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 detectors 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%.
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