Solving the problem of spatial rotation of 3D surfaces and their mapping on the plane

Abstract

The solution of the mathematical problem of rotation of a three-dimensional surface in space with an orthogonal basis and its mapping on a plane using simple geometric shapes is considered. This task arises when accompanying moving objects against the background of the surrounding environment. A design feature of such systems is that they contain functional additional elements that provide information about the maneuvering object of observation and generate control signals to work out the error that has occurred. This operation is performed continuously in real time. It is assumed that this problem is solved using a digital computer, i.e., the change in the angle of sight of the observed moving object will be recorded in separate time intervals — partial (discrete) ones. The initial state of the coordinate system can be represented in matrix form, respectively; the transition to the final state is carried out at discrete points in time. The problem is solved analytically. A number of restrictions on the magnitude of vectors and their mutual orientation in space are formulated. The proposed approach made it possible to increase the visibility and predictability of the operations performed due to the transition from nonlinear trigonometric equations to the simplest linear operations. To demonstrate the correctness of the implementation and clarity of the application of the proposed vector-algebraic approach, the background of the environment is presented in *.off format (geomview object file format). Finite expressions are obtained for the rotation of the coordinate system of an elastic body with a fixed center of mass. The solutions obtained are formalized on the basis of strict mathematical transformations and belong to the class of problems in which analytical relations accurately describe the data, that is, when, in the absence of measurement errors, the residual vector of the system is always zero. This approach allows you to avoid performing transformations on complex nonlinear mathematical expressions.

Keywords

• coordinate system
• orthogonal basis
• vector projection
• linear vector-matrix equations
• maneuvering