introduce the geometrical observation as the spatial distance, or wavelength, between two pulses (wave crests), where is the time elapsed between the emission of two pulses. The time elapsed between the passage of two pulses at the same point in space is the time period , and its inverse is the observed (temporal) frequency. The corresponding quantities in the emitters frame are endowed with primes. For light waves and the observed frequency is where is standard time dilation formula. Suppose instead that the wave is not composed of light waves with speed , but instead, for easy visualization, bullets fired from a relativistic machine gun, with velocity in the frame of the emitter. Then, in general, the geometrical observation is precisely the same. But now, , and is given by velocity addition, The calculation is then essentially the same, except that here it is easier carried out upside down with instead of . One finds Observe that in the typical case, the that enters is negative. The formula has general validity though. When , the formula reduces to the formula calculated directly for light waves above, If the emitter is not firing bullets in empty space, but emitting waves in a medium, then the formula still applies, but now, it may be necessary to first calculate from the velocity of the emitter relative to the medium. Returning to the case of a light emitter, in the case the observer and emitter are not collinear, the result has little modification, where is the angle between the light emitter and the observer. This reduces to the previous result for collinear motion when , but for transverse motion corresponding to , the frequency is shifted by the Lorentz factor. This does not happen in the classical optical Doppler effect. Hyperbolic geometry Associated to the relativistic velocity of an object is a quantity whose norm is called rapidity. These are related through where the vector is thought of as being Cartesian coordinates on a 3-dimensional subspace of the Lie algebra of the Lorentz group spanned by the boost generators . This space, call it rapidity space, is isomorphic to as a vector space, and is mapped to the open unit ball, , velocity space, via the above relation. The addition law on collinear form coincides with the law of addition of hyperbolic tangents with The line element in velocity space follows from the expression