Research On Irregular Equivolumetric Tubular Surface

The Tubular surface is one type of important geometric modeling, and has been widelyapplied into various fields, pipe surface and normal ringed surface are its two specialinstances, and base on which, the Equivolumetric tubular surface is proposed. Similar with theoffset curve surface, as a result of the limitation of space or material, two types of irregularproblems will occur in the practical application of the tubular surface, one is theself-intersection that occurs in the concave region of the spine curve when radius of thetubular surface is greater than the radius of curvature at a certain point of the spine curve andis called local self-intersection, the other is the self-intersection that occurs when the sum ofradii of two distinct points on the spine curve is greater than the distance between the twopoints and is called global self-intersection. The emergence of the irregular problem seriouslylimits the application range of the tubular surface, hence it is meaningful and urgent toanalyze the regularity of the tubular surface and investigate how to trim the self-intersectionregion. Since the radius of the pipe surface is constant, the radius of the normal ringed surfaceis the function of one variable, the radius of Equivolumetric is a bivariate function, theproblem of regularity of the Equivolumetric tubular surface is more complicated than that ofpipe surface and normal ringed surface.The regularity of pipe surface, normal ringed surface and Equivolumetric tubular surfaceis respectively analyzed from the perspectives of both local and global regularity, and theregularity condition of each is obtained. For the irregular case, the algorithm to detect andtrim the self-intersection points is investigate, the major contents and achievements of thestudy are as follows:Firstly, the regularity of pipe surface is analyzed, and its regularity condition is obtained.For the irregular pipe surface, a new algorithm called ball sweeping algorithm is proposed todetect and trim all the self-intersection points, that is, a ball whose centre of sphere is on thespine curve and radius is the radius is the radius of the pipe surface is used to sweep pipesurface along the spine curve and points locating into the ball are all the self-intersectionpoints and need to be trimmed. All the local and global self-intersection points are trimmedaway with the proposed algorithm.Secondly, the regularity of normal ringed surface is analyzed on the base of the analysisof regularity of pipe surface, and the regularity condition is obtained. For the irregular case ofnormal ringed surface, the normal plane of the spine curve is introduced to assist with theselection of the self-intersection points on the base of ball sweeping, which can detect andtrim the self-intersection regions of normal ringed surface effectively. At last, the volumetric ratio is introduced as an index to measure the regularity ofEquivolumetric tubular surface, integrating the analysis of the regularity of pipe surface andnormal ringed surface, the regularity condition is obtained. With the proof of the relationshipbetween Equivolumetric tubular surface and Equivolumetric offset surface, the irregularproblem of Equivolumetric tubular surface is converted into the problem how to detect andtrim self-intersection points of Equivolumetric offset surface, that is the volumetric ratio isintroduced to control the detection and trimming of the self-intersection points on the base ofthe algorithm to detect and trim the self-intersection points of irregular offset surfaces.All the algorithm proposed in this paper can solve the irregular problems of pipe surface,normal ringed surface and Equivolumetric tubular surface, which improve the theoreticalsystem of the tubular surface and broaden the application fields of that.