Stereographic projection mapping is typically introduced to explain the point at infinity in the complex plane. After this brief exposure in the context of complex analysis, students rarely get an opportunity to fully appreciate stereographic projection mapping as an elegant and powerful technique on its own with many fruitful applications in the physical sciences. Here, using a classical description of nuclear magnetic resonance in the rotating frame, I show how stereographic projection mapping to and from the Bloch sphere can be used for visualizing solutions to Bloch's equation and the Bloch–Riccati equation, respectively. After developing the fundamentals of stereographic projection mapping using examples drawn from nuclear spin precession in the rotating frame, the method is then applied to visualizations of composite pulse excitation of a spin-1/2 system and to radiation damping in a system of isolated spins-1/2. In the case of the radiation-damped system, these visualizations provide particularly vivid illustrations of loxodromic Möbius transformation dynamics.

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Since inversion is not defined at the origin (the center of inversion), inversion in this case must map to a punctured circle. This requirement can be dropped if the line L is extended to include the point at infinity (Ref. 22).
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Möbius transformations are also called “bilinear,” “homographic,” or “projective” transformations.
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These spiralling paths on the sphere are called loxodromes, named after the Greek words for “running obliquely.” The name is a navigational term, referring to a spherical path, which cuts every line of longitude at the same angle.
38.
If the spherical polar angles (ϑ,φ)=(π/4,π/2), there is a closed-form solution for ζ(t) (see the supplementary material, Ref. 26). This solution was used to generate this plot.
39.
The fixed points of a Möbius transformation are defined as solutions of z=M(z)=(az+b)/(cz+d).

Supplementary Material

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