On the role of circular sections of quadric surfaces

Abstract

As a part of new advanced course curriculum of Descriptive geometry, in the Faculty of Civil engineering, creative 3D modeling of specific structures is included. These structures, inspired by contemporary architecture achievements, relay on quadric surfaces (quadrics). Here, student's knowledge concerning properties of the quadrics: cone, sphere, ellipsoid, elliptic hyperboloid, etc. (also elaborated in the course - Mathematics 2, in the same year of the studies) is necessary. Throughout two given 3D modeling tasks, students elaborate two separate subtopics: discretization of the surface elliptic hyperboloid of one sheet (ELHY) and spiral surfaces, both connected by key elements circular sections of quadrics. The subtopic's presentation methodology designed for students is conducted by "steps" in 3D structures modeling process, aided by applications and tools of most common engineering software Auto CAD. The applications presented below are created on a base of analytical geometry methods. The first topic - discretization of ELHY surface, i.e. generation of polygonal mesh structure, has a rather simple explanation and graphical presentation (wireframe and surface models) of geometrical procedures, i.e. constructions in opposite to the available analytical geometry solutions. A method of generating the ELHY surface is based on two arbitrary circular sections and one generatrix. The accompanying CAD application, created in Auto lisp, as a tool for discretization has the purpose to enable creative 3D task solution - model of geometric structure ELHY shaped. Three different options/solutions of discretization are offered. The second topic regarding 3D models of spiral surfaces/structures is nowadays very popular in the architecture. Here included spiral surfaces are one which geometric origins are quadrics. Since the term "spiral" is directly connected to the rotation and circle, geometric concept relay again on the circular sections of quadrics. This topic is supported by a Visual basic application (VBA) procedure implemented in Auto CAD. It also requires advanced 3D modeling tools for the design and visualization process of the final product. If comparing two given tasks, the second one is more creative and optional, while the students role in designing is more significant. However, we are sure that both tasks have an important educational value in better understanding of theory regarding quadric surfaces. It is reasonable to expect that in a certain moment (according to the level of programming knowledge) student could create his own software solution, or modify the offered one (application), aimed for such geometric topics. According to the palette of the surfaces included, variety of final "products" - structures and their purposes should be resulted in buildings, furniture, or art design.