CONKHOIDAL TRANSFORMATION AS AN EXAMPLE OF AN ACTIVE TRANSFORMATION OF COORDINATES WHEN DISCRETE SURFACE MODELING

Abstract

The work discusses the possibilities of using active coordinate transformation using the example the Conchoidal transformation of space. There are many ways to form curved surfaces of different shapes. Transformation already well know and studied the geometrical objects is the most versatile way to obtain new curves of lines or surfaces. Synthesis of static-geometric method and geometric transformations will allow obtaining new forms of discrete models of various architectural surfaces.

The selection of a geometric transformation is related to the original data of the simulated surface and the properties of the transformation itself. This approach will extend the list of design and aesthetic properties that can be considered in surface modeling. The conversion of discrete wireframes is one example of the generalization of the static-geometric method. The coordinates of the nodes of the discrete frame of the surface are determined without taking into account the external form-building loading on the nodes. This allows you to define the boundaries of the existence of discrete framework nodes in a layer between two boundary surfaces.

Such surfaces, in shape, act as prototypes for new surfaces that are modeled. From a constructive point of view, boundary surfaces result from the selected spatial transformation of two planes. The formation of a discrete surface framework within the selected layer allows the shape of the simulated surface to be brought closer to the desired image, as evidenced by numerous examples.

The main idea of the presented work is to demonstrate the possibilities of using spatial conchoidal transformation of Cartesian coordinates into spherical coordinates when forming discrete frameworks. Using this kind of conchoidal transformation together with the static-geometric method, when forming discrete images in three-dimensional point space, it is possible to obtain a surface whose appearance will be similar to the surface of revolution with a generatrix in the form the Сonhoid of Nycomed’s.

Keywords: geometrical modeling, transformation of space, active transformation of coordinates, spatial conchoidal transformation, properties of transformation.