Investigation of Methods for Generating Smooth Continuous Curves Resulting from the Intersection of Conical Surfaces

Studied extensively in the present paper is a geometric problem related to the intersection of two conical surfaces with crossed axes. The goal was to derive a spatial intersection curve between the surfaces that resembles the classical Viviani curve but is defined as the inter-section of two rotational conical surfaces with crossed axes, forming an asymmetric spatial figure-of-eight shape. These unique spatial curves, as variations of the Viviani curve, possess intriguing geometric properties and are likely to have a wide range of potential applications in the fields of construction and engineering. Their presence can often be linked to the development of complex geometric shapes in architecture, particularly in the design of bridges with unconventional curves in their support structures or situations where inclined support beams intersect with other structural components. They are commonly employed in the construction of tunnels with conical and cylindrical shapes, in addition to various types of vaults and domes. These curves are critical in optimizing structures, such as roofs with intricate geometries or decorative façade elements, and can greatly enhance both their stability and aesthetic appeal.