The Monge theorem and its application in engineering practice

Engineering innovation activities cannot be performed without mathematical devices. The present paper describes the theoretical analysis of the reconstruction of the Monge theorem. The significance of the analysis is that we consider those Monge projections the same, which can be translated to one another by parallel shifting. Accordingly, in a fixed Descartes coordinate system in the space, every Monge projection is determined by its two projection lines that go through the origin point, which is described by three free parameters. The three free parameters determining the Monge projections make up the points of the Monge cuboid. We are looking for Monge projections to a fixed third-grade spatial curve, in which the reconstruction of the curve can be carried out only with two pictures without any further information. The inner points of the Monge cuboid—defined by the bijective Monge projections—are bordered by a surface. This surface is described in this article during our research work first. We have proved the engineering application of this theorem by calculating the correct camera position for the wearing examination of the hob’s cutting edge.