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Estimating the length of a bounded curve in three dimensions using total vertical projections

A new method is proposed to estimate the total length of a bounded, isolated linear feature in three dimensions from ‘total vertical projections’, obtained by rotating the curve about a fixed axis (arbitrarily called ‘vertical’) and projecting it onto a fixed vertical plane. No sections are required. Properly stained and embedded neuron dendrites, mycelial trees, fluorescent cytoskeletal filaments within a cell, etc., are candidate specimens for the method, especially in combination with the new devices for non‐invasive three‐dimensional microscopy. It is necessary that the specimen curve is rigid (i.e. of constant shape), that its length density is not too high (so that overlapping effects are not important) and that the embedding medium is fairly transparent. Given these requirements, the method can be very accurate and convenient to use, as is exemplified here.