Cyclographic Modeling of Surface Forms of Highways

A cyclographic approach towards the formation of a highway surface is suggested. The approach is based on mapping a point (x, y, z) of the space R ³ onto a cycle (a circle with direction) in the plane ∏₁ (xy). The centre of the cycle is a point (x, y) and its radius is defined as R = ±|z|. In case z > 0, the cycle radius R = z and the cycle has one direction. In case z < 0, the cycle radius R = −|z| and the cycle has the opposite direction. Thus, a one-to-one correspondence is established between the set of ∞³ points of the space R ³ and the set of ∞³ cycles of the plane (xy).

The main linear element of a road is its axis which in general is a spatial curve. The spatial forms of road elements are a roadway, road shoulders, slopes; they are determined by the main linear element. Proceeding from this principle of formation of geometric and mathematical models of road elements spatial forms, the use of road axis cyclographic projection is proposed. The parametric equations of the envelope of cycles modeling the axis point in the plane are obtained for the general case of cyclographic mapping. In a general case, unlike the classic one, the use of regular angle variation at the vertex of mapping cones is suggested. Taking the envelope of cycles and the axis of the road as the guidelines for the surface, the surface forms of road elements are obtained in the form of developable surfaces. The presence of two modifications of road elements' surface forms distinguishes the cyclographic approach from existing ones and gives additional possibilities for the choice of optimal surface forms in the design of highways. (paper)