NC-SA


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You may notice that circles with center in the origin are transformed into lines of latitude that move up to the north pole as the radius of the circle in the plane grows. Though it cannot be shown with this Demonstration, one can imagine that a circle in the plane with very large radius will be mapped onto a circle that shrinks around the north pole, while there is no point in the plane that has the north pole as an image. On the other hand, all mappings of lines contain the north pole; it is the image of the point at infinity which must be added to the plane to keep the mapping bijective. In a more descriptive way you could think of this "point" as a circle with infinite radius, but it is not part of the complex plane. One more thing that conflicts with the imagination: the interior of the unit circle in the plane is mapped onto the southern hemisphere, all other numbers outside the unit circle to the northern hemisphere.



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