CONSTRUCTING CURVED LINES AND SURFACES WITH THE HELP OF THE DARBU TRIANGLE

Abstract

In differential geometry, the Frenet trihedron is widely known, which is a companion for a spatial and, as a partial case, for a plane curve. Its three mutually perpendicular unit orthoses are defined uniquely for any point of the curve, with the exception of some special ones. For example, for the inflection point of a flat curve or for the straightening point of a spatial curve, the direction of the main normal becomes uncertain. The Darboux trihedron refers to the surface. Two of its single orths are located in a plane tangential to the surface, and the third is directed along the normal to the surface. It can also be a companion to a curve located on the surface. For this, one of the orthos in the plane tangent to the surface must be tangent to the curve. Then these two trihedra have a common ortho of the tangent to the curve, and there is a certain angle between the other two orthos. However, the direction of the orth, which is directed along the normal to the surface, can be chosen in one or the opposite direction. Thus, unlike the Frenet trihedron, the Darboux trihedron can have two positions at the point of the curve on the surface.

Frenet and Darboux trihedra are movable and change their position in relation to a fixed coordinate system due to movement and rotation. For a Frenet trihedron, the direction cosines of its angles are determined through the differential characteristics of the curve, to which the first and second derivatives of this curve are involved. At the point of the curve with a curvature equal to zero, the position of the orthos of the main normal and the binormal becomes uncertain. For a Darboux trihedron, one of the orthos is directed along the normal to the surface, that is, its direction is determined by the differential characteristics of the surface and is determined for a regular surface. The ortho of the tangent is also defined, therefore the third ortho, perpendicular to the first two, will also be defined. When studying the geometric characteristics of curves and surfaces with the help of accompanying trihedra, it is necessary to have formulas for the transition from the position of the elements of these objects in the moving trihedron system to the position in the stationary Cartesian coordinate system. For this, there are nine direction cosines - three for each ort. For the Frenet trihedron, they are completely determined through the first and second derivatives of the parametric equations of the directional curve\. For the Darboux trihedron - through the parametric equations of the surface. The article describes one of the possible ways to find them.

Key words: accompanying trihedron, direction cosines, parametric equations, normal to the surface, vector product.