Polar form approach to geometric modeling | |
| Author | Tang, Van To |
| Call Number | AIT Diss. no.CS-92-1 |
| Subject(s) | Computer-aided design Geometrical drawing |
| Note | A dissertation submitted in partial fulfillment of the requirements of the degree of Doctor of Engineering |
| Publisher | Asian Institute of Technology |
| Abstract | The polar form approach has been introduced in recent years to curve and surface modeling. In this study, this approach has been applied to curves, surfaces and solids. In this connection, all the polar models for curves and surfaces are reviewed and presented together in a systematic fashion, in order to make the advantages of the polar form approach when applied to them clearly seen. The study covers Bezier, B-spline, β-spline and Ball curves. Some typical surfaces as tensor product, ruled, revolution and Bezier triangular surfaces are also included. The extension of the polar form approach to solid modeling are finally introduced. It is shown that the polar form approach provides a very good tool to develop and explain a large number of basic operations on the aforementioned geometric objects. Using polar form, almost the same treatment can apply to both Bezier and B-spline curves. The control points for these curves can be represented directly by their polar values and the algorithms for computing the coordinates of a point on these curves can be obtained by mean of an affine rule. It is also shown that the conversion between these two models becomes very easy under the polar form approach. When applied to Ball curves, its true Bezier points can be determined quite easily. Having shown that any polynomial curve can be viewed as a Bezier curve, the study shows that for the same control points, the Bezier curve approximates the control polygon better than does the Ball curve. As a further application of the polar form approach to curve modeling, Bezier-based curves resulted from varying polar form arguments synchronically are also introduced and examined. The advantage of the polar form approach is therefore easily seen. For typical surfaces, a direct extension of the results obtained for curves can readily be made. The polar forms for all these surfaces are given. It is shown that the polar form approach can be used to easily process mixed-type tensor product surfaces. For solids modeling, the general parametric polynomial solid is considered and an efficient algorithm to calculate the coordinates of a point on it is described. Moreover, tetrahedral solids can be obtained as an extension of the polar form for Bezier triangular surfaces. All the results obtained have been derived mathematically and all algorithms devised have been given in pseudo codes or in the form conducive to implementation. |
| Year | 1992 |
| Type | Dissertation |
| School | School of Engineering and Technology (SET) |
| Department | Department of Information and Communications Technologies (DICT) |
| Academic Program/FoS | Computer Science (CS) |
| Chairperson(s) | Phien, H. N. |
| Examination Committee(s) | Vilas Wuwongse;Tabucanon, M. T.;Farin, Gerald; |
| Scholarship Donor(s) | Government of Japan |
| Degree | Thesis (Ph.D.) - Asian Institute of Technology, 1992 |