  The Jiangmen Neutrino Underground Observatory (JUNO) is a 20$\,$kton liquid
scintillator detector currently under construction near Kaiping in China. The
physics program focuses on the determination of the neutrino mass hierarchy
with reactor anti-neutrinos. For this purpose, JUNO is located 650$\,$m
underground with a distance of 53$\,$km to two nuclear power plants. As a
result, it is exposed to a muon flux that requires a precise muon
reconstruction to make a veto of cosmogenic backgrounds viable. Established
muon tracking algorithms use time residuals to a track hypothesis. We developed
an alternative muon tracking algorithm that utilizes the geometrical shape of
the fastest light. It models the full shape of the first, direct light produced
along the muon track. From the intersection with the spherical PMT array, the
track parameters are extracted with a likelihood fit. The algorithm finds a
selection of PMTs based on their first hit times and charges. Subsequently, it
fits on timing information only. On a sample of through-going muons with a full
simulation of readout electronics, we report a spatial resolution of 20$\,$cm
of distance from the detector's center and an angular resolution of
1.6$\,^{\circ}$ over the whole detector. Additionally, a dead time estimation
is performed to measure the impact of the muon veto. Including the step of
waveform reconstruction on top of the track reconstruction, a loss in exposure
of only 4% can be achieved compared to the case of a perfect tracking
algorithm. When including only the PMT time resolution, but no further
electronics simulation and waveform reconstruction, the exposure loss is only
1%.
