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The geometrical model of the filament during the fused deposition modeling (FDM) process was firstly proposed based on three different models, tractrix, parabola, and catenary. Comparing with the actual measured filament curves on the Stratasys 1600 FDM machine, it is indicated that the tractrix model had the best agreement with the actual measured curves. With the analytical simulation, the nozzle trajectories in the straight-line deposition road, circle road, and arbitrary continuous curve road were deduced, according to the traxtric based geometrical model of the filament.

Based on the border effect and relevant theories, it is found that measurement precision depends on resolution stability which is much higher than resolution itself. With the help of this new theory and the more sensitive discrimination of the border of a fuzzy area, it is possible to greatly improve the measurement precision. Since there are a lot of complicated frequency signals in quantum frequency standards, telecommunication, fundamental subjects and other fields, precise measurement is significantly important. In a great deal of measurement of complicated frequency signals, discrete fuzzy areas are common. In this paper, we put forward a feasible and high-precision frequency measurement scheme to make it easier to capture border information, which shows great advantages of border effect.

Limited detection resolution leads to fuzzy areas during the measurement, and the discrimination of the border of a fuzzy area helps to use the resolution stability. In this way, measurement precision is greatly improved, hence this phenomenon is named the border effect. The resolution fuzzy area and its application should be studied to realize high-resolution measurement. During the measurement of any frequency signal, the fuzzy areas of phase-coincidence detection are always discrete and irregular. In this paper the difficulty in capturing the border information of discrete fuzzy areas is overcome and extra-high resolution measurement is implemented. Measurement precision of any frequency-signal can easily reach better than 1 × 10−11/s in a wide range of frequencies, showing the great importance of the border effect. An in-depth study of this issue has great significance for frequency standard comparison, signal processing, telecommunication, and fundamental subjects.