Some Characterization of Spiral Surfaces

In this paper we will study special spiral surfaces in the three dimensional Euclidean space and we give some characterization of these surfaces. More specifically, we investigate the Chang-Yau operator acting on the Gauss map of spiral surfaces. We also give some results about canonical vector field of these surfaces, i.e., we study incomperssibility of canonical vector field in two types of spiral surfaces. Moreover, we give some necessary conditions for a spiral surface to be a Weingarten surface. Existence of umbilical point is another problem that we investigate about it for a special case of spiral surfaces of the first type.

Projection by Conical Helical Lines With Constant Pitch

This paper’s purpose is investigation of non-traditional projection systems and their projecting surfaces, the choice of such congruence parameters for conical helical lines, which allow cover the whole complex of requirements to the surface, obtained by projecting of an arbitrary flat or spatial line with congruence beams, as well as the use of computer graphics in surface visualization. In the paper has been presented an example of analytical interpretation for an image of curvilinear projection by conical helical lines with constant pitch, and a congruence example for conical helical lines located on coaxial cones with a common vertex and a variable angle of generatrix inclination to an axis. Have been investigated properties and defined parameters of the congruence helical line passing through a space arbitrary point which is not belonging to an axis. An approach for construction of spiral surfaces, which frame consists of beams projecting an arbitrary line. A form generation of surfaces by analytical methods and their visualization by means of computer graphics is one of applied geometry’s urgent problems in connection with the use of such methods in automated systems for scientific research, design, and manufacture on equipment with computer numerical control. The leading research method for this problem is the general analytical theory for surfaces’ applied form generation developed by Professor I.A. Skidan and formed a unique apparatus, based on mathematical support of computing technologies for design and creation of objects with complex forms. On examples of visualization for projecting surfaces by means of computer graphics it is possible to show applicability of analytical models in computer technologies for scientific researches, design and manufacturing.

Study of the Wheel of Abrasive Belt Based on Forming Polishing Technology

The wheel of abrasive belt is a special-shaped coated abrasive which has the advantage of good polishing quality, long life, and high efficiency, low cost and so on. Base on the features of the wheel of abrasive belt, through analyzing the basic motion relation of forming polishing spiral surface screw, the wheel of abrasive belt’s profile shape curve could be calculated by establishing the contact line equation of screw spiral surfaces with the wheel of abrasive belt rotating surfaces. Software to verify the correctness of the profile shape calculated, and in accordance with the wheel of abrasive belt profile shape curve to programming NC program. Start with the dressing experiments of the wheel of abrasive belt. Through analysis and research to get the wheel of abrasive belt’s profile shape curve meet the forming polishing standard.

A Milling Method of an Enveloping Toroidal Worm on a CNC Turning Center

To solve the problem of allowance equalization in the grinding process of an enveloping toroidal worm after turning by using the hindley worm turning method, a milling method was presented based on discretization of the spiral surfaces rather than the generating principle. The method is used on a CNC tuning center with a cylindrical end milling cutter rather than on a specialized machine tool. The material of the spiral grooves of the worm was removed layer by layer both in the tangential and the radial directions of the torus of the worm from the tooth top circle to the tooth root circle. The milling model was built, and the coordinates of the discrete points on the spiral surfaces were calculated according to the theoretical equations of the spiral surface of the worm. A series of feed positions X and Z of the milling cutter and the corresponding rotating angles C of the worm were determined for programming. The grinding allowance of a planar enveloping toroidal worm on axial sections was calculated and the trial manufacturing was performed. The allowance in the subsequent grinding process could be equalized and the manufacturing efficiency could be increased greatly.