How to plot Full spectrum with negative frequencies
Take the fft of x+i*y.

The phase angles in the picture you linked https://drive.google.com/file/d/0Byib6yrCQ9vnRUM2Yi1sMklXeFk/view are from the result of this fft. 

The image linked have an error, from the article it came from, where the phases of the negative frequencies of the fft result are Beta. The image uses Alpha in the negative and positive results. In the end, the image just shows mathematical prove that the amplitudes of the fft are derived from the amplitudes of the individual signals. 

The fullspectrum process is simple as the fft of x+i*y. The majoraxis of an ellipsis of one frequency will be the sum of the R+ and R-, and the direction will be the (Alpha+Beta)/2.

This paper could clarify it more "Use of directional spectra of vibration signals for diagnosis of misalignment in rotating machinery"