Surprising applications of Newton's hyperbolism transform of curves in Fourier-transform spectroscopy

The Fourier transform (FT) represents a key tool in modern spectroscopy which drastically reduces measurement times and helps to improve the signal-to-noise ratio in spectra. Fourier transforming exponentially decaying time domain signals gives Lorentzian line shapes which can be manipulated by apodization methods. The underlying transitions of spectral lines can be visualized by a Bloch vector or equivalent phase-space representations. Here, we study and generalize a surprisingly elegant geometric transform, the hyperbolism of curves originally found by Isaac Newton, which allows to transform ellipses into Lorentzian lines, and vice versa. With this, we show that the Bloch picture and especially corresponding phase-space representations are directly geometrically related to the Lorentzian line shape. We also introduce a novel continuous parametrization of Newton's transform which results in further interesting line shapes. In particular, we find that truncated parabolic lines with finite support can be obtained by the half transform and introduce a new apodization approach to replicate this line shape in experimental spectra. We discuss concrete applications in nuclear magnetic resonance spectroscopy.